Download Thermal Brane Probes

Università degli Studi di Perugia
Dottorato di Ricerca in Fisica e Tecnologie Fisiche
XXV Ciclo
Settore Scientifico Disciplinare FIS/02
Thermal Brane Probes
Candidato:
Andrea Marini
Relatori:
Coordinatore:
Prof. Gianluca Grignani
Prof. Maurizio Busso
Dott.ssa Marta Orselli
Anno Accademico 2011/2012
Ringraziamenti
In questa tesi ho raccolto il lavoro svolto ed i risultati ottenuti durante i tre anni di Dottorato in Fisica presso l’Università degli Studi di Perugia. Prima di iniziare la dissertazione
ritengo doveroso spendere alcune parole per dare il giusto riconoscimento a tutti coloro
che, da quando ho iniziato gli studi in fisica presso questo Ateneo, mi hanno sostenuto,
aiutato, guidato o semplicemente accompagnato in questa bella avventura.
Innanzitutto voglio ringraziare il mio supervisore, il Prof. Gianluca Grignani, la cui
costante guida, iniziata fin dalla mia tesi di Laurea Triennale, è stata determinante nella
mia formazione scientifica ed umana. Vorrei esprimere inoltre la mia riconoscenza verso
Marta Orselli, Troels Harmark e il Prof. Niels Obers, con i quali Gianluca ed io abbiamo
intrattenuto una collaborazione fruttuosa, stimolante e per me estremamente educativa,
che spero possa proseguire anche in futuro. In particolare sono immensamente grato a
Marta, per le innumerevoli volte in cui ha dedicato il suo tempo ad aiutarmi, sempre con
la massima gentilezza.
Un altro ringraziamento speciale mi sento in dovere di rivolgerlo ad Agnese, per la
disponibilità che ha sempre dimostrato ogni volta che le ho chiesto un aiuto o un consiglio.
Voglio ringraziare tutti gli amici che ho conosciuto in questi anni all’interno dell’Università, partendo dai miei compagni di Dottorato ed in particolare da coloro con i quali
ho “convissuto” nell’aula dottorandi B del Dipartimento di Fisica: Federico, Marialucia,
Aniello, Enrico, Matteo, Alessandro, Emanuele, Salem, compresi i frequenti ospiti Antonio,
Francesco, Stefania, Luca e Daniele. L’ambiente familiare, le quotidiane chiacchierate, gli
scherzi e le battute sono stati fondamentali per affrontare in maniera più serena questi tre
anni e per ridare la giusta dimensione ai vari ostacoli che abbiamo dovuto affrontare lungo
il tragitto.
In questo elenco non può ovviamente mancare Davide, con il quale condivido la passione
per la musica oltre che per fisica. Lo ringrazio in particolare per le stimolanti discussioni di
musica sopratutto durante le nostre escursioni pomeridiane alla Fonoteca Regionale Trotta.
Ringrazio Enrico e Dimitri, miei compagni di stanza al Residence “I Colli” di Firenze
rispettivamente durante le edizioni 2010 e 2011 della scuola LACES.
Voglio inoltre ringraziare Giuseppe, il più recente laureato magistrale in stringhe a
Perugia, la cui tesi ha riguardato il mio stesso ambito di ricerca. Lo ringrazio perché
confrontarmi con lui è stato un importante stimolo a chiarire molti dei dubbi che mi sono
presentati durante lo studio per il mio progetto.
Colgo l’occasione per ringraziare anche tutti i miei compagni di studio nei corsi di
Laurea Triennale e Specialistica (Daniele, Enrico, Antonia, Alessandra, Marco, Jacopo...),
che sono davvero felice di aver incontrato nel mio percorso.
i
Ringraziamenti
Ringrazio anche tutti i miei amici al di fuori del mondo della fisica, tra i quali non
posso esimermi dal citare i miei compagni di squadra di calcetto (nella ormai storica
“Longobarda”), in particolare Stefano e Cosimo.
Infine il ringraziamento più grande è rivolto a tutta la mia famiglia per avermi sempre
sostenuto e a Chiara per essere stata sempre al mio fianco negli ultimi cinque anni.
ii
Contents
Introduction
1
1 D-Branes
1.1 Fundamental strings . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 p-branes and anti-symmetric gauge fields . . . . . . . . . . . . . . .
1.3 Branes as supergravity solitons . . . . . . . . . . . . . . . . . . . .
1.3.1 Black branes solutions . . . . . . . . . . . . . . . . . . . . .
1.3.2 Extremal p-branes . . . . . . . . . . . . . . . . . . . . . . .
1.4 Open string view: Dirichlet branes . . . . . . . . . . . . . . . . . .
1.4.1 D-brane effective action . . . . . . . . . . . . . . . . . . . .
1.4.2 Super Yang-Mills theory from D-branes . . . . . . . . . . .
1.5 AdS/CFT correspondence . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 AdS/CFT at finite temperature . . . . . . . . . . . . . . . .
1.6 BIon solution of DBI . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 DBI action and setup . . . . . . . . . . . . . . . . . . . . .
1.6.2 BIon solution . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 DBI action at finite temperature . . . . . . . . . . . . . . . . . . .
1.7.1 Extrinsic embedding equations from DBI action . . . . . . .
1.7.2 Problems of the Euclidean DBI probe method . . . . . . . .
1.7.3 Towards a new method to describe thermal D-brane probes
2 Blackfold approach
2.1 Blackfolds: Motivations and definition . . . . . . . . .
2.2 Effective theory for black hole motion . . . . . . . . .
2.3 Effective worldvolume theory for a black brane . . . .
2.3.1 Collective coordinates . . . . . . . . . . . . . .
2.3.2 Effective energy-momentum tensor . . . . . . .
2.3.3 Fluid perspective . . . . . . . . . . . . . . . . .
2.4 Blackfold dynamics . . . . . . . . . . . . . . . . . . . .
2.4.1 Embedding and worldvolume geometry . . . . .
2.4.2 Blackfold equations . . . . . . . . . . . . . . . .
2.5 Stationary blackfolds . . . . . . . . . . . . . . . . . . .
2.5.1 Solution to the intrinsic equations . . . . . . .
2.5.2 Horizon geometry, mass and angular momenta
2.5.3 Action principle for stationary blackfolds . . . .
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Contents
2.6
Charged blackfolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6.1 Perfect fluids with conserved p-brane charge . . . . . . . . . . . . . . 50
2.6.2 Stationary solutions and action principles . . . . . . . . . . . . . . . 51
3 Heating up the BIon
3.1 Energy-momentum tensor for black D3-F1 brane bound state . . . . . . . .
3.1.1 More on DBI case . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Energy-momentum tensor for D3-F1 bound state from black brane
geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Thermal D3-brane configuration with electric flux ending in throat . . . . .
3.2.1 D3-F1 extrinsic blackfold equation . . . . . . . . . . . . . . . . . . .
3.2.2 Solution and bounds . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Analysis of branch connected to extremal configuration . . . . . . . .
3.2.4 Validity of the probe approximation . . . . . . . . . . . . . . . . . .
3.3 Separation between branes and anti-branes in wormhole solution . . . . . .
3.3.1 Brane-antibrane wormhole solution . . . . . . . . . . . . . . . . . . .
3.3.2 Diagrams for separation distance ∆ versus minimal radius σ0 . . . .
3.3.3 Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Equilibrium and stability of the brane-antibrane wormhole configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Comparison of phases in canonical ensemble . . . . . . . . . . . . . . . . . .
3.4.1 Choice of ensemble and measurement of the free energy . . . . . . .
3.4.2 Comparison of phases . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Energy in the extremal case . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Heuristic picture away from equilibrium . . . . . . . . . . . . . . . .
3.5 Thermal spike and correspondence with non-extremal string . . . . . . . . .
3.5.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Correspondence point for matching of throat to F-strings . . . . . .
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4 Thermal string probes in AdS and finite temperature Wilson loops
4.1 Finite temperature Wilson loops: standard method and new method . .
4.2 Thermal F-string probe . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Critical distance . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Probe approximation . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Small κ̃ expansion of solution . . . . . . . . . . . . . . . . . . . .
4.3 Physics of the rectangular Wilson loop . . . . . . . . . . . . . . . . . . .
4.3.1 Regularized free energy . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Free energy of rectangular Wilson loop for small LT . . . . . . .
4.3.3 Finite LT and Debye screening of charges . . . . . . . . . . . . .
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5 Thermal DBI action at weak and strong coupling
5.1 One-loop finite temperature DBI action . . . . . . . . . . . .
5.1.1 D3 with electric field . . . . . . . . . . . . . . . . . . .
5.1.2 D-branes with electric and magnetic fields . . . . . . .
5.2 DBI action from supergravity . . . . . . . . . . . . . . . . . .
5.2.1 D3-brane with electric field . . . . . . . . . . . . . . .
5.2.2 D3-brane with parallel electric and magnetic fields . .
5.2.3 D3-brane with orthogonal electric and magnetic fields
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Contents
Conclusions
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A Geometry of embedded submanifolds
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A.1 Extrinsic curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.2 Variational calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
B Hot BIon
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B.1 Analysis of branch connected to neutral configuration . . . . . . . . . . . . . 124
B.2 ∆ expansion for small temperature . . . . . . . . . . . . . . . . . . . . . . . 125
Bibliography
133
v
Introduction
The discovery of the AdS/CFT correspondence [1–3] has been one of the most outstanding
achievements in theoretical physics in recent years. The correspondence states the equivalence between two seemingly unrelated theories: On the one hand a theory of gravity living
in (d + 1)-dimensional Anti-de Sitter (AdS) space times some compact space and on the
other hand a conformally invariant gauge theory living in the d-dimensional boundary of
AdSd+1 .1
The most renowned concrete realization of such a correspondence was conjectured by
Maldacena in 1997, starting from symmetry considerations and from the twofold picture
through which one can describe the physics of a stack of Nc coincident D3-branes. According to the Maldacena’s conjecture, type IIB superstring theory on the AdS5 × S 5
background is equivalent to N = 4 supersymmetric Yang-Mills (SYM) theory with gauge
group SU (Nc ) in four space-time dimensions.
Taking into account how the (dimensionless) parameters of the two theories are related
among each other, one immediately understands the strength of the AdS/CFT correspondence. The parameters that specify the gauge theory are the rank of the gauge group,
2 N (g
Nc , and, the so-called ’t Hooft coupling λ = gYM
c
YM being the Yang-Mills coupling),
which is the true perturbative coupling constant of the theory. On the string side one
instead has the string coupling, gs , and the ratio R/`s between the radius of curvature of
AdS5 and the string length scale. The parameters are related as follows
R4
∼ λ,
`4s
gs ∼
λ
.
Nc
(1)
From the string point of view `s /R appears as the worldsheet coupling. Hence when
the string theory is weakly coupled (`s /R 1) the gauge theory is strongly coupled
(λ 1) and vice versa. In this sense the AdS/CFT correspondence is a weak/strong
coupling duality. This is, from a practical point of view, the most appealing feature of
the correspondence. Indeed, employing the usual perturbative techniques for N = 4 SYM
at weak coupling, one can extract non-trivial information about the quantum regime in
string theory. Conversely, studying the weakly coupled string theory one can learn how
N = 4 SYM behaves at strong coupling. Note that in the latter case, in order for the
“simple” classical description of string theory to be trustworthy, the condition `s /R 1,
which ensures that the stringy effects are negligible, it is not sufficient. Indeed, it must be
1
Indeed the acronym “AdS/CFT” stands for “Anti- de Sitter/Conformal Field Theory”. The correspondence is commonly referred to also as gauge/gravity duality or as holographic correspondence, being perhaps
the best actual implementation of the holographic principle.
1
Introduction
supplemented by the requirement that also quantum effects are small, which implies that
gs 1. In this regime, which on the gauge theory side corresponds to λ 1 and Nc 1,
the theory can be safely approximated by classical supergravity.
Up to now the AdS/CFT correspondence remains the only viable tool we have to
provide an analytic description of a gauge theory beyond the perturbative regime.2 For
this reason, immediately after its formulation, people started to think about clever ways
to exploit the duality for the study of real physical systems.
An extremely fascinating application which straightaway comes to mind concerns the
long-standing issue of reaching a sensible description of quantum chromodynamics (QCD)
in the strongly coupled regime. To this end, many efforts have been made in the development of new holographic models, based on the AdS/CFT correspondence, in the attempt
of making N = 4 SYM as similar as possible to QCD. For instance, exploiting suitable
D-brane systems, it is possible to introduce matter in the fundamental representation of
the gauge group in N = 4 SYM [4, 5],3 make the theory confining [6] and reproduce the
QCD chiral symmetry and its breaking [7–9] (for a review, see [10]).
In fact, a crucial issue in trying to use the gauge/gravity correspondence to explore
QCD is that the latter is very different from N = 4 SYM. This is true in particular at
zero temperature where QCD is a confining non-supersymmetric theory while N = 4 SYM
is conformally invariant and highly supersymmetric. However if one compares the two
theories at finite temperature, they are not so dissimilar. Finite temperature breaks both
the supersymmetries and the conformal invariance of N = 4 SYM and thus one may hope
that at least some properties of N = 4 SYM may be shared by QCD.
Moreover, at a critical temperature, Tdec ' 170 MeV, QCD is believed to undergo a
confinement/deconfinement phase transition. Above this critical temperature a new state
of matter is formed, in which quarks are no longer confined inside hadrons but are mixed
with gluons into a hot dense “soup” where they can freely move. This new phase is the
so-called quark-gluon plasma (QGP).
The results from the experiments at the Relativistic Heavy Ion Collider (RHIC) and
at the Large Hadron Collider (LHC) indicate that a QGP is indeed produced in ultrarelativistic heavy-ion collisions. There is also evidence that this plasma is actually strongly
coupled [11, 12]. Thus the QGP seems to be an extremely good candidate system to be
studied through the AdS/CFT duality.
As a matter of fact, the most famous and possibly important achievement in studying
holographically the strongly coupled QCD physics has been the calculation of the shear viscosity to entropy density ratio, η/s, of the QGP. The AdS/CFT predicts for this quantity,
in the limit of large Nc and λ = ∞, the celebrated result [13]
η 1
=
.
(2)
s λ=∞ 4π
Remarkably this value of η/s turns out to hold not only for N = 4 SYM but for any gauge
theory with a gravity dual in the regime of Nc 1 and infinite coupling. So, in this sense,
this is a universal result [14–17]. If large-Nc QCD should have a gravity dual, its η/s, in
the strict limit of infinite λ, would be given by 1/(4π). But even if this should not be the
2
A widely used approach to study strongly coupled gauge theory is lattice gauge theory. This is a
precious tool but of course it does not provide a deep theoretical understanding of the physics in play.
Moreover it has strict limitations, due to its inherent Euclidean nature, which makes it inadequate to
describe the dynamical properties of the theory.
3
N = 4 SYM has only degrees of freedom in the adjoint representation of the gauge group SU (Nc ).
2
Introduction
case, the result may still hold as a result of an enhanced universality linked to more generic
properties of strongly coupled theories [18].
The experimental data collected at RHIC showed for the QCD plasma a ratio η/s
very close to 1/(4π) [19, 20], confirming the actual possibility of exploiting the AdS/CFT
correspondence to yield insights into the phenomenology of hot QCD matter.
The use of the AdS/CFT duality as a tool to study strongly coupled gauge theory
has not being limited to QCD. For instance another “natural” field where the power of
the correspondence can be fully exploited is that of condensed matter physics. Indeed
many interesting applications have been considered also in this direction (an excellent
review on this topic can be found in Ref. [21]). A worthy aspect is that many effective
Hamiltonians can be used to describe valuable condensed matter systems. For this reason
condensed matter physics appears as a perfect arena in which developing an emergent
field theory with a known AdS dual and thus reaching experimental test of the AdS/CFT
correspondence.
Furthermore, the interest in this context is enhanced by the fact that many strongly
coupled condensed matter systems can be practically engineered and studied in detail in
laboratories. Even more importantly, some of these systems, such as graphene or high-Tc
superconductors, are of significant technological interest. The AdS/CFT correspondence
offers a promising way to gain insights into some of these unconventional materials.
What mentioned above gives an idea of the real revolutionary impact of the AdS/CFT
correspondence, linked essentially to the fascinating possibility of filling the discouraging
gap between real-world physics and string theory.
D-branes are key ingredients in this context. Not only they drove the formulation of
the correspondence but they also play a crucial role within the correspondence itself. We
already mention, for instance, their precious part in the construction of holographic models
of QCD.
The low energy effective dynamics of an extremal D-brane is efficiently captured by the
Dirac-Born-Infeld (DBI) action, which is obtained by integrating out the massive open
string degrees of freedom (DOFs) [22]. The highly non-linear nature of the DBI action
is responsible for many important D-brane phenomena. The first example in string theory where the full non-linear dynamics of the DBI action was exploited is that of the
BIon solution [23, 24]. The latter is a classical solution of the DBI equations of motions
describing the profile for a D-brane carrying a worldvolume electric flux, which can be
interpreted as an F-string dissolving into the brane. The new phenomena at play are that
one can describe multiple coincident F-strings in terms of D-branes and furthermore that
the F-strings go from being one-dimensional objects of zero thickness to be “blown up”
to a higher-dimensional brane wrapped on a sphere. Based on these phenomena, many
important applications of the DBI action were found in the context of the AdS/CFT correspondence. For gravitons satisfying a BPS bound by moving on the equator of the S 5
of the AdS5 × S 5 background it was found that they blow up to become Giant Gravitons, D3-branes wrapped on three-spheres, for sufficiently large energies [25–27]. Another
interesting application is the Wilson loop, originally considered in the AdS/CFT correspondence using the Nambu-Goto F-string action [28, 29]. Here, the “blown up” version
for a Wilson loop in a high-dimensional representation has been considered using the DBI
action, either for the symmetric representation using a D3-brane [30] or the antisymmetric
representation using a D5-brane [31].
3
Introduction
In a huge number of applications the DBI action has been successfully used to describe
D-branes probing zero temperature backgrounds of string theory. This, along with the
lack of its finite temperature generalization, motivated the application of the DBI action
also as a probe of thermal backgrounds, particularly in the context of the AdS/CFT correspondence with either thermal AdS space or a black hole in AdS as the background [6].
Applications include meson spectroscopy at finite temperature, the melting phase transition of mesons and other types of phase transitions in gauge theories with fundamental
matter (see [18] and references therein). Furthermore, the thermal generalizations of the
Wilson loop, the Wilson-Polyakov loop, in high-dimensional representations were considered [32, 33].
However, one can argue that using the DBI action to describe D-branes probing finite
temperature backgrounds is not accurate. In general the equations of motion (EOMs) for
any probe brane can be written as [34, 35]
Kab ρ T ab = J · F ρ
(3)
where Tab is the worldvolume energy-momentum (EM) tensor for the brane, Kab ρ is the
extrinsic curvature given by the embedding geometry of the brane and the right hand side,
J · F ρ , represents possible external forces due to the coupling between a charged brane
and an external field. Describing a D-brane probe in a thermal background through the
DBI action corresponds to using the EOMs (3) with the same EM tensor as in the zero
temperature case. Doing this one neglects the effect of the temperature of the background
on the physics of the brane. In a more accurate picture the thermal background acts
as a heat bath for the D-brane probe and the whole system attains thermal equilibrium.
Accordingly the DOFs living on the brane are “warmed up” and thus the EM tensor of the
brane changes with respect to the zero temperature case. This in turn changes the EOMs
(3) that one should solve for the probe brane.
The challenge is that one does not know what replaces the DBI action, which is a low
energy effective action for a single extremal D-brane at weak string coupling, when turning
on the temperature. The main aim of this thesis is precisely to address this issue. The key
observation is that in the regime of a large number N of coinciding D-branes we have an
effective description of the D-branes in terms of a supergravity solution in the bulk when
gs N 1,4 i.e. at strong coupling. Using this supergravity description one can determine
the EM tensor for the D-brane in the regime of large N . This EM tensor will then enable
one to write down the EOMs (3) for a non-extremal D-brane probe in the regime of large
N . In this way one replaces the DBI action, which provides a good description of a single
D-brane probing a zero-temperature background, by a new method that can describe N
coincident non-extremal D-branes probing a thermal background such that the probe is in
thermal equilibrium with the background [36, 37].
The new method for non-extremal D-branes probing thermal backgrounds is actually
based on the blackfold approach [35, 38–44], recently developed as a tool to build novel
approximate black hole solutions in more than four dimensions. In fact, the blackfold
approach is much more powerful and its applicability goes far beyond the original aim
it was developed for. In general, it provides an effective description of the black brane
dynamics, when the thickness of the black brane, r0 , is much smaller than the length scale
4
gs is the string coupling.
4
Introduction
(∼ R) of the embedding geometry.5 In this effective description the dynamical principle is
given by the conservation of the EM tensor of the brane which is that of a fluid living on the
brane worldvolume. The resulting EOMs are of hydrodynamic type – conservation of the
EM tensor – on the worldvolume along with elasticity equations of the form (3) describing
the extrinsic motion of the brane.6 To leading order in r0 /R the brane can be regarded as
a probe brane that does not backreact on the background geometry. Thus one can parallel
the probe approximation in the blackfold approach with the probe approximation that the
DBI action assumes and the only difference in the extrinsic EOMs (3) is that one should
replace the DBI EM tensor with that of the fluid EM tensor for the black brane.
Firstly we shall use our novel method for D-brane probes in finite temperature backgrounds to study the thermal generalization of the BIon solution [36, 37]. This, besides
being interesting in itself, serves as a test case of the method. The BIon solution is a configuration in flat space of a D-brane and a parallel anti-D-brane connected by a “wormhole”
with F-string charge. In the thermal generalization, obtained by putting this configuration in hot flat space, one finds that the finite temperature system behaves qualitatively
different than its zero-temperature counterpart.
One of the nice features of the new method is that it can be applied not only to Dbranes but more generally to any brane probing a thermal background, provided that we
know the corresponding non-extremal supergravity solution. Thus, for example, it is well
suited also for the description of fundamental string probes in thermal backgrounds.
The first application of the new method in the context of the AdS/CFT correspondence
concerns exactly thermal fundamental string probes, in order to study Wilson loops in the
N = 4 SU (Nc ) Super Yang-Mills at finite temperature [45]. Previously this problem has
been considered using extremal probes even though the background is at finite temperature
[46, 47]. As a result of our analysis we find a new term in the potential between static
quarks in the symmetric representation which for sufficiently small temperatures is the
leading correction to the Coulomb force potential.
It is worth emphasizing that this new method, being valid at strong coupling, works
in a regime opposite to that usually considered for D-brane probes. This is of course a
feature of the method and not a bug. However it would be interesting to have a way to
describe thermal D-brane probes in the weakly coupled regime, gs 1 and N = 1, which
is the same regime of validity of the DBI. This motivates the development of another
method to treat such probes, which consists in using a thermally corrected version of the
DBI action. The latter can be obtained perturbatively by computing small temperature
corrections to the DBI action. The leading order correction in temperature can be achieved
through the quantum computation of the one-loop effective action for the DBI [48], which
we dub “thermal DBI action at weak coupling”. In the last part of this thesis we derive
this correction for a Dp-brane with electric and magnetic fields.
The thesis is organized as follows. Chapter 1 provides a brief survey of string theory,
D-branes physics and the AdS/CFT correspondence, with a special focus on the aspects
5
The brane thickness scale is given by the length scale over which the brane backreacts on the surrounding spacetime.
6
According to these two sets of equations, black branes behave as liquids under strains parallel to their
worldvolume, and like elastic solids under orthogonal strains.
5
Introduction
that will be relevant for the following discussion. In particular we review the BIon solution
of the DBI action and we discuss the problems in the usage of the DBI action to describe
D-brane probing thermal backgrounds.
Chapter 2 contains a review of the blackfold effective worldvolume theory, which, as
already mentioned, is a key block for the development of main subject of the thesis.
In Chapter 3 we present in more detail the new method to describe thermal brane
probes, based on the blackfold approach. We do this by studying the thermal generalization
of the BIon solution, which we in fact use as a test case to study D-branes as probes of
thermal backgrounds [36, 37].
In Chapter 4 we apply the new method to a fundamental string probing the AdS black
hole background. In such a way we manage to study holographically finite temperature
Wilson loops in the N = 4 SYM theory [45].
Chapter 5 is devoted to the computation of the thermal DBI action at weak coupling.
For the specific case of D3-branes we also compare the latter action with the corresponding
one at strong coupling [48].
In the Conclusions we further discuss the main results described in this thesis along
with possible outlooks.
6
Chapter 1
D-Branes
D-branes play a pivotal role in string theory. Their study allowed to shed light on many
essential aspects of the theory and then contributed in a determinant manner to the development of the latter. Actually string theory is not a theory of strings only but it contains
more generic dynamical extended objects besides strings themselves, usually referred to as
branes. D-branes form an extremely important sub-class of these.
In this first chapter we present and briefly discuss some of the main features regarding
D-branes. We start in Section 1.1 by recalling some elementary notions about bosonic
and supersymmetric string theories, focusing in particular on the spectrum of type II
superstrings. This contains Ramond-Ramond gauge fields, which, as shown in Section 1.2,
are sourced by p-branes. In Section 1.3 we see how p-branes arise as classical solutions of
type II supergravities, which are the low energy limit of type II string theories.
In Section 1.4 we introduce D-branes from the open string point of view and we argue,
taking into account some of their properties, that they can be identified with the supergravity p-branes charged under the Ramond-Ramond gauge fields. We also write the low
energy effective action for D-branes, the so called Dirac-Born-Infeld (DBI) action and show
that a stack of coincident D-branes provide a beautiful way to realize non-Abelian gauge
theories.
In Section 1.5 we point out how the different ways in which one can describe the D-brane
physics lead “naturally” to the formulation of the AdS/CFT correspondence [1–3].
In the last part of the chapter we begin to approach the main subject of this thesis,
i.e. the description of brane probes at finite temperature. We do this by considering in
more detail the DBI action for D-branes. Firstly, in Section 1.6 we review the original
BIon solution [23, 24], which is a classical solution of the DBI equations of motion (at
zero temperature). This solution is interesting for our purpose, since in Chapter 3 we
will consider exactly the BIon system, generalized to finite temperature, as a test case
to explain our new method to treat brane probes in thermal backgrounds. The need to
introduce this new description is motivated by the inaccuracy of the “standard method”
used so far in the literature, which makes use of the DBI action and thus neglects the
thermal excitations on the brane. The problems and limitations in using the DBI action
at finite temperature are discussed in Section 1.7.
1.1
Fundamental strings
The first basic notion that a student learns when approaching string theory concerns the
dynamics of a relativistic one-dimentional object, a string, embedded in a D dimensional
7
Chapter 1. D-Branes
background (D > 2), with spacetime metric gµν [49–52]. This is captured by the NambuGoto action, which is defined, up to an overall factor, as the area of the string worldsheet,
i.e. the two-dimensional surface spanned by the string during its motion
Z
√
ING = −TF1 dτ dσ γ ,
(1.1)
where TF1 = 2π`2s is the string tension (mass per unit length),1 σ α = (τ, σ), with α =
0, 1, are the worldsheet coordinates and γ ≡ − det γαβ . The map X µ (τ, σ) defines the
embedding of the string in the target spacetime and γαβ = ∂α X µ ∂β X ν gµν is the induced
metric on the worldsheet.
A string-like object whose dynamics is described by the Nambu-Goto action (1.1) is
called fundamental string (F-string or F1 for short).2
In order to avoid the difficulties due to the presence of a square root in the action (1.1)
it is customary to replace it with the following action
Z
√
TF1
IP = −
(1.2)
dτ dσ λλαβ ∂α X µ ∂β X ν gµν ,
2
where λαβ is an auxiliary dynamical metric of the string worldsheet and λ ≡ − det λαβ . IP
is called Polyakov action and it is easy to prove using the equation of motion (EOM) for
λαβ that it is equivalent to the Nambu-Goto action (1.1). Let us assume the target space
to be flat, i.e. gµν = ηµν . Exploiting reparametrization and Weyl symmetries of the action
(1.2) we can also choose the worldsheet metric to be flat, λαβ = ηαβ , yielding
Z
TF1
IP = −
dτ dσ∂α X µ ∂ α Xµ .
(1.3)
2
The strings described by this action can be either closed or open: In the former case
they must satisfy periodic boundary conditions, in the latter Neumann or Dirichlet ones
Closed
Open
X µ (τ, 0) = X µ (τ, π)
∂σ X µ (τ, σ)|σ=0,π = 0 (Neumann)
∂τ X µ (τ, σ)|σ=0,π = 0 (Dirichlet)
(1.4)
where we assumed that the spatial worldsheet coordinate σ ∈ [0, π] and for open strings
σ = 0 and σ = π identify the endpoints.
From the two-dimensional worldsheet point of view Eq. (1.3) can be regarded as the
action for D scalar fields X µ . The theory built up from this action is then called Bosonic
String Theory: The critical dimension in which the theory can be consistently quantized
turns out to be D = 26.
The action (1.3) can be easily made supersymmetric from the worldsheet point of view
by introducing D fields ψ µ (τ, σ) which are fermionic superpartners of X µ (τ, σ)
Z
TF1
dτ dσ ∂α X µ ∂ α Xµ − i ψ̄ µ ρα ∂α ψ µ ,
(1.5)
I=−
2
`s is the characteristic string length scale. It is related to the celebrated Regge slope α0 through the
relation α0 ≡ `2s .
2
In general the action of string-like objects can contain also other terms besides the Nambu-Goto one,
which are related to their width: A fundamental string is then a string with zero width.
1
8
1.1 Fundamental strings
where ρα is a two-dimensional representation of the Clifford algebra and ψ̄ µ = (ψ µ )T ρ0 .
The fermionic fields ψ µ are two-component spinors on the worldsheet
µ
ψ−
µ
ψ =
(1.6)
µ
ψ+
and vectors on the target space.
The action (1.5), describing the dynamics of a supersymmetric fundamental string, is
the starting point in the development of the Superstring Theory, whose critical dimension,
which guarantees quantum consistency, is D = 10.
For open superstrings the fermionic fields ψ µ have to satisfy either Ramond (R) or
Neveu-Schwarz (NS) boundary conditions
µ
µ
ψ+
(τ, π) = ψ−
(τ, π)
µ
ψ+
(τ, π)
=
µ
−ψ−
(τ, π)
R
NS
(1.7)
µ
µ
where we choose ψ+
(τ, 0) = ψ−
(τ, 0). The R sector is fermionic and the NS is bosonic in
the target space.
For closed superstrings one has instead to impose for each component of ψ µ periodic
(R) or anti-periodic (NS) conditions
µ
µ
ψ− (τ, 0)
R
ψ+ (τ, 0)
R
µ
µ
(1.8)
ψ− (τ, π) =
ψ+ (τ, π) =
µ
µ
−ψ− (τ, 0) NS
−ψ+ (τ, 0) NS
Thus there are four sectors for closed strings: The R-R and NS-NS sectors which are
bosonic and the R-NS and NS-R which are fermionic.
In order to remove the unphysical tachyon states from the spectrum one has to perform the so called GSO (Gliozzi-Sherk-Olive) projection [53] which also makes the theory
supersymmentric in the target spacetime. Instead of starting from the action (1.5) supersymmetric in the worldsheet (Ramond-Neveu-Schwarz method) one can start directly
with imposing supersymmetry in the target space (Green-Schwarz method). The two approaches are equivalent in the light-cone gauge [49].
Unlike the bosonic theory which is unique, there exists instead five different consistent
superstring theories: Type I, type IIA, type IIB and Heterotic with either SO(32) or
E8 × E8 gauge group. These theories are five different limits of a more fundamental
underlying 11-dimensional theory, known as M-theory [54], and they are all related by
duality transformations.
The type I string theory is based on open superstring: It has only one spacetime
supersymmetry, as a consequence of the boundary conditions, which relate the left and
right-modes. In the case of only Neumann boundary conditions the massless bosonic
sector of the spectrum contains only a vector field Aµ . The most general case in which
also Dirichlet conditions are considered will be taken into account in Section 1.4.
In this thesis we will consider mainly the type IIA and IIB string theories, which are
theories of closed superstrings. The label II refers to the fact that these theories have two
independent supersymmetries and the A and B refer to whether the two ten-dimensional
supercharges have the opposite or the same chirality, respectively. This difference corresponds to a freedom of choice between two consistent GSO projections of the closed string
spectrum.
The massless bosonic fields of the type IIA and IIB superstring theories have in common
the NS-NS modes which are given by a symmetric metric tensor gµν , an anti-symmetric
9
Chapter 1. D-Branes
rank two tensor potential Bµν , called Kalb-Ramond field, and a scalar dilaton Φ. The
remaining massless bosonic fields come from the R-R sector and are different for the two
(1)
theories: In the type IIA theory the R-R fields are a vector potential Cµ , and an anti(3)
symmetric rank three potential Cµνλ , while in the IIB theory they are a scalar C (0) , the
(2)
(4)
anti-symmetric potentials of rank two and four Cµν and Cµνλσ .
1.2
p-branes and anti-symmetric gauge fields
As described above that type IIA and IIB superstring theories contain anti-symmetric
fields, which at the massless level are given by the NS-NS two form and the various R-R
forms. These fields are the generalization of the familiar electromagnetic gauge field. It is
then natural to ask whether there are, in the two theories, objects that carry charge under
these gauge fields. As a consequence of various string dualities, in order for the theories to
be consistent, such objects there must exists [54–56]. However if we restrict ourselves to
consider string theory as a theory of only strings we immediately run into problems. This
is because strings, having a two-dimensional worldsheet, cannot couple to a generic q-form
(with arbitrary q). They can couple only to a two-form and, in fact, as we will show in the
following section, they turn out to couple only to the NS-NS Kalb-Ramond field.3 So it
could seem that in the type IIA and IIB theories the sources for all the R-R gauge fields
are missing. Nevertheless this is not the whole story, since, as we already mentioned, in
string theory there are also other extended object besides strings themselves.
First let us observe that the natural object which can couple to an anti-symmetric
tensor field is a p-brane. The name “p-brane” generally indicates, in the context of a
theory containing gravity, a classical solution (a soliton) which is extended in p directions,
i.e. which has p spacelike translational Killing vectors. More loosely speaking we can
define a p-brane as an object which extends through p spatial direction and thus has a
(p + 1)-dimensional worldvolume Wp+1 .
Suppose we have in D dimensions an antisymmetric tensor field strength F (n) of rank
n, such that F (n) = dA(n−1) . Then it is trivial to see that it can couple minimally to the
worldvolume p-brane with p = n − 2 spatial extended directions:
Z
µn−2
Wn−1
A(n−1) = µn−2
Z
Aµ0 ···µn−2 dxµ0 ∧ · · · ∧ dxµn−2 .
(1.9)
Wn−1
Here µn−2 is the charge density of the (n − 2)-brane under the n-form field strength F (n) .
The term displayed in Eq. (1.9) is called electric coupling. The word “electric” refers to the
fact that F (n) has a time component.
However, in analogy with electromagnetism, we can also introduce magnetic charges.
Therefore we have to define the analog of the magnetic field, namely the magnetic dual
form Ã(D−n−1) . This (D − n − 1)-form is simply the potential of the Hodge dual of the
form F (n)
dÃ(D−n−1) = F̃ (D−n) = ?F (n) = ?dA(n−1) .
(1.10)
3
To be precise open strings can also couple to one-forms, since their ends are zero-dimensional points.
They are in fact sources of the electromagnetic gauge field, which is the bosonic massless mode of the open
string (type I) spectrum.
10
1.3 Branes as supergravity solitons
This magnetic potential can now couple to a p0 -brane, with p0 = D − n − 2, in the following
way
Z
Ã(D−n−1) .
µ̃D−n−2
(1.11)
WD−n−1
To summarize, every n-form field strength should imply the existence of an electric p-brane
and a magnetic p0 -brane, with p = n − 2 and p0 = D − n − 2. Note that p + p0 = D − 4,
and that the existence of both of these objects imposes a Dirac-like quantization condition
on their charges [57, 58]
µp µ̃D−p−4 = 2πk ,
k ∈ Z.
(1.12)
What we have shown so far hints that the stable p-branes charged under the R-R fields
which are allowed in type IIA string theory must have even p, while those allowed in type
IIB must have odd p. This is evident if we remember that the R-R massless field sector
of the theories contains odd rank forms for the type IIA and even rank forms for the type
IIB string theories. We will see in the following sections that this guess is indeed correct.
1.3
Branes as supergravity solitons
The low energy limit of type IIA and IIB string theories is D = 10 supergravity of type
IIA and IIB, respectively. In this regime only the massless modes of the original string
theories survive, since all the massive ones become infinitely heavy and thus decouple.
Hence, according to Section 1.1, the bosonic field content of the two supergravity theories
is the one shown in the Table 1.1.
NS-NS
R-R
IIA
gµν , Bµν , Φ
Cµ , Cµνλ
IIB
gµν , Bµν , Φ
C (0) , Cµν , Cµνλσ
(1)
(2)
(3)
(4)
Table 1.1: Bosonic field content of the type IIA and IIB supergravity theories.
In addition to the massless bosonic fields there are their fermionic superpartners, which
however are irrelevant for our purposes, so we do not take them into account.
The bosonic part of the effective action for type IIA supergravity in the string-frame is
Z
1
1
1
10 √
−2Φ
2
(3) 2
IIIA =
d x gS e
R + 4(∂Φ) −
(H ) − (F (2) )2
16πG10
2 · 3!
4
(1.13)
Z
1
1
(4) 2
(2)
(3)
(3)
−
(F ) −
B ∧ dC ∧ dC ,
2·4!
4(κ10 )2
where H (3) is the NS-NS three-form field strength with potential B (2) and F (2) and F (4)
are the R-R two-form and four-form with potential C (1) and C (3) respectively
H (3) = dB (2) ,
F (2) = dC (1) ,
F (4) = dC (3) + C (1) ∧ H (3) .
(1.14)
The 10-dimensional Newton constant is defined in terms of the string length G10 ∼ `8s .
We recall that the asymptotic value of the dilaton at infinity sets the string coupling
11
Chapter 1. D-Branes
eΦ∞ = gs . It is in general convenient to subtract from the dilaton field its constant part at
infinity Φ∞ and to insert it into the value of G10 , which becomes G010 ∼ e2Φ∞ `8s = gs2 `8s . If
we want to keep a unique coupling constant in front of the whole action, we also have to
rescale the R-R fields. The new dilaton Φ0 = Φ − Φ∞ is thus vanishing at infinity. Since in
the following we will always use this dilaton and Newton constant we accordingly drop the
primes. Taking into account also the constant numerical factor, the supergravity coupling
constant becomes
16πG10 = (2π)7 `8s gs2 .
(1.15)
The bosonic part of the type IIB supergravity is instead given by
Z
1
1
2
−2Φ
10 √
(3) 2
IIIB =
R + 4(∂Φ) −
d x gS e
(H )
16πG10
2 · 3!
2 1
1 (3)
1
(0)
(3)
(1) 2
(5) 2
− (F ) −
−
F +C ∧H
(F )
2 · 3!
2
4·5!
Z 1
1 (2)
(4)
(2)
∧ F (3) ∧ H (3) ,
−
C + B ∧C
4(κ10 )2
2
(1.16)
in which F (3) and F (5) are the R-R three-form and five-form with potential C (2) and C (4)
F (3) = dC (2) ,
F (5) = dC (4) − C (2) ∧ H (3) .
(1.17)
Note that in order to have the right number of bosonic DOFs the four-form C (4) has to be
self-dual. This condition cannot be implemented in any simple way in the action thus it
has to be imposed by hand in the equations of motion.
Both the type IIA and IIB actions (1.13) and (1.16) are written using the string-frame
metric. The relation which allows to convert them to the Einstein-frame, in which the
gravitational term has the canonical form, is
Φ
E
S
gµν
= e− 2 gµν
.
(1.18)
We now consider the case in which only one anti-symmetric potential, among the NSNS two-form and the R-R forms available in the two supergravity theories, is turned on.
Let us call this potential A(n−1) and the corresponding field strength F (n) = dA(n−1) .
Then the actions (1.13) and (1.16) can be combined in the Einstein-frame in the following
fashion
Z
1
1
1 aΦ (n) 2
10 √
µ
I=
d x gE R − ∂µ Φ∂ Φ −
e (F )
.
(1.19)
16πG10
2
2n!
where the constant a depends on the potential taken into account: a = −1 for the NS-NS
B (2) field and a = (5 − n)/2 for the R-R C (n−1) fields.
The equations of motion derived from (1.19) are
1 µ
1 aΦ
n−1 µ
µ,ξ2 ···ξn
(n) 2
µ
e
nF
Fν,ξ2 ···ξn −
δ ν (F )
R ν = ∂ Φ∂ν Φ +
2
2n!
8
1
a aΦ (n) 2
√
(1.20)
Φ = √ ∂µ ( gg µν ∂ν Φ) =
e (F )
g
2n!
√
∂µ ( geaΦ F µ,ν2 ···νn ) = 0 .
The Bianchi identity for F (n) is
∂[µ1 F µ2 ...µn+1 ] = 0.
12
(1.21)
1.3 Branes as supergravity solitons
The dual field strength F̃ (10−n) is defined as
F̃ (10−n) = eaΦ ? F (n) ,
(1.22)
with the Hodge dual given by
√
(?F )µn+1 ···µ10 =
g µ1 ...µn
F
µ1 ...µ10 ,
n!
(1.23)
µ1 ...µ10 being the 10-dimensional Levi-Civita tensor, with 01...9 = 1. The self-duality of
the four-form A(4) = C (4) then means F (5) = F̃ (5) .
1.3.1
Black branes solutions
The equations of motion (1.20) admits p-branes as classical solutions [59] (see also [60–63]).
In order to explicitly find them we have to look for solutions which have an SO(1, p) ×
SO(9 − p) isometry. We call the time t, the longitudinal coordinates x1 , . . . , xp and we
parametrize the (9 − p)-dimensional transverse space by spherical coordinates with radius
r and angles θ1 , . . . , θ8−p . According to what explained in Section 1.2 we can construct two
different solutions having either an electric or magnetic coupling with the field strength.
We only consider electric solutions since the magnetic ones can be obtained by electricmagnetic duality. This means that n = p + 2. An appropriate ansatz for the electric
p-brane has Einstein-frame metric
!
p
X
−1 2
p−7
2
i
2
2
2
2
ds = H 8
−f dt +
(dx ) + H f dr + r dΩ8−p
,
(1.24)
i=1
where dΩ28−p is the line element on the (8 − p)-sphere of unit radius.
Solving the equation of motions (1.20) using the ansatz (1.24) we obtain the solution
for the fields Φ and A(p+1) and the functions H and f . The dilaton is
a
eΦ = H 2
(1.25)
A(p+1) = coth α H −1 − 1 dt ∧ dx1 ∧ · · · ∧ dxp .
(1.26)
and the potential A(p+1) is given by
The harmonic functions are
H =1+
r07−p sinh2 α
,
r7−p
f =1−
r07−p
.
r7−p
The charge Q of the p-brane can be computed in this way
Z
Vp
Vp Ω8−p
Q=
F̃ (p+2) =
(7 − p)r07−p cosh α sinh α ,
16πG10 S 8−p
16πG10
(1.27)
(1.28)
where Ω8−p is the volume of the unit (8 − p)-dimensional sphere S 8−p . In general Ωm is
given by
m+1
2π 2
.
Ωm =
(1.29)
Γ m+1
2
13
Chapter 1. D-Branes
The overall factor (16πG10 )−1 has been chosen in order for the charge Q to have the
dimension of a mass.
We can use Eq. (1.28) to find an expression for sinh α
s
2(7−p)
h
1 1
2
sinh α =
+ − ,
(1.30)
r0
4 2
where we have defined h, proportional to the charge, as
h7−p ≡
Q 16πG10
= r07−p cosh α sinh α .
Vp Ω8−p
(1.31)
We can also compute the ADM mass of the solution which turns out to be
M=
Vp Ω8−p 7−p r0
8 − p + (7 − p) sinh2 α .
16πG10
(1.32)
Note that the p-brane defined in (1.24)-(1.27) is in general a black hole solution whose
event horizon is located at radius r = r0 . For this reason it is also referred to as black
p-brane.
We can thus examine the thermodynamics of such a solution. Its Hawking temperature
T and Bekenstein-Hawking entropy S are
T =
7−p
,
4πr0 cosh α
S=
Vp Ω8−p 8−p
cosh α ,
r
4G10 0
(1.33)
and the chemical potential is
µ = tanh α .
(1.34)
One can then prove that the Smarr formula
(7 − p)M = (8 − p)T S + (7 − p)µQ
(1.35)
and the first law of thermodynamics
dM = T dS + µdQ ,
M = M (S, Q)
(1.36)
are satisfied.
It is important to note that the point r = 0 is a singularity. The condition dictated
by the cosmic chensorship that this singularity is hidden inside the horizon implies the so
called Bogomol’nyi bound
M ≥ Q.
(1.37)
When the bound is saturated, i.e. when M = Q, the brane is referred to as extremal
p-brane.
1.3.2
Extremal p-branes
The extremality condition for which the Bogomol’nyi bound is saturated, is reached in
the limit r0 → 0, while keeping the charge (and then h) constant: Accordingly the limit
corresponds to sending α → ∞.
14
1.3 Branes as supergravity solitons
Extremal p-branes enjoy the remarkable property of being 1/2 BPS4 objects, since
they preserve half of the spacetime supersymmetries, namely 16 of the 32 supersymmetries
present in type II theories.
From Eq. (1.33) we also see that extremal branes have zero temperature.5
The solution found in (1.24)-(1.27) describes electric p-branes charged either under the
NS-NS Kalb-Ramond field or under one of the R-R potentials. We now distinguish the
two cases. Let us start with the p-branes carrying the R-R charge, since they are more
interesting for our purposes. In the extremal limit the p-brane solution is still given by the
einstein-frame metric (1.24) with
3−p
eΦ = H 2 ,
A(p+1) = H −1 − 1 dt ∧ dx1 ∧ · · · ∧ dxp ,
(1.38)
h7−p
H = 1 + 7−p ,
f = 1.
r
It follows that the string-frame metric is
"
#
p
X
ds2 = H −1/2 −dt2 +
(dxi )2 + H 1/2 dr2 + r2 dΩ28−p .
(1.39)
i=1
The mass (and then also the charge) is
M=
Vp Ω8−p 7−p
h
=Q
16πG10
(1.40)
For branes charged under R-R fields h is given by
h7−p =
(2π)7−p gs `s7−p
.
(7 − p)Ω8−p
(1.41)
Using Eq. (1.41) and Eq. (1.15) we can easily find the tension Tp = M/Vp of R-R p-branes:
Tp =
1
(2π)p gs `p+1
s
.
(1.42)
The extremality condition corresponds also to the fact that a couple of static p-branes
do not exert force between each other. This allows to write a generalized multicenter
p-brane solution characterized by the harmonic function
H =1+
N
X
i=1
h7−p
.
|~r − ~ri |7−p
(1.43)
This represents N different p-branes located at the position ri in their transverse space.
Let us now briefly examine the case of branes charged under the Kalb-Ramond field.
The NS-NS B (2) field, being a two-form, couples electrically to a 1-brane, namely a string.
The extremal solution in the string-frame metric reads
ds2 = H −1 −dt2 + (dx1 )2 + dr2 + r2 dΩ27
(1.44)
4
BPS stands for Bogomol’nyi-Prasad-Sommerfield [64, 65].
Note that in fact the extremal limit of black p-branes is well defined only when p < 5. This is related
to the fact that black branes of higher dimensionality are thermodynamically unstable, which then implies
a classical Gregory-Laflamme [66, 67] instability according to the “correlated stability conjecture” [68–71].
5
15
Chapter 1. D-Branes
and
32π 2 gs2 `6s
.
(1.45)
r6
The supergravity solution which one gets in this case describes a fundamental string.
This can be argued by looking at its tension which turns out to be 1/(2π`2s ) and thus it
exactly matches the fundamental string tension defined in Section 1.1. This confirms what
we already mentioned in Section 1.2, i.e. that a fundamental string is carries a NS-NS
charge and not a R-R charge. This can be also argued from the the structure of the vertex
operators associated to the NS-NS and R-R gauge fields [72].
The brane magnetically charged under the B (2) field is instead a 5-brane and it is
referred to as NS5-brane. The string frame metric for the extremal NS5-branes is
eΦ = H −1/2 ,
B01 = H −1 − 1 ,
ds2 = −dt2 +
5
X
H =1+
(dxi )2 + H dr2 + r2 dΩ23 ,
(1.46)
i=1
the dilaton and the NS-NS gauge potential are
eΦ = H 1/2 ,
B01 = H −1 − 1 ,
and its tension is
TNS5 =
1.4
H =1+
`2s
,
r2
1
.
(2π)5 gs2 `6s
(1.47)
(1.48)
Open string view: Dirichlet branes
Now let us see how “other” brane-like objects can be naturally introduced in the open string
context. Such objects, called D-branes, were discovered by Dai, Leigh and Polchinski [73],
and independently by Horava [74] in 1989.6 Remarkably there is a strict connection between
D-branes and the p-branes arising from type II supergravities, which we presented in the
previous section. Indeed, as we will argue shortly, D-branes couple to the R-R gauge field
of the closed type II strings [72].
We then start with a type II closed superstring theory, in which we add open strings.
As explained in Section 1.1, for open strings the equations of motion must be supplemented
by either Neumann or Dirichlet boundary conditions for the fields X µ . In general one can
have for each end a mixture of Neumann or Dirichlet boundary conditions, say7
(Neumann)
∂σ X a (σ)|σ=0,π = 0 ,
a = 0, . . . , p
(Dirichlet)
X I (σ)|σ=0,π = cI ,
I = p + 1, . . . , 9.
(1.49)
These conditions simply mean that the endpoints of the string are constrained to move on
a (p + 1)-dimensional hyperplane extending along the x0 , x1 , . . . , xp directions of the target
space and sitting at xI = cI in its transverse space. Such an hyperplane is called Dirichletbrane, D-brane for short, or, to be more specific Dp-brane, p indicating the number of
6
An exhaustive treatment of D-branes can be found in several books, such as [50, 52, 75], and reviews,
for instance [76–78] and of course the classical Polchinski’s TASI lectures [79].
7
Actually in the most general case the directions on which one imposes Dirichlet and Neumann boundary
conditions can be different for the two ends.
16
1.4 Open string view: Dirichlet branes
its spatial dimensions. Indeed we are led to the customary definition which states that
Dp-branes are (p + 1)-dimensional hypersurfaces on which open strings are allowed to end.
Up to this stage D-branes appear as static non-dynamical hyperplanes. However since
string theory is also a theory of gravity one can naively argue that this cannot be the case.
In fact it turns out that D-branes are dynamical objects in their own right in string theory.
By quantizing the open superstring with boundary conditions given by (1.49) we obtain
a spectrum of states with momentum only on the directions along the Dp-brane worldvolume. This means that the corresponding particles propagate only in the (p+1)-dimensional
worldvolume of the D-brane. At the massless level the bosonic states (NS sector) are a
gauge (p + 1)-vector Aa and 9 − p scalars φI . These fields along with their fermionic superpartners coming from the R sector fill out a U (1) vector multiplet in (p + 1) dimensions.
The scalar fields φI can be interpreted as the fluctuations of the brane in the transverse
direction. This is a first hint that D-branes are indeed dynamical objects.
Let us now examine some of the most important features of D-branes. As a first
observation we can note that such objects break half of the supersymmetries of the theory.
This is a direct consequence of the fact that the open strings attached to a brane, which
describe its fluctuations, have only half of the supersymmetry of the closed strings.
In order to analyze in more detail the supersymmetry preserved by D-branes we have
to make use of T-duality (for a review see [80]). This is a duality which relates a theory in
which one dimension is compactified on a circle of radius R with the same theory in which
the radius of compactification is `2s /R. So it essentially states the equivalence between
large and short distance physics. It can be easily shown that T-duality is a transformation
which acts as a parity reversal restricted only to the right-moving modes. Then it also
changes the chirality of half of the supersymmetry charges and therefore it maps the type
IIA theory into the type IIB, and vice versa. In open strings it corresponds to switching
the Neumann and Dirichlet boundary conditions. Starting with an open superstring theory
with only Neumann condition and applying T-duality transformations on 9 − p directions
xI we end up with the same theory described above in (1.49), and thus with a Dp-brane
into the game. T-duality plays a crucial role in the understanding of the D-brane physics
and it was exactly the study of T-duality that led to the discovery of D-branes.
Since T-duality exchanges Neumann and Dirichlet boundary conditions, its effect on
Dp-branes can be easily derived: A Dp-brane transforms into a D(p − 1)-brane if one Tdualizes along a direction longitudinal to the worldvolume of the original Dp-brane, or into
a D(p + 1)-brane if one T-dualizes along a direction transversal to it.
Now we are ready to find the supersymmetry projection associated with a D-brane. It
is easier to start with the D9-branes, i.e. D-branes filling the whole spacetime, which then
correspond to the “standard” (only Neumann conditions) open string theory. Let Qα and
Q̃α be the left- and right-moving spacetime supercharges of type IIB string theory. The
open string boundary conditions preserve only the sum of these supercharges Qα + Q̃α ,
leaving therefore half of the supersymmetry unbroken. We can now introduce Dirichlet
boundary conditions by the implementation of T-duality transformations. T-dualizing
along the directions xp+1 , . . . , x9 we get a Dp-brane lying along x1 , . . . , xp and the effect
on the supersymmetry generators is
Y
Q0α = Qα ,
Q̃0α =
Pm Q̃α ,
(1.50)
m∈Dp
/
where the product is over all the directions transverse to the D-brane. Pm is an operator
which anticommutes with Γm and commutes with all the other Γ’s. It can be chosen to
17
Chapter 1. D-Branes
be Pm = Γm Γ11 , with Γ11 = Γ0 · · · Γ9 being the matrix defining the
Q chirality. Hence we
can conclude that a Dp-brane preserves only the combination Qα + Pm Q̃α . Since we are
considering type II strings which have N = 2 spacetime supersymmetries, the introduction
of D-branes in the theory reduce the amount of supesymmetries to be only N = 1. This
proves that D-branes are BPS states.
Note furthermore that since T-duality interchanges type IIA and IIB theories we only
have supersymmetric D-branes of odd dimension in type IIB theory and of even dimension
in type IIA theory.
The fact that D-branes are BPS objects implies that they must carry conserved charges.
In 1995 Polchinski showed that indeed they carry R-R charges [72]. Polchinski computed
the static force exerted between two parallel D-branes, which is given by the amplitude
corresponding to a cylinder stretching between the two D-branes, without any insertion.
According to the open/closed string duality this can be interpreted either as a tree-level
closed string amplitude or as a one-loop amplitude from the open string point of view.
The result of the computation is that the amplitude vanishes, thus implying that there is
no static force between parallel D-branes. This is not surprising since we are calculating a
vacuum amplitude in a supersymmetric theory.
Moreover this result can be given an interpretation in the closed channel. The amplitude
actually vanishes because two terms cancel: The exchange of NS-NS particles (the graviton
and the dilaton) gives rise to an attractive force between the D-branes, while the exchange
of R-R gauge bosons determines a repulsive force. The balance of forces is a familiar feature
of BPS objects.
This proves that the D-branes carry R-R-charges, and that their charge is equal to
their tension. Furthermore, the tension can be extracted and it is given by
TDp =
1
(2π)p g
p+1
s `s
.
(1.51)
It is worth stressing the inverse dependence of the tension on the string coupling gs . This
is a sign of the non-perturbative nature of D-branes and it confirms that D-branes are
solitonic objects in the theory.8
In summary we have seen that D-branes are BPS objects charged under the R-R fields
whose tension, given by Eq. (1.51), is equal to their charge and it exactly matches the pbrane tension of Eq. (1.42). All these results lead naturally to the identification of D-branes
with the supergravity R-R p-branes presented in the previous section. This identification
provides a double description of the D-branes: The one that we developed here, based
on the open string point of view, which treats D-branes as topological defects; the other
instead coming from the closed string framework, in which the D-branes are described in
terms of the geometry they generate gravitationally. This twofold perspective will be the
crucial motivation behind the AdS/CFT correspondence, as we will see in Section 1.5.
1.4.1
D-brane effective action
The massless bosonic fields coming from the open string sector in presence of a Dp-brane are
a U (1) gauge field Aa and 9−p scalars φI . The scalars are the Goldstone bosons associated
to the breaking of the translational symmetries of the vacuum, caused by the Dp-brane.
8
Note however that the 1/gs scaling of the D-brane tension is not the typical one of solitons in field
theory, which is instead 1/g 2 .
18
1.4 Open string view: Dirichlet branes
For this reason the vacuum expectation values (VEVs) of these fields can be seen as the
coordinates of the brane in its transverse space. A non-trivial profile of these scalars, i.e. a
non-trivial dependence on the worldvolume coordinates of the brane φi (σ a ), represents the
fluctuations of the D-brane worldvolume in the target background. Therefore the effective
action for the massless open strings modes on the D-brane worldvolume corresponds to an
effective action for the D-brane, controlling the dynamics of its fluctuations as well as that
of the gauge field living on it.
There are two equivalent ways to derive this effective action. The first is to compute
scattering amplitudes in string theory and from these build up an effective action which
reproduces them. The second way is to couple the two-dimensional string worldsheet theory
to a general background of massless fields and then demand conformal invariance. The
equations obtained by requiring the vanishing of the one-loop beta functions, necessary
in order for the conformal invariance to be preserved at the quantum level, can be then
viewed as equations of motion for the spacetime fields, arising from a low energy effective
action [81].
Following one of these two procedures, one finds that in the low energy limit the effective
dynamics of a D-brane is described by the Dirac-Born-Infeld (DBI) action [22, 81, 82]
Z
q
IDBI = −TDp
dp+1 σe−φ − det(P [g + B]ab + 2π`2s Fab ) ,
(1.52)
Wp+1
where the integral is taken over the brane worldvolume Wp+1 and σ a are the worldvolume
coordinates, with a = 0, 1, . . . , p. Fab is the field strength of the U (1) gauge field Aa living
on the D-brane and P [· · · ] indicates the pull-back to the worldvolume.
In addiction to the DBI term (1.52), the effective action of open strings ending on
D-branes also has a Wess-Zumino (WZ) part which describes the coupling with the R-R
fields
Z
i
h
2
(1.53)
IWZ = TDp
P C (n) ∧ eB ∧ e2π`s F .
Wp+1
In the following we will often refer to DBI action to actually mean the sum of the DBI and
WZ terms.
The DBI action (1.52) is exact in `s and it is the lower order action in a derivative
expansion. Indeed it is valid in the regime of slowly varying fields.
For small value of the field F , which it also corresponds to small `s , the DBI term
reduces to the Maxwell action for the U (1) gauge field with coupling gU2 (1) ∼ gs .
Note that the effective action we wrote involves only bosonic fields. By including also
their fermionic superpartners, this can be generalized to a supersymmetric effective action
for the D-brane [83].
1.4.2
Super Yang-Mills theory from D-branes
One of the key features of D-branes is that they provide a simple and though beautiful
way to realize non-Abelian gauge theories [84].
Let us consider a stack of N Dp-branes, which, since they are BPS objects, can stay
statically at any distance from each other. For such a system it is natural to introduce
non-dynamical “quantum numbers”, m, n = 1, . . . , N , associated to each end of the open
strings. These quantum numbers, called Chan-Paton factors, are just labels that tell us
which brane the open string endpoints are attached to.
19
Chapter 1. D-Branes
Consider now the case in which the N Dp-branes coincide, i.e. they lie at the same
position on top of each other. Since each endpoint of an open string can lie on N different
branes, there are in total N 2 different “species” of open strings and thus the spectrum is
formed by N 2 copies of the states arising in presence of a single Dp-brane. Hence we can
arrange the fields coming from the open string spectrum to sit inside N × N Hermitian
matrices. Accordingly at the massless bosonic level we package the fields in the following
fashion
(Aa )m n ,
(φI )m n ,
with a = 0, . . . , p, I = p + 1, . . . , 9, and m, n = 1, . . . , N . These fields can be seen as
the bosonic part of the vector multiplet of a U (N ) Supersymmetric Yang-Mills (SYM)
theory in the worldvolume of the branes. Accordingly all the field transform in the adjoint
representation of the gauge group U (N ). The fact that the resulting gauge theory has
to be non-Abelian can be simply understood by considering the off-diagonal terms of the
gauge field (Aa )m n , which, being associated to massless, charged spin 1 particles, play the
role of W -bosons.
Thus the effective theory arising from a stack of N coincident D-branes is a U (N ) SYM
in p+1 dimensions. On the other hand, if all the D-branes are separated by a finite distance,
the effective theory at low-energies becomes the same SYM but now with a group U (1)N .
The mechanism driving the symmetry breaking from U (N ) to U (1)N exactly corresponds
to the spontaneous breaking of local symmetry in the low-energy effective action.
Let us examine in more details how this happens by looking at the action of a SYM
theory in p + 1 dimensions. In order to obtain this action we can perform a dimensional
reduction of the SYM action in ten dimensions, whose bosonic part is simply
Z
1
IYM10 = − 2
d10 x Tr Fµν F µν ,
Fµν = ∂µ Aν − ∂ν Aµ + i[Aµ , Aν ] ,
(1.54)
4 gYM10
where µ, ν = 0, . . . , 9. Now we split the ten coordinates into two sets, xµ = {σ a , xI }
(a = 0, . . . , p and I = p + 1, . . . , 9). The dimensional reduction is then obtained by
considering configurations that depend only on the first set. We thus have:
Fab = ∂a Ab − ∂b Aa + i[Aa , Ab ] ,
FaI = ∂a AI + i[Aa , AI ] ≡ Da AI ,
FIJ = i[AI , AJ ] .
The fields AI are now 9 − p scalars from the (p + 1)-dimensional point of view and hence we
rename them: AI ≡ φI . Finally the bosonic part of the action of SYM in p + 1 dimensions
becomes:
Z
1
1
1
1
p+1
ab
a I
2
IYMp+1 = − 2
d σ Tr
Fab F + Da φI D φ − [φI , φJ ] .
(1.55)
4
2
4
gYMp+1
The last term in (1.55) is the scalar potential: The vacua configurations which make this
term vanish are those with commuting φI . In this case all the scalar fields φI can be
simultaneously diagonalized, giving φI = diag{φI1 , · · · , φIN }.9 The diagonal component φIn
9
Note that the diagonalization of the φI is note unique: There is a residual gauge symmetry, the Weyl
group of U (N ), which permutes the diagonal entries. This symmetry is consequence of the fact that the
D-branes are identical, indistinguishable objects [84].
20
1.5 AdS/CFT correspondence
describes the position of the nth D-brane in its transverse space. Note that φIn have the
dimension of a mass, so, to be precise, the transverse coordinates of the branes (with the
right dimension of a length) are XnI = 2π`2s φIn . Pulling the D-branes apart is equivalent to
giving an expectation value to the scalars φIn . This corresponds to an Higgs mechanism in
the SYM effective theory, which breaks the gauge group from U (N ) to U (1)N . Thus the
W -bosons gain a mass which is proportional to the separation between the branes. In the
D-brane framework, in fact, the W -bosons are the lowest lying, now massive, modes arising
from the open strings connecting two separated D-branes. Therefore D-branes provide a
very beautiful heuristic picture for the spontaneous breaking of the gauge symmetry.
The action (1.55) can be related to the low energy limit, `s → 0, of the DBI action,
corresponding to small field strengths. This allows to express the SYM coupling constant
gYMp+1 in terms of string variables, simply by the comparison of the factors appearing
in front of the actions. Since the prefactor of the low energy limit of the DBI action is
(2π`2s )2 TDp it follows that the SYM coupling is
2
gYM
=
p+1
1
(2π`2s )2 TDp
= (2π)p−2 gs `p−3
s
(1.56)
where we use the expression of the Dp-brane tension given in Eq. (1.51).
1.5
AdS/CFT correspondence
One of the main reasons why D-branes are so important is certainly given by the central role
they play in the realization of dualities between gauge theories and gravity theories. We
shall now briefly present the most celebrated example of such gauge/gravity dualities, the
AdS/CFT correspondence [1–3]. The starting point of this correspondence conjectured by
Maldacena in 1997 is the double open/closed string picture through which one can describe
a stack of coincident D3-branes. This is nothing but the double picture valid for a generic
D-brane that emerges from what we have discussed so far. On the one hand, according
to what explained in Section 1.4, we have the open string description, in which D-branes
correspond to hyperplanes in a flat spacetime. The D-branes’ excitations are open strings
living on the branes, while closed strings propagate outside the branes. On the other hand,
in the description of Section 1.3, which is called the closed string description, D-branes
correspond to a spacetime geometry in which only closed strings propagate [18, 85].
The AdS/CFT correspondence can be motivated by considering the same decoupling
limit for a stack of a Nc coincident D3-branes in both descriptions. Let us start with
the open string point of view. The decoupling limit we consider is the low energy limit,
`s → 0, with gs and Nc fixed. In this limit the open string sector decouples from the closed
string sector, since the coupling constant regulating their interactions is the 10-dimensional
Newton constant G10 ∼ gs2 `8s which goes to zero in the limit. Also the interactions between
closed strings is governed by the Newton constant and thus they can be neglected. This
corresponds to the fact that gravity is infrared free. As a consequence of what we discussed
in Section 1.4.2 the dynamics of the open string sector is captured by the 4-dimensional
N = 4 SU (Nc ) SYM theory.10
10
According to Section 1.4.2 the gauge group realized in the worldvolume of Nc D-branes should be
U (Nc ) = SU (Nc ) × U (1): Here we discard the overall U (1) since it corresponds simply to the center of
mass of the set of D3-branes.
21
Chapter 1. D-Branes
We now have to take the same limit in the closed string description. In this case,
as argued in Section 1.3, the low energy regime is described by type IIB supergravity.
The supergravity solution of Nc coincident (extremal) D3-branes can be obtained from
Eqs. (1.38),(1.39),(1.41) and (1.43). In the string-frame it reads
ds2 = H −1/2 −dt2 + dx21 + dx22 + dx23 + H 1/2 dr2 + r2 dΩ25 ,
A0123 = H −1 − 1 ,
Φ = 0,
H =1+
R4
.
r4
(1.57)
R is the characteristic length scale of the gravitational effects of the Nc D3-branes and it
is given by
R4 = 4πgs Nc `4s .
(1.58)
The decoupling limit, `s → 0, has to be performed carefully since one has to take care
of keeping fixed the characteristic gauge theory quantities, such as a typical VEV of a
massless Higgs field. This VEV has, in analogy to what we showed in Section 1.4.2, a
nice realization in terms of D-branes. One can imagine to start with Nc + 1 coincident
D3-branes and then pull one of them apart at distance r from the others. This implies
that the gauge group SU (Nc + 1) is broken by an Higgs mechanism to SU (Nc ) × U (1).
The separated D3-brane can be regarded as a probe for the geometry of the Nc D3-branes.
We can relate the Higgs VEV u, which is the ground-state energy of the string stretched
between the two sets, to the radius r at which the separated brane sits. This gives the
relation between the scalar fields of N = 4 SYM and the transverse coordinates of the
branes, i.e.
r
u= 2.
(1.59)
`s
Hence u is a quantity to be held fixed and the right decoupling limit is
`s → 0
while keeping gs , Nc and u =
r
fixed.
`2s
(1.60)
Note that this limit corresponds to taking r → 0 and for this reason is usually called
near-horizon limit. Its effect is that one can neglect the ‘1’ on the harmonic function H,
so that H ' R4 /r4 and consequently the D3-brane metric becomes
ds2 =
R2 2
r2
2
2
2
2
2
2
−dt
+
dx
+
dx
+
dx
1
2
3 + 2 dr + R dΩ5 .
R2
r
(1.61)
The resulting geometry is that of AdS5 × S 5 , the product of a 5-dimensional Anti-de Sitter
(AdS) space and a 5-dimensional sphere. The curvature radius of both AdS5 and S 5 is R.
It is worth noting that in the opposite regime r R the spacetime is nearly flat (in
this case H ' 1). One can then build the following picture for the geometry sourced
by the stack of D3-branes: In the asymptotic region far from the branes the geometry is
well approximated by 10-dimensional Minkowski spacetime; on the other hand close to the
branes a throat geometry of the form of AdS5 × S 5 develops.
Observe furthermore that the limit r → 0 is a sensible way to restrict to the low
energy regime in supergravity. This is clear if we take into account the gravitational redshift which affects the energy of a signal originating at the
p radius r as measured by an
asymptotic observer: This energy is decreased by a factor gtt (r) = H −1/4 , corresponding
to the gravitational potential that the signal has to climb to reach the asymptotic region.
22
1.5 AdS/CFT correspondence
In the near horizon limit the red-shift factor blows up to infinity, and thus the throat
decouples from the asymptotic region.
To summarize, the decoupling limits in the open and closed string descriptions of a set
of Nc D3-branes correspond respcetively to:
1. N = 4 SYM theory with gauge group SU (Nc ) living on the branes worldvolume plus
free gravity away from the branes;
2. Type IIB string theory in AdS5 × S 5 plus free gravity in the asymptotic region.
Therefore we are naturally lead to conjecture the equivalence of the two descriptions:
Type IIB supestring theory on AdS5 × S 5 is equivalent to N = 4 SU (Nc )
Super Yang-Mills theory in four dimension.
This is the Maldacena conjecture for the AdS/CFT correspondence [1–3].
Both descriptions are characterized by two dimensionless parameters. On the gauge
theory side these are the rank of the gauge group Nc and the Yang-Mills coupling gYM ,
while on the string theory side the string coupling gs and the ratio R/`s . We can now
express the relation between these parameters. The relation between the string coupling
and the Yang-Mills coupling follows directly from Eq. (1.56) and is
2
gs = 2πgYM
.
(1.62)
From Eq. (1.58) we can find the gauge theory parameter corresponding to the curvature
radius of the AdS5 × S 5 geometry R:
R4
= 4πgs Nc = 2λ ,
`4s
(1.63)
where we have defined the ’t Hooft coupling λ as
2
λ = gYM
Nc .
(1.64)
Let us now discuss the limits of effective applicability of the two descriptions involved
in the correspondence. From the analysis of loop diagrams in the field theory one can
prove that the perturbative regime of the SYM theory can be trusted only when λ 1,
2 N . On the other hand supergravity gives
thus not just gYM has to be small, but also gYM
c
a sensible effective description of string theory only when the curvature of the geometry
is small, i.e. when R/`s is large. According to Eq. (1.63), this corresponds to large λ.
Thus we immediately see that the regime in which one theory can be studied in practice
corresponds to the regime in which the other theory is intractable, and vice versa. Hence,
since such a correspondence relates two theories in opposite coupling regimes, it is usually
called “weak/strong coupling duality”. This makes the correspondence hard to test but
at the same time useful and interesting, making it possible to obtain information on the
strongly coupled physics of the two theories.
It has to be specified that the condition λ 1 is not sufficient for the supergravity
limit to be a good approximation of the string theory: This condition must, in fact, be
supplemented by the requirement that gs 1, which in turn implies that Nc has to be
23
Chapter 1. D-Branes
large, since otherwise D-strings become light and render the gravity approximation invalid.
Equivalently this can be seen by considering the radius of curvature in Plank units
R4
R4
√
∼ Nc ,
=
`4p
G10
(1.65)
`p being the Planck length. So the classical limit in which the string loops are negligible
corresponds to Nc → ∞. Quantum correction on the string side are suppressed by powers
1/Nc .
A first simple check of the correspondence concerns the symmetries on the two sides.
N = 4 SYM is a conformal field theory (CFT). Its global bosonic symmetries are generated
by the conformal group SO(4, 2) and the R-symmetry group SU (4). On the dual string
theory side we can find exactly the same symmetries: Indeed the isometry group of AdS5
is SO(4, 2) and the isometry group of S 5 is SO(6), which has covering group SU (4).
Taking into account also the fermionic symmetries, on the gauge theory side the N = 4
supersymmetry implies the conservation of 16 supercharges, the supersymmetry preserved
by the stack of D3-branes. However these are enhanced to a total of 32 supersymmetries
because of the invariance under the superconformal group. These supersymmetries match
exactly the ones of the maximally supersymmetric AdS5 × S 5 background. Therefore the
gauge theory and the string theory have the same symmetries.
1.5.1
AdS/CFT at finite temperature
The discussion above relates type IIB string theory on AdS5 × S 5 to N = 4 SYM theory
at zero temperature. This can be naively argued recalling that we started our discussion
with a stack of Nc extremal D3-branes. We showed in Section 1.3.2 that extremal branes
have zero temperature. However the correspondence can be generalized to a nonzero
temperature T [6, 86]. In order to do this one has to consider the solution of type IIB
supergravity corresponding to Nc non-extremal D3-branes (see Section 1.3.1):
ds2 = H −1/2 −f dt2 + dx21 + dx22 + dx23 + H 1/2 f −1 dr2 + r2 dΩ25 ,
Φ = 0,
A0123 = coth α H −1 − 1 ,
(1.66)
H =1+
r04 sinh2 α
,
r4
f =1−
r04
,
r4
r04 cosh α sinh α = 4πgs `4s Nc .
Performing the near-horizon limit (1.60), keeping also u0 ≡ r0 /`2s fixed, one gets the nearextremal D3-brane geometry:
ds2 =
R2 2
r2
2
2
2
2
−f
dt
+
dx
+
dx
+
dx
+ 2 dr + R2 dΩ25 ,
1
2
3
R2
r f
where
f (r) = 1 −
r04
.
r4
(1.67)
(1.68)
The metric (1.67) is that of a product of a black hole in AdS5 with S 5 . The horizon is
located at r = r0 .
Thus the finite temperature generalization of the AdS/CFT duality relates N = 4
SU (Nc ) SYM to type IIB string theory on the AdS black hole background (1.67)-(1.68) [6].
24
1.6 BIon solution of DBI
The temperature T of the gauge theory is set by the global temperature of the AdS black
hole background which can be derived from Eq. (1.33)
T =
r0
√
π`2s
2λ
.
(1.69)
The entropy and the free energy on the supergavity side are
S=
π2 2
N V3 T 3 ,
2 c
F =
π2 2
N V3 T 4 ,
8 c
(1.70)
where V3 is the 3-dimensional spatial volume of the D3-branes. These can be compared to
the entropy and the free energy in large Nc N = 4 SYM at weak coupling λ 1:
S=
2π 2 2
N V3 T 3 ,
3 c
F =
π2 2
N V3 T 4 ,
6 c
(1.71)
We see that between the strong and weak coupling values of the entropy and the free
energy the only difference is an overall factor of 3/4 [86, 87]. Note that this factor is not a
discrepancy since the two computations are carried out in different regimes.
The Maldacena conjecture of the AdS/CFT duality has been one of the most striking
breakthrough in the theoretical physics of the last fifteen years. Here we only sketched its
derivation in connection with D-branes. For a more detailed discussion see for instance
[52, 88–92].
1.6
BIon solution of DBI
The remaining two sections of this chapter are essentially devoted to a deeper analysis of
the Dirac-Born-Infeld effective theory. Here we start by revisiting an interesting application
of the latter, which was first discussed in Refs. [23, 24].
In the linearized regime the DBI action reduces to Maxwell electrodynamics for the
gauge field while the scalars are free and massless. In this linear approximation there exist
worldvolume solutions describing Maxwell point charges with delta function sources. In the
bulk theory these are interpreted as F-strings ending on a point charge on the D-brane.
However, this picture is changed when one takes into account the non-linearities of the
DBI action. It has been found that the brane curves before one reaches the point charge.
The solution of DBI theory describing this is called a BIon, and its bulk interpretation is
that of F-strings dissolving into the D-brane [23, 24]. Furthermore, the BIon solution can
describe a configuration of two parallel D-branes (with one being an anti-brane) connected
by F-strings.
In this section we review the BIon solution which has an important role in this thesis
since in Chapter 3 we shall exploit exactly this solution to introduce a new method to
describe brane probes at finite temperature. In particular we consider only the D3-brane
BIon solution. However, all the results and considerations can readily be extended to
general Dp-branes.
1.6.1
DBI action and setup
Consider a D3-brane embedded in a 10-dimensional spacetime with the only background
flux turned on being the Ramond-Ramond five-form flux. We furthermore assume the
25
Chapter 1. D-Branes
background to have a constant dilaton. The DBI action for the D3-brane then takes the
form
Z
Z
p
4
2
d σ − det(γab + 2π`s Fab ) + TD3
ID3 = −TD3
P [C(4) ]
(1.72)
W4
W4
where the integrals are performed over the four-dimensional worldvolume W4 . Here we
have defined the induced worldvolume metric
γab = gµν ∂a X µ ∂b X ν
(1.73)
where gµν is the background metric, X µ (σ a ) is the embedding of the brane in the background with σ a being the worldvolume coordinates, a, b = 0, 1, 2, 3 are worldvolume indices and µ, ν = 0, 1, ..., 9 are target space indices. Fab is the field strength of the U (1)
gauge field Aa living on the D-brane, C(4) is the R-R four-form gauge field of the background and P [C(4) ] is its pull-back to the worldvolume. Finally, the D3-brane tension is
TD3 = [(2π)3 gs `4s ]−1 .
Embedding
To describe the BIon we specialize to an embedding of the D3-brane world volume in 10D
Minkowski space-time with metric
2
2
2
2
2
2
2
ds = −dt + dr + r (dθ + sin θdφ ) +
6
X
dx2i ,
(1.74)
i=1
without background fluxes. Choosing the worldvolume coordinates of the D3-brane as
{σ a , a = 0 . . . 3} and defining τ ≡ σ 0 , σ ≡ σ 1 , the embedding of the three-brane is given
by
t(σ a ) = τ , r(σ a ) = σ , x1 (σ a ) = z(σ) , θ(σ a ) = σ 2 , φ(σ a ) = σ 3
(1.75)
and the remaining coordinates xi=2,...,6 are constant. There is thus one non-trivial embedding function z(σ) that describes the bending of the brane. The induced metric on the
brane is then
2
γab dσ a dσ b = −dτ 2 + 1 + z 0 (σ) dσ 2 + σ 2 dθ2 + sin2 θdφ2 .
(1.76)
p
so that the spatial volume element is dV(3) = 1 + z 0 (σ)2 σ 2 dΩ(2) .
To get the appropriate F-string flux on the brane we turn on the worldvolume gauge field
strength component F01 . With this, the DBI action (1.72) gives the following Lagrangian
Z ∞
q
2
(1.77)
L = −4πTD3
dσσ 1 + z 0 (σ)2 − (2π`2s F01 )2
σ0
Note that we assumed F01 to depend only on σ since this is required for spherical symmetry.
Boundary conditions
We have two boundary conditions on the BIon solution. The first one is
z(σ) → 0 for σ → ∞ .
26
(1.78)
1.6 BIon solution of DBI
This condition ensures that far away from the center at r = 0 the D3-brane is flat and
infinitely extended with x1 = 0. Decreasing σ from ∞ the brane has a non-trivial profile
x1 = z(σ). In general we have a minimal sphere with radius σ0 in the configuration. For a
BIon geometry z(σ) is naturally a decreasing function of σ where at σ = σ0 the function
z(σ) reaches its maximum. Thus, σ takes values in the range from σ0 to ∞. At σ0 we
impose a Neumann boundary condition
z 0 (σ) → −∞ for σ → σ0 .
(1.79)
The rationale of this condition is that if z(σ0 ) < ∞ the brane system cannot end at
(r = σ0 , x1 = z(σ0 )) because of charge conservation and this boundary condition, as we
describe below, enables us to attach a mirror of the solution, reflected in the hyperplane
x1 = z(σ0 ). In line with this, we define
∆ ≡ 2z(σ0 )
(1.80)
as the separation distance between the brane and its mirror. Fig. 1.1 illustrates our setup
and the definitions of σ0 and ∆.
z
∆
2
σ0
σ
D3
Figure 1.1: Illustration of the setup, showing the embedding function z(σ) and the
definition of the parameters σ0 and ∆.
1.6.2
BIon solution
We now consider the Hamiltonian corresponding to the Lagrangian (1.77). To derive this
we need the canonical momentum density 4πσ 2 Π(σ) = δL/δ(∂τ A1 ) associated with the
worldvolume gauge field component A1 . Using F01 = ∂τ A1 − ∂σ A0 this gives
(2π`2s )2 F01
Π(σ) = TD3 q
1 + z 0 2 − (2π`2s F01 )2
(1.81)
so that the Hamiltonian can be easily constructed as
Z
Z
2
HDBI = 4π dσσ Π(σ)∂τ A1 (σ, τ ) − L = 4π dσ σ 2 Π(σ)F01 − ∂σ (σ 2 Π(σ))A0 (σ) − L
(1.82)
27
Chapter 1. D-Branes
Here we have in the second step integrated by parts the term proportional to ∂σ A0 , showing
that A0 can be considered as a Lagrange multiplier imposing the constraint ∂σ (σ 2 Π(σ)) = 0
on the canonical momentum. Solving this constraint gives
Π(σ) =
k
TD3 κ
= 2
2
4πσ
σ TF1
(1.83)
where k is an integer, TF1 = 1/2π`2s is the tension of a fundamental string and we have
kTF1
defined κ ≡ 4πT
= kπgs `2s . Using (1.83) in (1.82) the Hamiltonian becomes
D3
r
Z
p
κ2
2
HDBI = 4πTD3 dσ 1 + z 0 (σ)2 FDBI (σ) , FDBI (σ) ≡ σ 1 + 4
(1.84)
σ
The resulting EOM for z(σ), obtained by varying (1.84), is
!0
z 0 (σ)FDBI (σ)
p
=0
1 + z 0 (σ)2
Solving for z(σ) subject to the boundary conditions stated above, we obtain
p
− 12
σ04 + κ2
FDBI (σ)2
0
p
− z (σ) =
−
1
=
FDBI (σ0 )2
σ 4 − σ04
(1.85)
(1.86)
where we recall that σ0 is the minimum value of the two-sphere radius σ. The explicit
solution for z can be obtained by integrating the expression in (1.86)
p
Z ∞
σ 4 + κ2
0
(1.87)
z(σ) =
dσ p 0
σ 04 − σ04
σ
In particular for non-zero σ0 this represents a solution with a finite size throat, as illustrated
in Fig. 1.2.
Figure 1.2: Sketch of solution with finite size throat.
From the expression in (1.86) we can compute the energy density by evaluating the
integrand of (1.84), yielding
p
dH
σ 4 + κ2
= 4π TD3 (1 + z 0 (σ)2 ) (σ 4 + κ2 ) = 4π TD3 p
dσ
σ 4 − σ04
(1.88)
From this, dividing by the derivative of the solution z 0 (σ), we compute the energy density
along the brane
dH
1 dH
σ 4 + κ2
= 0
= 4π TD3 p 4
.
(1.89)
dz
z (σ) dσ
σ0 + κ2
which is finite for σ in the range [σ0 , ∞).
28
1.6 BIon solution of DBI
Spike solution
For σ0 = 0 the integral in Eq. (1.87) gives the Coulomb-charge type of solution
z(σ) =
κ
σ
(1.90)
i.e. the spike solution (see Fig. 1.3). In Ref. [23] it was shown that the energy corresponding
to this solution is the energy of a fundamental string of a given length. This was done
by comparing the integral appearing in (1.84) with the explicit form of the solution z(σ)
at a point near the end of the spike, in the linear approximation and with a suitable
regularization of the integral providing the energy. In the non-linear case, and also to
avoid divergences, it is more convenient to compute the energy density along the brane as
in (1.89). In particular setting σ = σ0 = 0 in (1.89) we find that the energy density at the
tip of the spike is given by
dH = 4π TD3 κ = kTF1
(1.91)
dz σ=σ0 =0
where we used κ defined below (1.83). We thus find that this is the tension of the fundamental string times the number of strings k, as expected.
Figure 1.3: Sketch of the spike configuration.
Wormhole solution
For σ0 = 0 we showed above that the solution (1.87) corresponds to a spike. However, as
explained in Sec. 1.6.1 for more general values of σ0 one can use the solution to construct
a configuration representing strings going between branes and anti-branes [23], to which
we refer as the wormhole configuration (see Fig. 1.4).
Figure 1.4: Attaching a mirror solution to construct a wormhole configuration.
Obviously, a system of a D-brane separated from an anti-D-brane is unstable since
the branes attract each other both gravitationally and electrically. However, the time
29
Chapter 1. D-Branes
scale of this is very large for small string coupling gs 1 since the tension of a D-brane
goes like 1/gs while the gravitational coupling goes like gs2 . Indeed, for a Dp-brane and
anti-Dp-brane system the time scale t is of order
9−p
t
1
∆ 2
∼√
(1.92)
`s
gs `s
where ∆ is the distance between the brane and anti-brane.
Turning back to the wormhole configuration, the separation (1.80) between the D3brane and anti-D3-brane can be computed from (1.87) and is given by
p
√
2 πΓ( 54 ) σ04 + κ2
(1.93)
∆ ≡ 2z(σ0 ) =
Γ( 34 )σ0
A plot of this quantity as a function of σ0 is given in Fig.1.5. It is clear that there is a
√
minimum value of the distance between the two branes, the minimum occurs at σ0 = κ
and its value is
√
√
2 2πΓ 45
κ
∆min =
(1.94)
3
Γ 4
Since κ is related to the worldvolume gauge field, we see that only for zero electric field
the two branes can annihilate. For large σ0 the distance ∆ between the two branes grows
D
20
15
10
5
0
1
2
3
4
5
Σ0
Figure 1.5: ∆ for κ = 1 in the Callan-Maldacena case
linearly with σ0 . We can now solve (1.93) for σ0 by keeping fixed the distance between
the branes ∆ and the number of strings, which is done by keeping the charge parameter κ
fixed. We obtain
√
∆2 ± ∆4 − 4a4 κ2
2
(1.95)
σ0 =
2a2
√
2 πΓ( 5 )
where the numerical constant a is given by a2 = Γ 3 4 . There are two solutions which,
(4)
for large ∆, behave as [23]
aκ
∆
σ0 '
,
σ0 ' .
(1.96)
∆
a
In the first case, the “thin throat” branch, the radius of the throat goes to zero as ∆ → ∞.
In the second case, the “thick throat” branch, the radius of the throat grows linearly with
∆ as ∆ → ∞.
30
1.7 DBI action at finite temperature
1.7
DBI action at finite temperature
In this section we examine the validity of the DBI action as an effective action to describe Dbranes probing thermal backgrounds. Before doing that it is convenient to write a general
expression for the EOMs of the DBI action, which takes the form of a set of extrinsic
embedding equations.
For the sake of simplicity, we shall refer explicitly only to the case of D3-brane, however
our considerations are easily generalized to Dp-branes.
1.7.1
Extrinsic embedding equations from DBI action
We start from the DBI action for the D3-brane given in (1.72). Before considering the
EOMs we first obtain the worldvolume EM tensor.11 This can be done by varying the
action (1.72) with respect to the worldvolume metric γab in (1.73), i.e.
2 δIDBI
T ab = √
γ δγab
(1.97)
We find
T
ab
TD3
=−
2
p
i
− det(γ + 2π`2s F ) h
2
−1 ab
2
−1 ba
((γ
+
2π`
F
)
)
+
((γ
+
2π`
F
)
)
√
s
s
γ
(1.98)
where we defined the determinant γ = − det(γab ).
We now consider the EOMs for the DBI action (1.72). These are found by variation
of the embedding map X µ (σ a ). We first notice that the Dirac-Born-Infeld term (the first
integral) in (1.72) only contributes through the variation of the worldvolume metric. We
compute
δγab = gµν,λ ∂a X µ ∂b X ν δX λ + gµλ (∂a X µ ∂b δX λ + ∂b X µ ∂a δX λ )
(1.99)
Hence we can write the variation of the Langrangian density of (1.72) as
1 √ ab
γT (gµν,λ ∂a X µ ∂b X ν δX λ + 2gµλ ∂a X µ ∂b δX λ )
2
TD3 abcd
+
∂a X µ ∂b X ν ∂c X ρ (4∂d δX λ Cµνρλ + ∂d X α Cµνρα,λ δX λ )
4!
(1.100)
giving the EOMs
1 √ ab
√
√
γT gµν,λ ∂a X µ ∂b X ν − ∂b ( γT ab )gµλ ∂a X µ − γT ab gµλ,ν ∂a X µ ∂b X ν
2
TD3 abcd
√
− γT ab gµλ ∂a ∂b X µ +
∂a X µ ∂b X ν ∂c X ρ ∂d X α (Cµνρα,λ − 4Cµνρλ,α ) =(1.101)
0
4!
We define now the projector hµν along the tangent directions to the D3-brane
hµν = γ ab ∂a X µ ∂b X ν
(1.102)
along with the projector ⊥µν along the orthogonal directions, defined as ⊥µν = g µν − hµν .
Using these we can define the extrinsic curvature tensor12 for the embedding
Kab ρ =⊥ρ λ (∂a ∂b X λ + Γλµν ∂a X µ ∂b X ν )
11
12
Energy-momentum tensor, as defined in the Introduction.
A more detailed discussion of this tensor will be given in Section 2.4 and in the Appendix A.
31
(1.103)
Chapter 1. D-Branes
We define furthermore the partial pull-back of the R-R five-form field strength F(5) =
dC(4)
Fλabcd = ∂a X µ ∂b X ν ∂c X ρ ∂d X α Fλµνρα
(1.104)
and the D3-brane R-R charge current
1
J abcd = TD3 √ abcd
γ
(1.105)
Projecting now the EOMs (1.101) with ⊥ρλ we can write the resulting EOMs as
T ab Kab ρ =
1 ρλ abcd
⊥ J
Fλabcd
4!
(1.106)
using the above definitions.13 This is the extrinsic equation for the D3-brane, with the EM
tensor given by (1.98). Generalizing to a Dp-brane the EOMs of the DBI action are
T ab Kab ρ =
1
⊥ρλ J a0 ···ap Fλa0 ···ap
(p + 1)!
(1.107)
with the EM tensor obtained from the variation of DBI action for the Dp-brane with
respect to the induced metric, according to Eq. (1.97). The fact that the EOMs of the DBI
can be recasted in the form (1.107) should not surprise, since this is exactly the equation
governing the extrinsic dynamics of any probe brane as derived by Carter in [34]. We
accordingly refer to Eq. (1.107) as the Carter equation. It is basically the Newton’s second
law of brane mechanics, with T ab replacing the mass and Kab ρ the acceleration of the point
particle, while the right hand side generalizes the Lorentz force for a charged particle.
1.7.2
Problems of the Euclidean DBI probe method
As reviewed in the Introduction, a number of papers in the literature (see [32,33,93,94] and
later works) have used the classical DBI action to probe finite temperature backgrounds in
string theory. In short, this method consists in Wick rotating both the background as well
as the classical DBI action, then finding solutions of the EOMs from the classical Euclidean
DBI action and finally identifying the radius of the thermal circle of the background with
the radius of the thermal circle in the Euclidean DBI action. From the classical solution
one can then evaluate physical quantities for the probe such as the energy, entropy and
free energy. We dub this method the “Euclidean DBI probe” method.
Let us discuss in more detail the Euclidean DBI probe method. As we said, we stick
to a D3-brane (thus in type IIB string theory) but all the considerations apply to any
Dp-brane. Consider a type IIB string theory background with metric gµν and with a R-R
five-form field strength F(5) turned on (for simplicity we do not consider other R-R field
strengths and we also assume a constant dilaton). As shown above, the EOMs for the
D3-brane DBI action (1.72) take the form
ab
TDBI
Kab ρ =
1 ρλ abcd
⊥ J
Fλabcd
4!
(1.108)
ab is given by (1.98) and J abcd by (1.105). We now perform a Wick rotation
where TDBI
on the background t = itE where t is the time coordinate of the background and tE the
13
As we will result clear in the following chapter, if we instead project the EOMs (1.101) with hρλ we
obtain the equation for conservation of the EM tensor Tab on the brane.
32
1.7 DBI action at finite temperature
corresponding direction in the Euclidean section of the background. Similarly we also
perform a Wick rotation for the worldvolume time τ = iτE . Then the EOMs (1.108)
become
1
ρ
(TE )ab
(⊥E )ρλ (JE )abcd (FE )λabcd
(1.109)
DBI (KE )ab =
4!
where the subscript E means that it is the Wick rotated quantity where the Wick rotation
in both the bulk and on the worldvolume are treated as simple linear transformations
00
on the tensors, e.g. (TE )00
DBI = −TDBI and so on. It is now easily shown that one also
obtains the equations (1.109) as the EOMs of the Euclidean DBI probe in the Wick rotated
background, i.e. by varying the Wick rotated DBI action in the Wick rotated background.
We can conclude from this that there is a one-to-one map between solutions of the EOMs
for a DBI probe in a thermal background and the solutions of the EOMs for a Euclidean
DBI probe in the Wick-rotated thermal background.
To solve the EOMs corresponds to solving certain differential equations under the
restriction of certain boundary conditions. The above equivalence between solving the
EOMs for a DBI probe in a thermal background and the EOMs for an Euclidean DBI
probe in the Wick-rotated thermal background only means that the differential equations
are the same, instead the boundary conditions are different. Thus, the equivalence works
only locally. Indeed there are global differences in being in the Wick-rotated frame or not
since one imposes different boundary conditions, in particular regarding the thermal circle
direction tE . In the Euclidean probe method one uses the Wick rotated version of the
usual static gauge tE = τE and one imposes that the size of the thermal circle is the same
for the probe as for the (Wick rotated) thermal background. Thanks to the static gauge
this boundary condition does not enter in the EOMs for the Euclidean probe and is in
that sense a global condition on the solution. Therefore, one could think that this global
condition is enough to ensure that the probe, using the Euclidean DBI probe method,
is in thermal equilibrium with the background. However, we shall now argue that this
global condition is not enough since the requirement of thermal equilibrium between the
probe and the background changes the EOMs of the probe by changing the EM tensor,
and hence the requirement of thermal equilibrium changes the probe not only globally but
also locally.
Consider the example of a D3-brane with zero worldvolume field strength Fab = 0 in
the AdS5 ×S 5 background. In the probe approximation this is described by the DBI action.
ab = −T γ ab . We notice that the EM tensor
The EOMs are of the form (1.108) with TDBI
D3
locally is Lorentz invariant. This conforms with the fact that the electromagnetic field on
the D3-brane is in the vacuum state.
We now turn on the temperature in the background. Thus, the background is either
hot AdS space, or an AdS black hole, depending on the temperature, times the S 5 . The
D3-brane should be in thermal equilibrium with the background. This means in particular
that the DOFs living on the D3-brane get thermally excited, acquiring the temperature
of the background. Among the DOFs are the ones described by the electromagnetic field
Fab living on the brane. The quantum excitations of Fab are described by the maximally
supersymmetric quantum electrodynamics locally on the brane (since one can see by expanding the DBI action that one locally has Maxwell electrodynamics for small Fab which
then is supplemented by the superpartners from the N = 4 supersymmetry). This means
that for small temperatures, near extremality, the EM tensor takes the form of the Lorentz
invariant piece plus the EM tensor corresponding to a gas of photons and their superpartners. Consider a particular point q on the brane. We can always transform the coordinates
33
Chapter 1. D-Branes
locally so that γab = ηab at that point. Then the EM tensor at q takes the form
(NE)
Tab = −TD3 ηab + Tab
(NE)
, T00
(NE)
= ρ , Tii
= p , i = 1, 2, 3
(1.110)
(NE)
where Tab
is the contribution due to the gas of photons and superpartners, having the
equation of state ρ = 3p = π 2 T 4 /2. Note that the power of T 4 follows from the fact that
N = 4 supersymmetric quantum electrodynamics is conformally invariant. Thus, we see
that the fact that we have local DOFs living on the D3-brane means that the EM tensor
is changed once we turn on the temperature. In terms of the EOMs for the probe we see
that they are given by (1.106) with the EM tensor (1.110) (one can easily find this EM
tensor for general worldvolume metric γab ). Therefore, the EOMs are clearly not the same
ab = −T γ ab . Indeed, the EM tensor (1.110) is no longer
as those of (1.108) where TDBI
D3
locally Lorentz invariant, which is in agreement with the fact that the brane has an excited
gas of photons and superpartners on it.
In conclusion, the above example clearly illustrates that the requirement of thermal
equilibrium affects the probe not only globally but also locally in that the EOMs change
from those given from the DBI action. This shows that the thermal D-brane probe is not
accurately described by the Euclidean DBI probe method.14
1.7.3
Towards a new method to describe thermal D-brane probes
Above we showed that the EOMs of the DBI action are equivalent to the Carter equation
with the EM tensor given by (1.98). The significance of rewriting the DBI EOMs as (1.106)
is that the generalization to a thermally excited D3-brane becomes more evident. In fact,
as we shall see in more details in Chapter 3, all we have to do is simply to replace the
EM tensor (1.98) in Eq. (1.106) with that of a thermally excited brane and then we have
the EOMs for a thermally excited D3-brane [36]. However, in order to do this we have to
shift to a regime in which the EM tensor of a thermally excited D3-brane is known: When
we have a large number N of coincident D3-branes and gs N is large.15 In this regime
the D3-branes backreact on the geometry of the background and we can compute the EM
tensor from the supergravity solutions of non-extremal black D3-branes. Doing this, we
are essentially employing the blackfold approach [35, 38–44].
So we pause for the moment our discussion on the thermal branes and present in the
following chapter the blackfold approach. We will then come back to the main subject
of the thesis in Capther 3, by examining in detail the new method we propose and in
particular applying it to the thermal generalization of the BIon solution considered in
Section 1.6.
Before doing this let us note that the new description of the thermal D-brane probe is
in accordance with the above example of D3-brane with zero worldvolume field strength
Fab = 0 in the AdS5 × S 5 background. Indeed, if we compute the EM tensor for a non14
Note that the difference between the thermal D-brane probe and the Euclidean DBI probe is not
due to backreaction. A backreaction would mean that Kab ρ should change. Instead, demanding thermal
equilibrium means that the EM tensor changes. Thus, even in the probe approximation the Euclidean DBI
probe is not accurate.
15
As in the AdS/CFT correspondence, the requirement that gs N should be large follows from demanding
that the curvature length scale of the supergravity solution of N coincident extremal D3-branes, which is
(N TD3 G)1/4 ∼ (gs N )1/4 `s , should be larger than the string length `s .
34
1.7 DBI action at finite temperature
extremal D3-brane16
T 00 =
π2 2 4
π2 2 4
TD3 r0 (5 + 4 sinh2 α) , T ii = −γ ii TD3
r0 (1 + 4 sinh2 α) , i = 1, 2, 3 (1.111)
2
2
and then we expand it for small temperatures we get
T 00 = N TD3 +
π2
3π 2 2 4
N T , T ii = γ ii (−N TD3 + N 2 T 4 ) , i = 1, 2, 3
8
8
(1.112)
We see that this is precisely of the form (1.110) (with an extra factor of N in the leading
part) with the leading part being locally Lorentz invariant and the correction corresponding
to a gas of gluons and their superpartners (with the usual factor of 3/4 since we are at strong
coupling [86]). This is thus in accordance with our general arguments above. Finally, we
note that whereas in the regime of validity of the DBI action the near-extremal correction
to the EM tensor (1.110) should be computed quantum mechanically, we can use classical
supergravity to compute the full EM tensor at finite temperature in the large N and large
gs N regime since the classical approximation is reliable in this regime.
16
The EM tensor can be read off from the non-extremal black brane solution (1.24) with p = 3, using,
for instance, the procedure outlined in [95].
35
Chapter 2
Blackfold approach
This chapter contains an introduction to the blackfold effective worldvolume theory [35,38–
44]1 which is a key building block of the new method to describe thermal brane probes that
will be presented in the following chapters. Indeed, as we will see, the blackfold approach
provides an efficient (approximate) way to describes the dynamics of black branes, i.e.
non-extremal branes possessing black hole horizon and thus having non-zero temperature
(see Section 1.3.1).
The blackfold approach was originally developed as an approximate tool to find new
black hole solutions in more than four spacetime dimensions. However, as we will see, its
applicability is much wider and more generally we can think of it as a framework that
allows to describe black branes in the probe approximation regime.
2.1
Blackfolds: Motivations and definition
In this section, following the “historical” derivation, we present the original motivations
that led to the development of the blackfold approach and we accordingly provide the
definition of blackfold.
One of the main novel features that appear in General Relativity in more than four
spacetime dimensions is given by the existence of black objects with extended horizons [96],
such as the black p-branes, which we introduced in Chapter 1. These new objects suggest
the possibility to have other black hole solutions without any four-dimensional counterpart.
Indeed one could expect that it should be possible to build new solution with finite horizon
area simply by bending a black brane into the shape of a compact hypersurface. For
instance one can obtain a black ring by bending a black string (black 1-brane) into the
shape of a circle and spinning it up to balance forces due to its tension that otherwise
would make it collapse.
Unfortunately, a consistent way to construct exact black hole solutions in higher dimensions is still missing. Nevertheless it is at least sensible to expect that smoothly bent
black brane, could be obtained as a perturbation of a straight one. This expectation is
supported by the success obtained in many areas of physics in the application of approximate methods to describe other extended objects. An illuminating example, as shown in
the previous chapter, is given by the D-branes: Although there are no exact methods to
treat the way in which D-branes bend and vibrate, when the scale of the deformations is
1
The contents of this chapter derive essentially from [35, 42, 43]. For a more detailed discussion we refer
the reader to the original papers cited in the text.
36
2.2 Effective theory for black hole motion
sufficiently large that we can locally regard the brane as being flat, the Dirac-Born-Infeld
worldvolume theory captures in an efficient manner the dynamics of these deformations.
With the same spirit one can look for an effective theory for describing black branes
whose worldvolume is not exactly flat, or not in stationary equilibrium, but where the
deviations from the flat stationary black brane occur on scales much longer than the brane
thickness. This is exactly the core of the blackfold approach. A blackfold is then defined as
a black brane whose worldvolume spans a curved submanifold of the background spacetime.
Therefore the application of blackfold method to black branes which bend over compact submanifolds allows to find many new horizon topologies. However, as we already
mentioned, its applicability is not limited to that. In fact, the approach is in general well
suited to describe the bending of black branes into geometries that are not necessarily
compact, which is the case we are interested in. In particular we will see that the method
is perfectly suited for the description of D-branes probing finite temperature background.
Notation:
Before starting with the discussion of the blackfold effective theory it is useful specifying
the notation we will use.
Consider a blackfold extending through p spatial dimensions embedded in a D-dimensional
background spacetime. We denote the (p + 1)-dimensional blackfold worldvolume as Wp+1
and its p-dimensional spatial section as Bp . It is convenient to define
n = D − p − 3,
(2.1)
so that the codimension of the blackfold worldvolume is n + 2.
Background (spacetime) and worldvolume quantities are denoted and distinguished as
follows:
Background spacetime
Worldvolume Wp+1
µ, ν . . . = 0, . . . , D − 1
a, b . . . = 0, . . . , p
Coordinates
Xµ
σa
Metric
gµν
Connection
Γσµν
γab (induced)
a
Covariant derivative
∇µ
Indices
bc
Da
Spacetime indices µ, ν, . . . are lowered and raised with gµν , worldvolume indices a, b, . . .
with γab . We moreover define g = − det gµν and γ = − det γab .
We use the same letter to indicate a background tensors tangent to the worldvolume,
tµ... ν... , and its pullback onto the worldvolume, ta... b... (the only exception is the first fundamental form hµν and the induced metric γab ).
2.2
Effective theory for black hole motion
In order to better understand the general relativistic aspects of the blackfold effective
theory for the dynamics of a black p-brane it is helpful considering first the simpler case of
37
Chapter 2. Blackfold approach
p = 0 (an “ordinary” black hole) [42]. So we shall start discussing the effective theory of a
black hole whose horizon radius r0 is much smaller than the background curvature radius
∼ R in which it moves
r0 R .
(2.2)
The existence of two such separated scales allows to identify two regions in the spacetime
in which a well approximated solution for the geometry can be very easily found. To be
definite we call r a radial coordinate centered in the black hole. The first region is that
surrounding the black hole, r R, which we will refer to as near-zone. Here at the leading
order in r/R the metric is given simply by the Schwarzchild-Tangherlini solution
ds2(near) = ds2(Schwarzschild) + O(r/R) .
(2.3)
The corrections to the Schwarzschild metric are the distortions that the background curvature creates on the black hole.
The second region in which we can have a simple description of the spacetime is the
far-zone, where r r0 . In practice we are far enough from the black hole that its effect
on the background geometry is very mild and can be treated as a small perturbation, thus
ds2(far) = ds2(background) + O(r0 /r) .
(2.4)
At these distances one looses sight of the black hole size and so we are led to formulate
an effective long-distance theory in which the black hole is replaced by a point-like source.
In practice we integrate-out the short-wavelength DOFs obtaining in this way an effective
worldline theory of a point particle. The trajectory of this particle is described by X µ (τ )
(τ denotes its proper time) and its velocity is Ẋ µ = ∂τ X µ with gµν Ẋ µ Ẋ ν = −1. In
order for the effective theory to be sensible we need to impose the matching condition
that the point-like particle should be such as to reproduce the gravitational effects of the
black hole on the far-zone background geometry. This means that we have to assign to
it a suitable effective EM tensor. It is natural to approximate this tensor by neglecting
the acceleration and other higher derivatives of the particle’s velocity. These are in fact
related to the deviations away from flatness of the background in the region where the
black hole moves and thus they are suppressed by powers of r0 /R. The EM tensor of
the effective source to the lowest order in the r0 /R expansion is fixed by symmetry and
worldline reparametrization invariance to have the form
T µν = m Ẋ µ Ẋ ν .
(2.5)
Note that the near and far zones overlap in r0 r R, which we dub overlap region.
The existence of this intermediate region is fundamental since the matching condition can
be practically imposed only there. This is because in this region both the descriptions
are valid and the matching condition simply consists in the request that the respective
geometries match. In this region, the near-zone Schwarzschild solution can be linearized
around Minkowski spacetime and on the other hand, the background curvature of the farzone geometry can be neglected, so the far field is the linear perturbation of Minkowski
spacetime sourced by (2.5). These two fields are the same if m is equal to the ADM mass
of the Schwarzschild solution with horizon radius r0 , that is
m=
(D − 2)ΩD−2 D−3
r0
.
16πG
38
(2.6)
2.3 Effective worldvolume theory for a black brane
A consistent formulation that allows to compute corrections to any perturbative order
is given by the method of matched asymptotic expansions [97]. Actually the procedure
outlined before, in which the black hole is treated as a “test particle” that does not affect
the background geometry, constitutes only the first step in this method. According to the
matched asymptotic expansion method one can go beyond the test particle approximation
by solving the Einstein equations perturbatively in the far-zone taking into account the
backreaction from the particle with suitable asymptotic conditions. This solution evaluated
in the overlap region provides the asymptotic condition for near-zone geometry. Then
one perturb linearly the near-zone Schwarzschild metric (2.3) looking for a solution that
preserves the regularity of the horizon and that matches the far-zone solution in the overlap
region. This process can be iterated to any order and the corrections one finds involve
higher derivatives of the velocity and of r0 .
This matching construction allows to write a subset of the Einstein equations as equations for r0 (τ ) and X µ (τ ). Their derivation however is in practice very complicated. Nevertheless, the leading order equations can be obtained in a much more simpler manner
using symmetry and conservation principles. In this case the equations are dictated by
general covariance, which imposes that
∇µ T µν = 0 .
(2.7)
This equation guarantees the consistency of the coupling between short- and long-wavelength
DOFs. The symbol ∇µ in (2.7) denotes the covariant derivative projected along the effective particle trajectory,
∇µ = −Ẋµ Ẋ ν ∇ν ,
(2.8)
since the covariant derivative along an arbitrary direction does not make sense. Nevertheless Eq. (2.7) has components both orthogonal and parallel to the particle’s worldline,
which are respectively
(gρν + Ẋρ Ẋν )∇µ T µν = 0
Ẋν ∇µ T
µν
=0
⇒ maµ = 0 ,
⇒ ∂τ m(τ ) = 0 ,
(2.9)
(2.10)
with aµ = Dτ Ẋ µ = Ẋ ν ∇ν Ẋ µ being the effective particle’s acceleration. The meaning
of these two equations can be understood straightforwardly. Eq. (2.9) is the geodesic
equation that describes the trajectory of a test particle. Eq. (2.10) tells us that m must
remain constant along the trajectory.
Actually in this first example of the black hole effective motion the leading order equations are not so interesting: Geodesic motion is in fact the usual motion for any probe
particle in the background. In order to find non-trivial effects one has to take into account
higher order corrections in the matched asymptotic expansion. Conversely, as we will see
in the following sections, the effective blackfold theory for black p-branes with p > 0 leads
to non-trivial results also at the leading order. In fact, in our analysis of the blackfold
approach and in its subsequent applications, we will always remain at the leading order
approximation where the black brane is a “test brane” in a background spacetime.
2.3
Effective worldvolume theory for a black brane
Our aim is to extend the effective worldline theory of black holes to a worldvolume theory
that describes the collective dynamics of a black p-brane.
39
Chapter 2. Blackfold approach
2.3.1
Collective coordinates
As we already said, we are looking for an effective theory that describes slightly curved
black branes in which deviation from flatness occurs on length scale R much larger than the
brane thickness r0 . Like in the example considered in the previous section, we again have
two separated scales. Accordingly the DOFs of General Relativity can be split into longwavelength components, living in the far-zone r r0 , and short-wavelength components,
living in the near-zone r R. Schematically we have
(long) (short)
gµν = {gµν
, gµν
}.
(2.11)
In order to obtain a long-wavelength effective theory we have to integrate-out the shortwavelength gravitational DOFs. By doing this the Einstein-Hilbert action2 becomes
Z
Z
q
1
1
D √
D
(long)
IEH =
d x gR ≈
d x g (long) R(long) + Ieff [gµν
, φ] .
(2.12)
16πG
16πG
(long)
Ieff [gµν , φ] is an effective action which, through a set of collective coordinates that we
denote schematically by φ, take into account the coupling between the short- and longwavelength components of the gravitational field.
In order to identify these effective field variables we consider a flat static black p-brane
in D = 3 + p + n spacetime dimensions, whose geometry is
ds2p−brane
p
X
r0n
dr2
2
= − 1 − n dt +
(dz i )2 +
+ r2 dΩ2n+1 .
r0n
r
1
−
i=1
rn
(2.13)
The (p + 1)-dimensional worldvolume Wp+1 of the brane is spanned by the coordinates
σ a = (t, z i ), with a = 0, . . . , p. A more general form of the metric can be obtained by
performing a boost with velocity field ua such that ua ub ηab = −1 along the worldvolume:
r0n
dr2
2
dsp−brane = ηab + n ua ub dσ a dσ b +
+ r2 dΩ2n+1 .
(2.14)
rn
r
1 − r0n
The parameters that determines the black brane configuration are the horizon thickness r0 ,
the D − p − 1 coordinates identifying the position of the brane in directions transverse to
the worldvolume, which we denote collectively by X ⊥ , and the p independent components
of the velocity u (say, its spatial components ui ). Constant shifts in these parameters still
give solutions of the Einstein equations. So, in total, one has D zero-modes that yield D
collective coordinates of the black brane. Since we want to describe curved black branes
we allow the collective coordinates to depend on the worldvolume coordinates
φ(σ a ) = {X ⊥ (σ a ), r0 (σ a ), ui (σ a )} .
(2.15)
The long-wavelength effective theory requires ∂X ⊥ , ln r0 and ui to vary slowly on the
worldvolume Wp+1 , over a large length scale R r0 . Typically R is set by the smallest
intrinsic or extrinsic curvature radius of Wp+1 .
Instead of using only the transverse coordinates X ⊥ it is convenient to enlarge the
embedding coordinates of Wp+1 to include all the spacetime coordinates X µ (σ a ). In this
2
For simplicity we only consider DOFs obeying vacuum gravity dynamics, Rµν = 0.
40
2.3 Effective worldvolume theory for a black brane
way, paying the price of introducing some gauge redundancy, one manages to preserve
manifest diffeomorphism invariance.
In the far-zone the long-wavelength metric describes the background geometry in which
the thin brane lives. The metric induced on the brane worldvolume is
(long)
γab = gµν
∂a X µ ∂b X ν .
(2.16)
On the opposite regime, in the strict limit where R → ∞, the near-zone solution is (2.14),
but when R is large but finite, the collective coordinates depend on σ. The requirement
that the near- and far-zone metrics matches in the overlap region r0 r R implies that
the near-zone metric for the black brane must be of the form
r0n (σ)
dr2
µ
2
a
b
ds(near) = γab (X (σ)) +
u
(σ)u
(σ)
dσ
dσ
+
+ r2 dΩ2n+1 + . . . (2.17)
a
b
r0n (σ)
rn
1 − rn
where the dots indicate additional terms, of order O(r0 /R), required for this to be a solution
to Einstein’s equations.
2.3.2
Effective energy-momentum tensor
The EM tensor plays the same role of the mass m of the black hole in the example of
Section 2.2. The procedure of “integrating out the short-distance dynamics” actually means
that one has to solve the Einstein equations in the near-zone and then has to cook up a
EM tensor that encodes all the effects of the black brane on the far-zone geometry. This
tensor has to depend only on the collective coordinates.
The effective action (2.12) yields the following equations of motion
1
(long)
(long)
eff
Rµν
− R(long) gµν
= 8πGTµν
,
2
(2.18)
eff is the effective worldvolume EM tensor
where Tµν
eff
Tµν
δIeff =√
µν
g(long) δg(long)
2
.
(2.19)
Wp+1
It turns out that this effective EM tensor should be given by the Brown and York
quasilocal EM tensor [98]. This can be computed by considering a timelike hypersurface
lying away from the black brane which encloses it by extending along the worldvolume
directions and the angular directions Ω(n+1) , i.e. the hypersurface acts as a boundary.
The angular directions are irrelevant to the lowest order we are considering and this allows
to simplify the discussion by focusing only on the worldvolume directions of the boundary
(see [35]). Assuming the boundary metric along worldvolume directions to be γab , the
quasilocal EM tensor is
2 δIcl
(ql)
Tab = √
,
(2.20)
γ δγ ab
where Icl is the classical on-shell action of the solution. For our purposes the action Icl is
the one in which the short-distance gravitational DOFs are integrated-out and so it is the
same function of the collective variables as Ieff . Exploiting the relation (2.16) we can thus
identify (2.19) with (2.20). Henceforth to simplify notation we drop the superscripts from
them. We also drop the superscript (long) from the background metric gµν .
41
Chapter 2. Blackfold approach
In the overlap region r0 r R, where the gravitational field is weak, the quasilocal
EM tensor T ab matches, to leading order in r0 /R, the ADM EM tensor. For the boosted
black p-brane (2.14) the result of the calculation is
Ω(n+1) n a b
T ab =
r0 nu u − η ab .
(2.21)
16πG
Allowing for a small variation of the collective coordinates the EM tensor becomes
Ω(n+1) n
T ab (σ) =
r0 (σ) nua (σ)ub (σ) − γ ab (σ) + . . .
(2.22)
16πG
where the dots indicate terms containing gradients of ln r0 , ua , and γab , which are taken
to be small and are neglected to the order we are working in.
2.3.3
Fluid perspective
The long-wavelength effective theory for any kind of brane is described by a fluid living on
the worldvolume. The equation capturing its dynamic takes the form of the conservation
equations Da T ab = 0 and the the EM tensor T ab , to lowest derivative order, is that of an
isotropic perfect fluid,
T ab = (ε + P )ua ub + P γ ab ,
(2.23)
with energy density ε, pressure P and velocity ua satisfying ua ub γab = −1.
Fluid dynamics provides an effective long-wavelength description of fluctuations around
equilibrium configurations, whose macroscopic description is given by thermodynamics. So
in general there will be an equation of state, P (ε), and the system will obey locally the
laws of thermodynamics
dε = T ds
(2.24)
and Euler-Gibbs-Duhem relation
ε+P =Ts
(2.25)
where T is the local temperature and s the entropy density of the fluid in its rest frame.
The fluid may also carry additional conserved charges.
It is straightforward to see that the EM tensor (2.22) is exactly of the isotropic perfect
fluid form. By comparing (2.22) and (2.23) we obtain that for a black brane the effective
fluid has energy and pressure given by
Ω(n+1)
1
ε=
(n + 1)r0n ,
P =−
ε.
(2.26)
16πG
n+1
In the rest frame of the fluid the entropy can be computed by means of the BekensteinHawking identification with the horizon area AH = 4 G S
Ω(n+1) n+1
r
,
(2.27)
s=
4G 0
and the temperature by means of the identification with the surface gravity κg = T /2π
n
.
(2.28)
4πr0
One can easily check that Eqs. (2.26)-(2.28) satisfy the correct thermodynamic relations
(2.24) and (2.25).
Going beyond the perfect fluid approximation (2.23), the EM tensor will acquire dissipative terms proportional to gradients of ln r0 , ua , γ ab , which, as we already said, we
neglect and are, at any rate, absent for stationary configurations.
T =
42
2.4 Blackfold dynamics
2.4
Blackfold dynamics
The general effective theory of classical brane dynamics can be formulated as a theory of a
fluid on a dynamical worldvolume. The effective variables must satisfy “intrinsic” equations
for the fluid and “extrinsic” equations for the dynamics of the worldvolume geometry.
We shall start by introducing a few notions about the geometry of worldvolume embeddings. More details and proofs can be found in the Appendix A.
2.4.1
Embedding and worldvolume geometry
The worldvolume Wp+1 of the black brane is embedded in a background with metric gµν
and its induced metric γab is given by Eq. (2.16). The background spacetime indices µ, ν
are raised and lowered with gµν , while the worldvolume indices a, b with γab . The first
fundamental form of the submanifold
hµν = ∂a X µ ∂b X ν γ ab
(2.29)
acts as a projector onto Wp+1 . The projector ⊥µν along directions orthogonal to Wp+1 is
then defined as
⊥µν = gµν − hµν .
(2.30)
The pull-back map ∂a X µ allows to convert the background tensors tµ... ν... with support
on Wp+1 into worldvolume tensors ta... b... and viceversa. A relevant example is given by
the EM tensor
T µν = ∂a X µ ∂b X ν T ab .
(2.31)
The covariant differentiation ∇µ along an arbitrary direction of the background spacetime is not well defined; only its projection along tangential directions, which we denote
by an overbar
∇µ = hµ ν ∇ν ,
(2.32)
is meaningful. As we noted in Section 2.2 in the case of the black hole (p = 0), in general
∇ρ tµ... ν... has both orthogonal and tangential components.
The tangentially projected part of the divergence of the EM tensor satisfies (see (A.19))
hρ ν ∇µ T µν = ∂b X ρ Da T ab .
(2.33)
The extrinsic curvature tensor is defined as
Kµν ρ = hµ σ ∇ν hσ ρ ,
(2.34)
and t is tangent to Wp+1 along its (symmetric) lower indices µ, ν, and orthogonal to Wp+1
along ρ. Its trace gives the mean curvature vector
K ρ = hµν Kµν ρ = ∇µ hµρ .
(2.35)
Explicit expressions for the extrinsic curvature tensor in terms of the embedding functions
X µ (σ a ) can be found in the Appendix A.
43
Chapter 2. Blackfold approach
2.4.2
Blackfold equations
Carter in [34] showed that the general extrinsic dynamics of any classical brane can be
expressed in terms of a EM tensor with support on the worldvolume Wp+1 and satisfying
the tangentiality condition
⊥ρ µ T µν = 0 .
(2.36)
As a consequence of the underlying conservative dynamics of General Relativity (even if the
macroscopic effective dynamics may be dissipative) and of the spacetime diffeomorphism
invariance, the EM tensor must satisfy the conservation equations
∇µ T µρ = 0 .
(2.37)
These provide the EOMs for the entire set of worldvolume collective coordinates φ(σ a ).
In order to separate the intrinsic and extrinsic contributions we decompose (2.37) along
directions parallel and orthogonal to Wp+1 as
∇µ T µρ = ∇µ (T µν hν ρ ) = T µν ∇µ hν ρ + hν ρ ∇µ T µν
= T µν hν σ ∇µ hσ ρ + hν ρ ∇µ T µν
=T
µν
ρ
ρ
Kµν + ∂b X Da T
(2.38)
ab
where in the last line we exploited (2.33) and (2.34). In this way we split the D equations
(2.37) into D −p−1 equations in directions orthogonal to Wp+1 and p+1 equations parallel
to Wp+1 ,
T µν Kµν ρ = 0
Da T
ab
=0
(extrinsic equations) ,
(2.39)
(intrinsic equations) .
(2.40)
For the specific EM tensor of a neutral black brane (2.26), the extrinsic equations
(2.39), using (A.14), become
K ρ = n⊥ρ µ u̇µ ,
(2.41)
and the intrinsic equation (2.40)
u̇a +
1
ua Db ub = ∂a ln r0 ,
n+1
(2.42)
with u̇ = uν ∇ν u and u̇b = uc Dc ub . Thus the temporal and spatial worldvolume gradients
of r0 determine the worldvolume acceleration and expansion of u, respectively.
Eqs. (2.41) and (2.42) are the blackfold equations which describe the general collective
dynamics of a neutral black brane.
It is interesting to observe that the extrinsic equations (2.39), when written explicitly
in terms of the embedding X µ (σ a ) become
T ab Da ∂b X ρ + Γρµν ∂a X µ ∂b X ν = 0
(2.43)
(see eqs. (A.26) and (A.28)). These can be regarded as generalizations to p-branes of the
geodesic equation for free particles, or more simply, of “mass×acceleration= 0”.
One of the characteristic features of the branes that can be described through the
blackfold approach is that they possess black hole horizons. In the effective theory we
44
2.5 Stationary blackfolds
integrate over distances which are of the order of the brane thickness and then we loose track
of the horizon. However the effective theory is still reminiscent of the presence of the horizon
as proved by the existence of an entropy and by the thermodynamic equilibrium of the
effective fluid. An important request is therefore that the long-wavelength perturbations
preserve the regularity of the horizon. We shall assume that this requirement is satisfied
when the blackfold equations are satisfied. The correctness of this assumption has been
proved explicitly in [99]. In the same paper, it was shown also that the blackfold equations
(2.41) and (2.42) can be derived directly from the Einstein equations.
The set of field variables that describes the effective long-wavelength theory is given by
the collective coordinates (the intrinsic and the extrinsic coordinates of the black brane)
and the background gravitational field gµν . The equations controlling the dynamics of such
variables are the intrinsic equations (2.40), the extrinsic equations (2.39) and backreaction
equations (2.18). Since these equations follow from general symmetry and conservation
principles, they retain their form at any perturbative order. Of course what changes at
each order is the form of the EM tensor, as well as that of the background metric.
In principle the blackfold effective theory can be carried out at any perturbative order
in r0 /R through the method of the matched asymptotic expansion [97] outlined (for the
simpler case of a p = 0 black hole) in Section 2.2.3 However for our purposes it is enough
considering it to the lowest order. At this level, as in the black hole example, the only
equations one has to solve are the intrinsic and the extrinsic ones, since the backreaction of
the brane on the background geometry is neglected. This means that using the blackfold
effective theory at the zeroth order one is in the probe approximation regime, i.e. the
branes are treated as “test” (slightly bent) black branes, that do not affect the spacetime
metric gµν .
2.5
Stationary blackfolds
Stationary blackfolds deserve particular attention for two reason: First they correspond
to stationary black holes and furthermore for such configurations it is possible to solve
explicitly the intrinsic blackfold equations for the thickness r0 and velocity u. Hence one
is left only with the extrinsic equations for the worldvolume embedding X µ (σ) which, as
we will shortly show, can be derived from an action principle.
2.5.1
Solution to the intrinsic equations
For stationary fluid configurations the dissipative effects must be absent. One can prove
that this requires the velocity field to be proportional to a worldvolume Killing field l = la ∂a
[100]:
u = l/|l|
(2.44)
where
|l| =
p
−γab la lb
(2.45)
and l satisfies the worldvolume Killing equation
D(a lb) = 0 .
3
(2.46)
In the blackfold context this was considered in [38] and applied to the specific case of higher-dimensional
black rings.
45
Chapter 2. Blackfold approach
We assume the existence of a timelike Killing vector lµ ∂µ in the background
∇(µ lν) = 0 ,
(2.47)
whose pull-back onto the worldvolume Wp+1 gives exactly the worldvolume Killing field
l, i.e. la = ∂a X µ lµ . The existence of a timelike Killing vector field is in fact a necessary
assumption if we intend to describe stationary black holes.
The Killing equation (2.47) contracted with lµ lν implies lµ ∂µ |l| = 0. From this it
follows that
u̇µ = ∂ µ ln |l| ,
(2.48)
which together with the intrinsic blackfold equation (2.42) yields
so that
∂a ln |l| = ∂a ln r0
(2.49)
r0
= constant .
|l|
(2.50)
We shall now see how to fix the proportionality constant.
The background Killing vector l is a timelike vector with norm l2 = −|l|2 . We have to
keep in mind that, according to our discussion, the background metric used to compute
this norm is that of the far-zone, r r0 . When r R the geometry is different and it
is well described by the near-zone metric (2.17). As we said the near- and far-zone metric
should match in the overlap region r0 r R. Thus it is natural to extend l as a Killing
vector in the whole geometry. In the near-zone the norm of l computed with the metric
(short)
gµν
(2.17) turns out to be
rn
rn
(short) µ ν
(2.51)
gµν
l l = γab + 0n ua ub la lb = − 1 − 0n |l|2 ,
r
r
where we used (2.44) and (2.45). Since l goes to zero while approaching the horizon,
r → r0 , it is the null Killing generator of the horizon. The surface gravity κg is
κg =
n|l|
.
2r0
(2.52)
and then, because of (2.50), it is a constant over the worldvolume of the blackfold.
Eqs. (2.44) and (2.52) provide the general solution to the intrinsic equations for stationary blackfolds. Let us decompose l as a linear combination of orthogonal commuting
vectors of the background geometry, ξ timelike and χi spacelike, namely
X
l=ξ+
Ωi χi ,
(2.53)
i
with constant Ωi and i running at most up to p. We define the worldvolume functions
Ra (σ) as the norms of ξ and χi on the worldvolume
q p
R0 = −ξ 2 ,
Ri = χ2i .
(2.54)
Wp+1
Wp+1
The Ra must be regarded as part of the embedding coordinates X µ (σ). It is convenient to
choose ξ as the generator of asymptotic time translations, and χi as the Cartan generators
46
2.5 Stationary blackfolds
of asymptotic rotations with closed orbits of periodicity 2π. With this choice R0 is a
redshift factor between infinity and the blackfold worldvolume and Ri are the proper radii
of the orbits generated by χi along the worldvolume. The Ωi are the horizon angular
velocities relative to observers that follow orbits of ξ.
From ξ and χi we can define the corresponding set of orthonormal vectors (with respect
to the worldvolume metric γab ) as
∂
1
ξ,
=
∂t
R0
∂
1
=
χi
i
∂z
Ri
(no sum in i).
(2.55)
As we did for l, we can extend their definition to the whole geometry, including the nearzone (2.17). The vector l can be expressed in terms of these vectors as
!
X
∂
∂
l = R0
+
Vi i ,
(2.56)
∂t
∂z
i
where Vi is the worldvolume spatial velocity field
Vi (σ) =
Ωi Ri (σ)
u · ∂z i
.
=
−u · ∂t
R0 (σ)
(2.57)
Accordingly the norm of l is
s
p
X
|l| = −ξ 2 −
Ω2i χ2i = R0 1 − V 2 ,
(2.58)
i
where
V2 =
X
i
Vi2 =
1 X 2 2
Ω R .
R02 i i i
(2.59)
it follows that |l| can be interpreted as the relativistic Lorentz factor at a point in Wp+1 ,
with a local redshift, all relative to the reference frame of ξ-static observers. Plugging
(2.58) into (2.52) and solving for r0 we get
r0 (σ) =
nR0 (σ) p
1 − V 2 (σ) .
2κg
(2.60)
This relation gives the thickness r0 as a function of the embedding coordinates (through
the Ra ) for given values of κg and Ωi .
It is worth noting that actually one can avoid considering the short-wavelength geometry of the horizon, and remain exclusively in the framework of the effective theory. Indeed
exploiting only the fluid and thermodynamics equations one can derive that the variation
of the local temperature T (2.28) along the worldvolume is dictated by the local redshift
T =
T
n
√
=
.
4πr0
R0 1 − V 2
(2.61)
The integration constant T has a natural interpretation as the global temperature of the
black hole. We will see that this is indeed confirmed by the discussion of Section 2.5.3.
Therefore, using the usual relation between the temperature and the surface gravity, T =
κg /2π, we recover exactly Eq. (2.60).
47
Chapter 2. Blackfold approach
2.5.2
Horizon geometry, mass and angular momenta
From the metric (2.17) one sees that in the blackfold approach a “small” transverse sphere
sn+1 with Schwarzschild radius r0 (σ) sits at every point in the spatial section Bp of Wp+1 .
Accordingly the blackfold is a black hole with horizon geometry that is the product of Bp
and sn+1 : The product is warped since the radius of the sn+1 varies along Bp . As we saw,
the null generators of the horizon are proportional to the velocity field u. If r0 is non-zero
everywhere on Bp then the sn+1 are trivially fibered on Bp and the horizon topology is
given by the product of the topology of Bp with sn+1 .4
The horizon geometry can be studied by considering the metric (2.17) that locally
describes the geometry of the blackfold to lowest order in r0 /R in the region r R.
There we can choose a local orthonormal frame (∂t , ∂z i ) on the worldvolume, such that ∂t
coincides in the overlap zone r0 r R with the timelike unit normal nµ to Bp ,
nµ = (∂t )µ .
(2.62)
Since to lowest order in r0 /R the worldvolume metric is flat we can write for the spatial
metric on the horizon at r = r0
ds2H = (δij + ui uj ) dz i dz j + r02 dΩ2(n+1)
(2.63)
with ui = u·∂z i . Then it is straightforward to compute the local area density of the horizon
aH
p
aH = Ω(n+1) r0n+1 1 + δij ui uj .
(2.64)
For a stationary blackfold we can use the choice for ∂t and ∂z i we made in (2.55), so that
1 µ
ξ ,
R0
(2.65)
ΩR
√i i
.
R0 1 − V 2
(2.66)
nµ =
and
ui =
Hence the area density is
Ω(n+1) r0n+1
aH = √
= Ω(n+1)
1−V2
n
2κg
n+1
n
R0n+1 (σ a )(1 − V 2 (σ a )) 2 ,
(2.67)
and the total area of the horizon is
Z
AH =
dV(p) aH (σ a ) ,
(2.68)
Bp
where dV(p) is the volume form in Bp .
Note that the same result can be obtained computing the total entropy of the blackfold,
which can be deduced from the entropy current sa = sua ,
Z
Z
Z
R0
s
S=−
dV(p) sa =
dV(p) s =
dV(p) √
.
(2.69)
|l|
1−V2
Bp
Bp
Bp
4
It can be proved that if Bp has boundaries then r0 will shrink to zero size at them [35], resulting in a
non-trivial fibration and different topology.
48
2.5 Stationary blackfolds
Indeed using the entropy density of the blackfold fluid given in (2.27) we get
Ω(n+1)
S=
4G
n
2κg
n+1 Z
n
Bp
dV(p) R0n+1 (σ a )(1 − V 2 (σ a )) 2 =
AH
,
4G
(2.70)
in perfect agreement with the geometric area computed from (2.68). Thus again, one can
avoid any reference to the near-geometry and work exclusively with quantities defined in the
effective fluid theory. This shows that the blackfold construction gives precise information
about the entire horizon geometry, including the size of the sn+1 .
The mass and angular momenta are conjugate to the generators of asymptotic time
translations and rotations, which we assume are the vectors ξ and χi that we introduced
in Section 2.5. Then
Z
Z
µ ν
M=
dV(p) Tµν nµ χνi .
(2.71)
dV(p) Tµν n ξ ,
Ji = −
Bp
Bp
Plugging here (2.65) and the results from Section 2.5 we obtain
Z
Ω(n+1)
n−2
n n
M=
dV(p) R0n+1 (1 − V 2 ) 2 n + 1 − V 2 ,
16πG 2κg
Bp
and
2.5.3
Ω(n+1)
Ji =
16πG
n
2κg
n
Z
nΩi
Bp
dV(p) R0n−1 (1 − V 2 )
n−2
2
Ri2 .
(2.72)
(2.73)
Action principle for stationary blackfolds
In Section 2.5 we showed that for a stationary blackfold a general solution of the intrinsic equations (2.42) can always be found: Indeed, given a timelike Killing vector field l,
Eqs. (2.44), (2.58) and (2.60) allow to express r0 (σ) and ua (σ) in terms of the embedding
functions X µ (σ). Thus, for stationary configurations, we are left only with the extrinsic
equation (2.41) that determines the embedding of the blackfold. Exploting Eq. (2.48), this
becomes
K ρ =⊥ρµ ∂µ ln |l|n .
(2.74)
Since r0 /|l| is constant, using (2.26), we can rewrite it as
K ρ =⊥ρµ ∂µ ln r0n =⊥ρµ ∂µ ln(−P ) .
(2.75)
Using the result (A.38) from the Appendix A, this can be seen as the EOM derived from
the action
Z
√
µ
˜
I[X (σ)] =
dp+1 σ γ P
(2.76)
Wp+1
Xµ
by considering variations of
in directions transverse to the worldvolume.
The action (2.76) can be put in a simpler form. Writing the proper time t on the
worldvolume as the asymptotic time, conjugate to ξ, times the redshift factor of R0 , and
performing the trivial integration over the asymptotic time we get
Z
Z
p+1 √
d σ γ P = ∆t
dV(p) R0 P ,
(2.77)
Wp+1
Bp
49
Chapter 2. Blackfold approach
where we denoted with ∆t the finite length interval of integration.
Contracting the EM tensor (2.23) with na lb , then using (2.25) and (2.61), we find
!
X
a
b
b
Tab n ξ +
Ωi χi + T sua na = na la P = −R0 P ,
(2.78)
i
so, integrating over Bp and using (2.69), (2.72) and (2.73),
!
X
I˜ = −∆t M − T S −
Ωi Ji .
(2.79)
i
We can rotate to Euclidean time, it → τ , with periodicity ∆τ = β = 1/T , and find that
the Euclidean action, iI˜ → −I˜E is
X
IE = β(M − T S −
Ωi Ji ) ,
(2.80)
i
i.e. it is related to the thermodynamic grand-canonical potential W at constant T and Ωi
through the relation IE = βW [T, Ωi ].
The identity (2.79) holds for any embedding, not necessarily a solution to the extrinsic
equations. Thus, if we regard M , Ji and S as functionals of X µ (σ), and consider variations
at fixed surface gravity and angular velocities, we have
X
δ I˜ = 0 ⇔ dM − T dS −
Ωi dJi = 0 .
(2.81)
i
Hence, the extrinsic equations are equivalent to the requirement that the first law of black
hole thermodynamics is satisfied.
In the Euclidean quantum gravity approach to black hole thermodynamics one can
interpret (2.80) as the effective action that approximates, in the blackfold regime r0 /R 1,
the gravitational Euclidean action of the black hole [101].
2.6
Charged blackfolds
Up to now we have only considered the simplest case of neutral blackfolds. We now extend
the approach to black branes with charges [43]. This extension is particularly interesting
for us also in view of the applications described in the following chapters, since it allows
to consider the branes relevant for string theory, such as D-branes (charged under the R-R
fields), F-strings and NS5-branes (charged under the NS-NS Kalb-Ramond field).
2.6.1
Perfect fluids with conserved p-brane charge
We start by considering a perfect fluid that lives in a (p + 1)-dimensional worldvolume
Wp+1 and carries a p-brane current
J = Qp V̂(p+1) ,
(2.82)
where V̂(p+1) is the volume form on Wp+1 . It is natural to assume that this current is
conserved since it sources a gauge field in the target background spacetime of the brane.
Hence the charge must be constant along the worldvolume,
∂a Qp = 0 .
50
(2.83)
2.6 Charged blackfolds
This means that there are no modes in the worldvolume that describe local fluctuations
of this charge and thus Qp is not a collective variable of the fluid. Then the collective
coordinates of the charged blackfold are the same as those for the neutral case: The
intrinsic variables are the local fluid velocity u and the energy density ε and the extrinsic
variables are formed by the embedding coordinates X µ (σ a ), which determine the induced
metric γab = gµν ∂a X µ ∂b X ν . The presence of Qp manifests itself in the equation of state
of the fluid, where it enters as a parameter.
The perfect fluid is characterized by its isotropic EM tensor (2.23) and the equation of
state, which will be specified later. Locally it satisfies the thermodynamic relations (2.24)
and (2.25).
The general analysis is very similar to that of perfect neutral fluids: The intrinsic fluid
equations are again Da T ab = 0. They can be decomposed into components parallel (timelike) and transverse (spacelike) to ua . The former gives the energy continuity equation,
which for a perfect fluid, and using (2.25), is equivalent to the conservation of entropy
Da (sua ) = 0 .
(2.84)
The (spacelike) Euler force equations relate the fluid acceleration along the worldvolume,
u̇a = ub Db ua , to the pressure gradient and, through the thermodynamic identity dP =
sdT , to the temperature gradient so that
(γ ab + ua ub )(u̇b + ∂b ln T ) = 0 .
(2.85)
The extrinsic equations have to be slightly generalized with respect to that of the
neutral blackfolds (2.39), in order to take into account the possible coupling between the
black brane and the background fields (if present). Thus the extrinsic dynamics is captured
by Carter’s equations [34]
Kµν ρ T µν =
1
⊥ρ σ Jµ0 ...µp F µ0 ...µp σ ,
(p + 1)!
(2.86)
where the right-hand side represents exactly a force from the coupling of the current J to
a background (p + 2)-form field strength F (p+2) . However, since in the applications that
we will consider this coupling it is not present, for the sake of simplicity, we set this term
to zero. Then, for the perfect fluid EM tensor (2.23) these equations become
− P K ρ =⊥ρ
2.6.2
µ sT
u̇µ .
(2.87)
Stationary solutions and action principles
We now take into account the stationary configurations. Many of the considerations made
for the neutral case are still valid now. In particular we take again the fluid velocity
aligned with a Killing vector l along the worldvolume, u = l/|l|, and we assume that
l = lµ ∂µ generates isometries not only of the worldvolume but also of the background
spacetime. Then the relation (2.48) still holds. This determines completely the solution to
the intrinsic fluid equations, since the local temperature is obtained by simply redshifting
the global temperature T , which is uniform on the worldvolume,
T (σ a ) =
51
T
.
|l|
(2.88)
Chapter 2. Blackfold approach
Since now sT u̇µ = −s∂µ T = −∂µ P , the extrinsic equation (2.87) is
K ρ =⊥ρµ ∂µ ln(−P ) .
(2.89)
We thus recover also for the charged blackfolds the extrinsic Eq. (2.76). We can therefore
proceed exactly in the same way as in Section 2.5.3 and obtain the correct action for the
extrinsic equation, which is obviously again given by Eq. (2.79),
I˜ = −∆t(M −
X
Ωi Ji − T S) .
(2.90)
i
The Euclidean action I˜E is
I˜E = β(M −
X
Ωi Ji − T S) ,
(2.91)
i
with β = 1/T , i.e. it is again the thermodynamic potential at constant T and Ω, but now
with the further request that Qp should be fixed.
Since T and Ω are integration constants, the extrinsic equations (2.89) are equivalent
to requiring that
X
dM = T dS +
Ωi dJi
(fixed Qp ) .
(2.92)
i
Namely we extremize the action for variations among configurations with the same value
of T , Ω, and the charge Qp .
Even if Qp is a global quantity for the fluid on the brane and there is no local, intrinsic
fluid degree of freedom on the worldvolume associated to this charge, it is one of the
conserved total charges of the configuration, along with M and J. Thus one can consider
variations among stationary fluid brane configurations with different charges. Hence there
(p)
must exist a global potential ΦH conjugate to Qp , which can be identified by observing
that the local equation of state that relates ε to s does also depend on Qp as a parameter
that specifies the entire fluid, ε(s; Qp ). This allows to introduce on the fluid worldvolume
a local potential Φp (σ a ) as
∂ε(s; Qp )
Φp (σ a ) =
.
(2.93)
∂Qp
Since Qp couples to the field strength F (p+2) of the background, Φp actually corresponds
to the potential of F (p+2) on the worldvolume. We introduce the global potential
(p)
ΦH
Z
=
dV(p) R0 Φp (σ a )
(2.94)
Bp
by integrating over spatial directions of the worldvolume taking into account the local
redshift R0 relative to infinity.
We can now reformulate the variational principle for stationary solutions using this
potential. Locally, we introduce the Gibbs free energy density
G = ε − T s − Φp Qp = −P − Φp Qp ,
52
(2.95)
2.6 Charged blackfolds
which, for hydrodynamic fluctuations for which Qp is necessarily constant along the worldvolume, satisfies5
dG = −sdT − Qp dΦp = −dP − Qp dΦp .
(2.96)
Thus we can rewrite the extrinsic equations (2.89) as
(G + Φp Qp )K ρ =⊥ρµ (∂µ G + Qp ∂µ Φp ).
(2.97)
We now consider variations of the worldvolume embedding where, instead of Qp , we
(p)
keep constant the potential ΦH in (2.94). this implies that the local potential Φp (σ a ) must
vary in such a way that
⊥ρµ ∂µ Φp (σ a ) = Φp (σ a )K ρ
(2.98)
(see Eq.(A.38)). Therefore the extrinsic equations (2.97) become
K ρ =⊥ρµ ∂µ ln G
(2.99)
and they can derived by extremizing the action
Z
√
I=−
dp+1 σ γ G
(2.100)
Wp+1
for variations of the embedding among stationary fluid configurations where now T , Ω
(p)
and ΦH are kept fixed. By integrating (2.95) and going to Euclidean time we recover the
expected Legendre-transform-type relations
X
(p)
(p)
IE [T, Ωi , ΦH ] = β(M −
Ωi Ji − T S − ΦH Qp )
i
(2.101)
(p)
˜
= IE [T, Ωi , Qp ] − βΦH Qp ,
(p)
(p)
and βΦH = ∂ I˜E /∂Qp which, consistently, justifies the way we have defined ΦH in (2.94)
in terms of worldvolume quantities. The extrinsic equations are now equivalent to the
complete form of the global first law,
(p)
dM = T dS + ΩdJ + ΦH dQp .
(2.102)
5
While T is a true independent local variable of the fluid, which characterizes local fluctuations in the
energy density of the fluid, Φp (σ a ) is not. In fact, Φp (σ a ) measures the response of the fluid to an external
source that globally changes the fluid’s Qp , which however remains constant over the worldvolume. So this
“local grand-canonical ensemble” is a phony one since G is only a function of T and not of Φp , which is
determined by the condition that Qp remains constant.
53
Chapter 3
Heating up the BIon
The blackfold approach was the last missing block that we needed for our purpose. Thus
we are now ready to resume the discussion interrupted at the end of the Chapter 1 and
develop our new method to describe branes probing finite temperature background.
In this chapter we do this by considering, as a test case, the thermal generalization
of the BIon solution [36, 37]. In Section 1.6 we found the latter as a solution of the DBI
action for a D-brane probing ten-dimensional flat space-time, with an electric flux in the
worldvolume, interpreted in the bulk as an F-string. Here we take into account the same
system when the temperature is turned on. As explained in Section 1.7 the DBI action does
not provide an accurate description of a D-brane probe at finite temperature. We showed
that the DBI EOMs are equivalent to the Carter equation (1.107) with the EM tensor
being that obtained from the variation of the DBI action with respect to the worldvolume
metric (1.98). The crucial point is that in this EM tensor the thermal DOFs are completely
neglected (see Section 1.7.2) and thus this makes the DBI inaccurate. This motivates our
proposal for a new method to treat such thermal probes.
Since the Carter equation (1.107) gives the general EOMs for any brane probe all
we need to do in order to obtain the EOMs for a thermal D-brane probe is to find the
corresponding EM tensor. According to what we saw in the previous chapter we can use the
blackfold approach to get an effective description of a stack of N coincident non-extremal
(i.e. at finite temperature) D-branes in terms of a supergravity solution in the bulk when
gs N 1. Using this supergravity description one can determine the EM tensor for the Dbrane in the regime of large N . Then, in this regime, the EOMs for a non-extremal D-brane
probe can be simply obtained by plugging this EM tensor in the Carter equation (1.107). In
this way we are actually replacing the DBI action, which furnishes a good description of a
single D-brane probing a zero-temperature background, by a new approach that describes
N coincident non-extremal D-branes, with N large, probing a thermal background such
that the probe is in thermal equilibrium with the background.
In the application of our new method the first thing that we need is therefore the EM
tensor of the appropriate supergravity solution. For the BIon, this we attack in Section
3.1. Since the BIon solution includes an F-string charge on the brane we find the EM
tensor for the non-extremal D3-F1 brane bound state.
In Section 3.2 we consider the BIon solution in the background of hot flat space. We
first write down the EOMs (1.107) explicitly for the setup for the BIon solution introduced
in Section 1.6. This results in a single equation for the profile of the solution. Based on the
blackfold approach, we propose an action from which the EOMs can be derived. For our
setup this gives the same equation as before, but written in a form that makes it readily
54
3.1 Energy-momentum tensor for black D3-F1 brane bound state
solvable. Using this we find the general solution for the D3-F1 blackfold configuration that
generalizes the BIon solution of the DBI action to non-zero temperature.
In Section 3.3 we examine more closely the brane-antibrane wormhole solution at finite
temperature. This solution consists of N D3-branes and N anti-D3-branes, separated by
a distance ∆, connected by an F-string charged wormhole with minimal sphere radius
σ0 . We consider ∆ as a function of σ0 when σ0 is varied for a given temperature and
compare in this way our configuration at non-zero temperature with the corresponding
extremal configuration. At zero temperature, the wormhole solution is characterized by a
thin branch with small σ0 and a thick branch with large σ0 for fixed ∆. We discover that,
when the temperature is turned on, the separation distance ∆ between the brane-antibrane
system develops a local maximum in the region corresponding to the zero temperature thin
branch. This is a new feature compared to the zero temperature case. The existence of
this maximum gives rise to three possible phases with different σ0 for a given ∆. For small
temperatures and/or large σ0 the ∆ as a function of σ0 resembles increasingly closely the
zero temperature counterpart.
The last two sections are devoted to the analysis of the thermodynamics of our newly
found solution. In Section 3.4 we study the physics of the three different phases found in
Section 3.3 in the brane-antibrane wormhole configuration by comparing the free energy
in the appropriate thermodynamical ensemble to see which phase is the dominant one. In
Section 3.5 we examine whether it is possible to construct a thermal generalization of the
infinite spike solution.
3.1
Energy-momentum tensor for black D3-F1
brane bound state
In this section we consider the EM tensor of D3-branes. The EM tensor (1.98), derived
from the DBI action in Section 1.7.1, corresponds to a single D3-brane at zero temperature
with a gauge field strength Fab turned on. We work in the regime in which γab and Fab vary
so slowly over the brane that their derivatives can be ignored. The aim of this section is to
find the EM tensor of a D3-brane still with the gauge field strength turned on, but now at
non-zero temperature and with a large number N 1 of parallel stacked D3-branes and
with gs N large.
3.1.1
More on DBI case
Before turning to the non-zero temperature case we take another look at the zero temperature EM tensor (1.98). Consider the special case in which the worldvolume metric γab is
diagonal, γ00 = −1, and the only non-zero component of the two-form gauge field strength
is F01 . Then the EM tensor (1.98) becomes
T 00 = √
p
TD3
TD3
, T 11 = −γ 11 √
, T ii = −γ ii TD3 1 − E 2 , i = 2, 3 ,
1 − E2
1 − E2
(3.1)
where we defined
2π`2
E ≡ √ s F01 .
γ11
55
(3.2)
Chapter 3. Heating up the BIon
In the bulk, turning on the electric flux F01 on the brane corresponds to having a number
of F-strings along the σ 1 direction of the brane. Call this number of F-strings k. From the
worldvolume point of view we can say that we have k units of electric flux. We can relate
k to F01 as
p
Z
Z
2
TD3
E
√
2
3 ∂ − det(γ + 2π`s F )
dσ dσ
k = −TD3
=
dσ 2 dσ 3 γ22 γ33 √
. (3.3)
∂F
T
1 − E2
01
F1 V23
V23
If we consider the case
√ in which γab and F01 are constant the above equation can be written
as kTF1 = V⊥ TD3 E/ 1 − E 2 where V⊥ is the area in the σ 2,3 directions perpendicular to
the F-strings. Using this we see that the mass density of the brane can be written as
s
2 2
2 + k TF1 .
T00 = TD3
(3.4)
V⊥2
We recognize this as the 1/2 BPS mass density formula for the D3-F1 brane bound state
in the case of a single D3-brane and k F-strings.
3.1.2
Energy-momentum tensor for D3-F1 bound state from black brane
geometry
We now turn to obtaining the EM tensor for N D3-branes with an electric field on, corresponding to k units of electric flux, at non-zero temperature. We can obtain this from a
black D3-F1 brane bound state geometry assuming that we are in the regime of large N
and gs N .
The D3-F1 black brane bound state background has the string frame metric [102, 103]
1
1
1
1
1
1
ds2 = D− 2 H − 2 (−f dt2 + dx21 ) + D 2 H − 2 (dx22 + dx23 ) + D− 2 H 2 (f −1 dr2 + r2 dΩ25 ) , (3.5)
where
r04
r04 sinh2 α
,
H
=
1
+
, D−1 = cos2 ζ + sin2 ζH −1 ,
(3.6)
r4
r4
and with dilaton field φ, Kalb-Ramond field B(2) , and two- and four-form Ramond-Ramond
gauge fields C(2) and C(4) given by
f =1−
e2φ = D−1 , B01 = sin ζ(H −1 − 1) coth α
(3.7)
C23 = tan ζ(H −1 D − 1) , C0123 = cos ζD(H −1 − 1) coth α .
We now proceed to read off the EM tensor, the D3-brane and F-string currents, and the
thermodynamical parameters as seen by an asymptotic observer. However, we first need
to consider how the non-trivial worldvolume metric γab can enter in this. Reading off the
EM tensor from
find it in the (t, xi ) coordinates for which the worldvolume metric
P(3.5)i we
2
2
is just −dt + i (dx ) . Instead we want the EM tensor with a worldvolume metric of the
form
γab dσ a dσ b = −dσ02 + γ11 dσ12 + γ22 dσ22 + γ33 dσ32 ,
(3.8)
since this is the most general form needed for our computations in this chapter, i.e. a
diagonal metric without red-shift factor. To transform the resulting EM tensor in the (t, xi )
coordinates to the above worldvolume coordinates σ a we can simply make the rescaling
56
3.2 Thermal D3-brane configuration with electric flux ending in throat
√
t = σ 0 , xi = γii σ i , i = 1, 2, 3. One could infer here that such a rescaling is problematic
since in general the worldvolume metric γab varies according to where we are situated on the
brane. However, to the order we are working in we are precisely suppressing the derivative
of the metric in the EM tensor, just as they are suppressed in the DBI EM tensor (1.98).
With this in mind we read off the EM tensor in the σ a worldvolume coordinates
π2 2 4
π2 2 4
T 00 =
TD3 r0 (5 + 4 sinh2 α) , T 11 = −γ 11 TD3
r0 (1 + 4 sinh2 α) ,
2
2
π2 2 4
π2 2 4
TD3 r0 (1 + 4 cos2 ζ sinh2 α) , T 33 = −γ 33 TD3
r0 (1 + 4 cos2 ζ sinh2 α) ,
2
2
(3.9)
using for example [95]. The D3-brane current is
T 22 = −γ 22
J 0123 =
2
2π 2 TD3
cos ζr04 cosh α sinh α .
√
γ
(3.10)
The number of D3-branes in the bound state is N . Thus, using (3.10) we find
N
.
2π 2 TD3
Furthermore, imposing that we have k F-strings gives
Z
k
TD3
√
=
dσ 2 dσ 3 γ22 γ33 tan ζ .
N
TF1 V23
cos ζr04 cosh α sinh α =
(3.11)
(3.12)
We also give the thermodynamic quantities associated to the horizon, that will be used in
what follows. The temperature T and entropy density s are
1
2 5
, s = 2π 3 TD3
r0 cosh α ,
(3.13)
T =
πr0 cosh α
while the local D3-brane and F-string chemical potentials are
(local)
µD3
(local)
= tanh α cos ζ , µF1
= tanh α sin ζ .
(3.14)
Extremal limit
As a check we consider here the extremal limit of the EM tensor of the black D3-F1 brane
bound state. We see from (3.11) that the extremal limit is to take r0 → 0 keeping ζ and
r04 cosh α sinh α fixed. This gives T = 0 as it should and the EM tensor
T 00 =
N TD3
N TD3
, T 11 = −γ 11
, T ii = −γ ii N TD3 cos ζ , i = 2, 3 ,
cos ζ
cos ζ
(3.15)
along with the formula (3.12) for k. We see that the above formulas match (3.1) and (3.3)
provided we identify E = sin ζ and put N = 1.
3.2
Thermal D3-brane configuration with electric
flux ending in throat
In this section we study the D3-F1 configuration at finite temperature in hot flat space.
We derive the EOMs for the embedding directly and supplement this with an action-based
derivation. We then proceed by solving the equations and comment on the relation to the
DBI solution and Hamiltonian. Finally, we discuss the regime of validity of the solution.
57
Chapter 3. Heating up the BIon
3.2.1
D3-F1 extrinsic blackfold equation
Our setup is specified by exactly the same type of embedding and boundary conditions as
discussed in Section 1.6.1 for the extremal case. We briefly recall it. The background is
the (hot) 10D Minkowski spacetime with metric
ds2 = −dt2 + dr2 + r2 (dθ2 + sin2 θdφ2 ) +
6
X
dx2i .
(3.16)
i=1
Denoting the worldvolume coordinates of the D3-brane as {σ a , a = 0 . . . 3} and defining
τ ≡ σ 0 , σ ≡ σ 1 , the embedding of the three-brane is given by
t(σ a ) = τ , r(σ a ) = σ , x1 (σ a ) = z(σ) , θ(σ a ) = σ 2 , φ(σ a ) = σ 3
(3.17)
and the remaining coordinates xi=2,...,6 are constant. Thus the only non-trivial embedding
function is z(σ) that describes the bending of the brane. The induced metric on the brane
is then
2
γab dσ a dσ b = −dτ 2 + 1 + z 0 (σ) dσ 2 + σ 2 dθ2 + sin2 θdφ2 .
(3.18)
p
so that the spatial volume element is dV(3) = 1 + z 0 (σ)2 σ 2 dΩ(2) .
In order to get the BIon solution we have to impose two boundary conditions. The
first is
z(σ) → 0 for σ → ∞ ,
(3.19)
i.e. far from the center at r = 0 the D3-brane is flat and infinitely extended with x1 = 0.
Decreasing σ from ∞ the brane has a non-trivial profile x1 = z(σ). In general we have a
minimal sphere with radius σ0 in the configuration. For a BIon geometry z(σ) is naturally
a decreasing function of σ where at σ = σ0 the function z(σ) reaches its maximum. Thus,
σ takes values in the range from σ0 to ∞. At σ0 we impose a Neumann boundary condition
z 0 (σ) → −∞ for σ → σ0 .
(3.20)
An illustration of the setup is given in Figure 1.1.
With the results of the previous section, we are ready to compute the D3-F1 extrinsic
blackfold equation (1.106), where the right-hand side is zero since our background is 10D
Minkowski spacetime so there is no five-form field strength. For the left-hand side we
need to compute the extrinsic curvature tensor (1.103) for the embedding described in
(3.16)-(3.18). The resulting non-vanishing components are given by
z 00 (σ)
σz 0 (σ)
σ sin2 θz 0 (σ)
x1
x1
,
K
=
,
K
=
, Kiir = −z 0 (σ)Kiix1 ,
22
33
1 + z 0 (σ)2
1 + z 0 (σ)2
1 + z 0 (σ)2
(3.21)
x1
r
with i = 1, 2, 3. Since Kii and Kii are proportional, the EOMs (1.106) for this case
becomes simply T ab Kab x1 = 0. Here Tab is the EM tensor (3.9) of the black D3-F1 brane
system, where the collective coordinates r0 , α, ζ are now promoted to be functions of σ.
We then find the following EOM for the D3-F1 blackfold
K11 x1 =
z 00
2 1 + 4 cos2 ζ sinh2 α
=
−
.
σ
1 + 4 sinh2 α
z 0 (1 + z 0 2 )
(3.22)
The precise form of the functions ζ(σ) and α(σ) entering this equation (as well as r0 (σ))
follow from a number of constraints on the solution, as we will now discuss.
58
3.2 Thermal D3-brane configuration with electric flux ending in throat
Constraints on solution
Beyond the EOM (3.22) we also have further constraints that follow from charge conservation, of both D3-brane and F-string charge, and constancy of the temperature. From
the point of view of the general blackfold construction, as discussed in Section 2.5.1, the
constancy of the temperature and angular velocity is a consequence of stationarity. Since
in our case the blackfold is static only the temperature is relevant. Furthermore, charge
conservation can be seen to follow from additional EOMs [43].
We start by considering the F-string charge, given in eq. (3.12), which should be conserved along the σ direction. Inserting the induced metric (3.18), this gives that
κ = σ 2 tan ζ .
(3.23)
kTF1
4πN TD3
(3.24)
where the constant κ is defined as1
κ≡
in terms of the conserved charges N and k. Note that we can also write (3.23) as
1
cos ζ = q
1+
κ2
σ4
.
(3.25)
We also need to ensure the conservation of F-string chemical potential [43] (the quantity
R √
(local)
(local)
defined as dσ γ11 µF1
with µF1
the local F-string chemical potential given in (3.14))
in the worldvolume directions transverse to σ. This is automatic since we impose spherical
symmetry for the two-spheres parameterized by the coordinates θ, φ transverse to the Fstring direction σ on the worldvolume of the D3-brane.
The equation (3.25) ensures F-string charge conservation by conserving the number of
F-strings k assuming the conservation of the number of D3-branes N . The latter is imposed
using the formula given in Eq. (3.11), while constancy of the temperature is imposed using
Eq. (3.13). Thus, from Eqs. (3.11), (3.13) and (3.25) we can eliminate r0 (σ) and ζ(σ)
yielding the constraint
r
sinh α
π2 N T 4
κ2
=
1
+
.
(3.26)
2 TD3
σ4
cosh3 α
Below we solve this equation explicitly for cosh α.
Summarizing, we note that for given (T, N, k) the σ-dependence of the two functions
ζ(σ) and α(σ) is determined by the two equations (3.25) and (3.26). Given the solution of
α(σ), one can then compute the thickness r0 (σ) from (3.13).
Action derivation
It is also instructive and useful to consider an action derivation of the EOMs (3.22). In
fact, as we shall see later, the action derived below helps in finding an analytic solution
of the equations of motion. Moreover, it can be viewed as the action that replaces the
DBI action when thermally exciting the D3-F1 system in the regime of large number of
D3-branes and large gs N .
1
Note that, in a slight (but meaningful) abuse of notation this differs by a factor of 1/N from κ used
in the DBI analysis in Section 1.6.
59
Chapter 3. Heating up the BIon
To write down the action we use the result of Section 2.6.2 in which we showed that
for stationary blackfolds the extrinsic blackfold equations can be derived from an action
which is proportional to the free energy. We therefore use the thermodynamic action2
F = M − TS ,
(3.27)
where F = F(T, N, k) is the free energy appropriate for the ensemble where the temperature T and number of D3-branes N and F-strings k are fixed. Here the total mass M and
entropy S are found by integrating the energy density T 00 in (3.9) and the entropy density
s in (3.13) over the D3-brane worldvolume.
Z
Z
π2 2
4
2
3 2
M=
T
dV(3) r0 (5 + 4 sinh α) , S = 2π TD3 dV(3) r05 cosh α .
(3.28)
2 D3
These are thus regarded here as functionals of the embedding function z(σ), that give the
actual total mass and entropy of the system when evaluated on-shell. Using (3.28) in (3.27)
we then find the action functional
Z
π2 2
T
dV(3) r04 (1 + 4 sinh2 α) .
(3.29)
F=
2 D3
Eliminating r0 (σ) using (3.13) and using the induced metric (3.18) we get
Z ∞ p
2T 2
dσ 1 + z 0 (σ)2 F (σ) ,
F = D3
πT 4 σ0
(3.30)
where we introduced the function
F (σ) = σ 2
4 cosh2 α − 3
.
cosh4 α
(3.31)
Note that we integrate from σ0 to infinity in (3.30) according to the boundary conditions
discussed above.
The function F (σ) defined in (3.31) is a specific function of σ for given T , N and k, as
seen from (3.26). This means that when we vary the action (3.30) with respect to z(σ) the
function F (σ) does not vary. Performing this variation in the action (3.30) we then find
that the EOMs take the form3
!0
z 0 (σ)F (σ)
p
= 0,
(3.32)
1 + z 0 (σ)2
which we will use below when solving the system. It is noteworthy that the EOM (3.32)
is exactly as that of (1.85) for the DBI case with FDBI replaced by F in (3.31). The same
holds in fact when comparing the free energy (3.30) and the DBI Hamiltonian (1.84). In
Section 3.2.3 we will further comment on their relation.
As a check on the action approach, we now show that (3.32) obtained from the thermodynamic action is consistent with the D3-F1 extrinsic blackfold equation (3.22). From
(3.32) we find
z 00
F0
=
−
.
(3.33)
z 0 (1 + z 02 )
F
2
3
More properly, the action
√ is I = βF, but0 since β = 1/T is constant we directly use the free energy F.
This is computed as δ( 1 + z 0 2 F ) = √z F 0 2 δz 0 and adding a total derivative to the last formula.
1+z
60
3.2 Thermal D3-brane configuration with electric flux ending in throat
Computing F 0 (σ) from (3.31) and using (3.25) and (3.26) to eliminate α0 (σ) and κ, we get
i
2σ h
2
2
F0 =
1
+
4
cos
ζ
sinh
α
.
(3.34)
cosh4 α
Using this with (3.31) we indeed obtain (3.22) from (3.32). Note that the equivalence of
these two equations implies that the blackfold EOMs are equivalent to requiring the first
law of thermodynamics, since the latter follows from extremizing the free energy functional.
3.2.2
Solution and bounds
We now proceed solving the EOM (3.32) subject to the constraints (3.25) and (3.26).
The latter imply a bound on the temperature as well as the worldvolume coordinate σ
parameterizing the size of the two-sphere, which we will first discuss. Then we turn to the
explicit solution of the constraint (3.26) in terms of the constants (T, N, k) and subsequently
present the solution of the EOMs for the profile z(σ).
Bounds on the temperature and σ
Considering the left hand side of the constraint (3.26)
check that sinh α/ cosh3 α
√ it is easy to
is bounded from above, with a maximal value 2 3/9 for cosh2 α = 3/2. Using (3.25) we
thus get an upper bound for the temperature
4
cos ζ
T 4 ≤ Tbnd
for given (N, k) and σ, where we defined the temperature
!1
√
4 3TD3 4
Tbnd ≡
.
9π 2 N
(3.35)
(3.36)
From (3.35) it is obvious that we furthermore have the weaker upper bound T ≤ Tbnd
which only depends on N and thus not on k and σ. The temperature Tbnd is the maximal
temperature for N coincident non-extremal D3-branes, which can never be reached in the
presence of non-zero F-string charge. For future convenience we furthermore define the
rescaled temperature
T
T̄ ≡
,
(3.37)
Tbnd
which will simplify expressions below. Note from (3.35) that we have T̄ ≤ 1.
Since σ takes values in [σ0 , ∞) we see that cos ζ in (3.25) is minimized for σ = σ0 . It
then follows that the upperbound (3.35) can be stated more accurately as
− 18
κ2
(3.38)
T ≤ Tbnd 1 + 4
σ0
for given N , k and σ0 of the brane profile z(σ). Note however that generically this bound
cannot be saturated for a given N , k and σ0 , so that generically the maximal temperature
is lower than this bound. Finally, we remark that using (3.25) the upper bound on the
temperature (3.35) can be turned into a lower bound on σ
− 1
√
σ ≥ σmin ≡ κ T̄ −8 − 1 4
(3.39)
for given (T, N, k).
61
Chapter 3. Heating up the BIon
Solving cosh α in terms of σ
We now present the explicit solution of the constraint (3.26). Using the upper bound (3.35)
on the temperature along with (3.25) we can define the angle δ(σ) by
r
κ2
cos δ(σ) ≡ T̄ 4 1 + 4 ,
(3.40)
σ
where we restrict the angle to be in the interval 0 ≤ δ(σ) ≤ π/2. In terms of this angle the
constraint (3.26) can be written as
4 cos2 δ
cosh6 α − cosh2 α + 1 = 0 ,
27
(3.41)
which is a cubic equation in cosh2 α. Thus, it has three independent solutions for cosh2 α
in terms of δ. The first solution is
√
3 cos 3δ + 3 sin 3δ
2
.
(3.42)
cosh α =
2
cos δ
As δ goes from 0 to π/2, the right hand side increases monotonically from 3/2 to infinity.
Substituting δ → −δ in (3.42) we find the second solution
√
3 cos 3δ − 3 sin 3δ
2
cosh α =
.
(3.43)
2
cos δ
Here the right hand side decreases monotonically from 3/2 to 1 as δ goes from 0 to π/2.
Finally, by substituting δ → δ − 2π in (3.42) we find the third solution
√
δ+π
3 cos δ+π
2
3 + 3 sin 3
cosh α = −
.
(3.44)
2
cos δ
This solution can be immediately discarded since it decreases from −3 to −∞ as δ goes
from 0 to π/2, and α has to be real.
Turning to the two solutions (3.42) and (3.43) we note that both of them respect that
cosh2 α ≥ 1. In the extremal limit one takes α → ∞. Therefore, the solution branch
connected to the extremal solution is the first solution (3.42). For this branch the energy
density (T 00 in (3.9)) for each value of σ increases as the temperature T increases holding
(N, k) fixed. Thus, this is the thermodynamically stable branch with positive heat capacity.
Instead in the second solution branch (3.43) the energy density at each σ decreases with
increasing temperature, thus resulting in a negative heat capacity. This branch is connected
to the neutral 3-brane with α = 0. The two branches meet at the point cosh2 α = 3/2.
The solution
Finally, we turn to the solution of the EOM (3.32). Imposing the boundary condition
(1.79) we find that the solution takes form4
0
− z (σ) =
F (σ)2
−1
F (σ0 )2
4
− 12
.
(3.45)
Notice that it follows from the expression (3.45) that the derivatives dn σ/dz n of the inverse function
σ(z) vanish for odd n at z = z(σ0 ). This is a necessary requisite for the smoothness of the wormhole
solution of Section 3.3.
62
3.2 Thermal D3-brane configuration with electric flux ending in throat
Imposing the boundary condition (3.19) we find
Z
∞
dσ
z(σ) =
0
σ
F (σ 0 )2
−1
F (σ0 )2
− 12
, F (σ) = σ 2
4 cosh2 α − 3
,
cosh4 α
(3.46)
with σ ≥ σ0 and where we repeated the definition of F (σ) from (3.31) given in terms of
cosh2 α. Furthermore, cosh2 α(σ) is given by one of the two branches (3.42) and (3.43).
The result (3.46) describing the heated up BIon solution is one of the central results of
this chapter. Just as the BIon solution, it is obtained in a probe approximation, which is
discussed in more detail in Section 3.2.4.
It should be emphasized that for each of the two branches we have an explicit expression
for the derivative of the brane profile z 0 (σ) so that the mass and entropy for each solution
branch can be obtained explicitly from the integrands in (3.28), (3.30). This gives
2T 2
M = D3
πT 4
Z
∞
F (σ)
4 cosh2 α + 1
dσ p
σ2
,
cosh4 α
F 2 (σ) − F 2 (σ0 )
σ0
2T 2
S = D3
πT 5
Z
∞
σ0
4
F (σ)
σ2
.
dσ p
2
2
F (σ) − F (σ0 ) cosh4 α
We also give the integrated chemical potentials [43] that follow from (3.14)5
Z ∞
F (σ)
µD3 = 4πTD3
dσ p
σ 2 tanh α cos ζ ,
2
F (σ) − F 2 (σ0 )
σ0
Z ∞
F (σ)
µF1 = TF1
dσ p
tanh α sin ζ .
2
F (σ) − F 2 (σ0 )
σ0
(3.47)
(3.48)
(3.49)
(3.50)
These satisfy the first law of thermodynamics and Smarr relation
dM = T dS + µD3 dN + µF1 dk , 4(M − µD3 N − µF1 k) = 5T S .
(3.51)
The integrations in (3.47)-(3.50) can be performed numerically or analytically in certain
limits.
The solution presented above gives the profile of a configuration of N coincident infinitely extended D3-branes with k units of F-string charge, ending in a throat with minimal
radius σ0 , at temperature T . The configuration is sketched in Figure 1.2. As explained in
Section 1.6.1 we can construct a corresponding wormhole solution by attaching a mirror
solution as illustrated in Figure 1.4. This will be discussed in Section 3.3.
In the next subsection we examine two limits of the branch (3.42) connected to the
extremal configuration, which will be used in Section 3.3. Appendix B.1 presents some
features of the branch (3.43) connected to the neutral 3-brane, which will not be further
discussed in this work.
3.2.3
Analysis of branch connected to extremal configuration
Our main focus is the solution (3.46) for the branch (3.42) connected to the extremal
configuration, which we start analyzing here by considering two physically relevant limits.
5
We redefine these for convenience by an extra factor of the D3-brane and F-string tension respectively.
63
Chapter 3. Heating up the BIon
We then go on to showing the equivalence of our thermal action (free energy) (3.30) in the
zero temperature limit to the DBI Hamiltonian.
One interesting limit is the limit of small temperature, which enables to see in a small
temperature expansion the effect of heating up the BIon as compared to the extremal case.
The temperature enters the solution through the function α(σ), which is given by (3.42)
for the branch connected to the extremal configuration. From (3.40) it is seen that for
small temperatures δ is close to π/2, so that
√
√
3 3
1
3
2
2
cosh α =
− −
cos δ −
cos2 δ + O(cos3 δ) .
(3.52)
2 cos δ 2
12
27
This gives for the function F (σ) defined in (3.31) the expansion
√
√
F
8 3
4
4 3
2
=
cos δ −
cos δ −
cos3 δ + O(cos4 δ) .
σ2
9
27
243
(3.53)
Keeping the first two terms in the expansion we then find from the solution (3.45) the
result
s
s
"
!
#
r
√
κ2 + σ04
3 4 κ2 + σ 4
κ2
κ2
0
8
T̄ 4
− z (σ) =
1+
1 + 4 − 1 + 4 + O(T̄ ) . (3.54)
18
σ
σ 4 − σ04
σ − σ04
σ0
Taking the zero temperature limit, we see that the first term agrees with the BIon solution
(1.86). The second term describes the leading order effect due to heating up the BIon.
Note that the factor in front of T̄ 4 does not blow up as σ → σ0 . This expression
is a good
p
approximation for all σ ≥ σ0 when (T, N, k) and σ0 are given such that T̄ 4 1 + κ2 /σ04 1.
√
Another interesting limit is σ/ κ 1. This limit describes the profile for large σ, i.e.
in the region close to the flat D3-brane at infinite σ. In this limit we have to leading order
√
cos δ = T̄ 4 . Therefore, from (3.42) we see cosh2 α is a function of T̄ only. For large σ/ κ
we thus have
F (σ) = σ 2 g(T̄ ) + O(κ/σ 2 ) ,
(3.55)
where g(T̄ ) is a function that increases from 0 to 4/3 as T̄ goes from 0 to 1. For sufficiently
large σ we find therefore from (3.46) the behavior
z(σ) =
F (σ0 ) 1
+ O(σ −5 ) .
g(T̄ ) σ
(3.56)
√
√
If we require σ/ κ 1 for all σ, the limit corresponds to σ0 / κ 1, which can be
regarded as the limit in which the F-string charge is taken to be small. In this case it
follows from (3.55) that F (σ)/F (σ0 ) ' σ 2 /σ02 and hence
− z 0 (σ) ' p
σ02
σ 4 − σ04
,
(3.57)
√
which agrees with (1.86) in the limit σ0 / κ 1.
Comparison with DBI Hamiltonian
Finally, we point out the relation of the DBI action with our thermal D3-F1 brane action
(free energy) in (3.30). To this end we compute the T → 0 limit of our action. In
64
3.2 Thermal D3-brane configuration with electric flux ending in throat
particular, using the small temperature expansion (3.53) for the branch connected to the
extremal configuration along with the expression (3.40) for cos δ and the definition (3.36)
we find by inserting in the action (3.30) that
lim F = HDBI |TD3 →N TD3 .
T →0
(3.58)
This shows that our thermodynamic action (3.27) can be viewed as the thermalization of
the DBI Hamiltonian (1.84). The replacement TD3 → N TD3 induces an extra factor of N
in front of the DBI Hamiltonian (1.84), and at the same time includes an extra factor of
1/N in κ, yielding the one defined in (3.24). It is satisfying to recover these multiplicative
factors of N since the DBI action is valid for a single D3-brane only and our thermal
system contains instead N D3-branes. Moreover, this relation implies that the systems are
properly connected also off-shell and makes manifest that the extremal limit of the branch
(3.42) of our solution gives the BIon solution reviewed in Section 1.6.
3.2.4
Validity of the probe approximation
In the blackfold approach used to obtain the EOM (3.22), the D3-F1 system is considered
in the probe approximation. We therefore need to determine the conditions for which this
is a valid approximation. If we denote the transverse size scale of the D3-F1 geometry by
rs , there are two conditions that have to be met across the embedded surface in order for
gravitational backreaction to be negligible:
• The size scale should be much less than the S 2 radius σ of the induced geometry:
rs σ
• The size scale should be much less than the length scale Lcurv of the extrinsic curvature of the induced geometry: rs Lcurv .
The transverse size scale rs of the D3-F1 geometry (3.5) is determined by the harmonic
functions f , H and D in (3.6). The three corresponding scales are r0 , r0 sinh1/2 α and
r0 (sin ζ sinh α)1/2 respectively. Since sin2 ζ ≤ 1, we only need to consider the first two scales
and we let rs be the largest of the two. For the main solution branch (3.42) connected to the
extremal configuration the largest scale is then rs4 = r04 sinh2 α, since 1/2 ≤ sinh2 α < ∞.
Moreover, it is in fact sufficient to consider the charge radius rc4 = r04 sinh α cosh α instead.
Near extremality this is obviously true, but, more generally we have on this branch 1/3 ≤
tanh2 α ≤ 1, so that rc differs by a factor of order 1 from rs . For the other branch (3.43)
connected to the neutral 3-brane, where we have 0 ≤ sinh2 α ≤ 1/2, similar reasoning
implies that it is sufficient to consider the radius rs = r0 . Focusing on the main branch,
the above conditions can thus be written as
rc (σ) σ , rc (σ) Lcurv (σ) ,
where, using (3.11) and (3.25), the charge radius is found to be
r
N
κ2
4
rc ∼
1+ 4 .
TD3
σ
(3.59)
(3.60)
Considering the first condition rc (σ) σ in (3.59) we see that rc (σ) is largest for σ = σ0
thus to satisfy rc (σ) σ for all σ ≥ σ0 it is sufficient to demand rc (σ0 ) σ0 .
65
Chapter 3. Heating up the BIon
To understand the second condition rc (σ) σ in (3.59) we first need to determine
ρ
Lcurv (σ). For this we need to compute the mean curvature vector K ρ = γ ab Kab
. The
−1
ρ
curvature size is then given by Lcurv = |K| with K = K nρ , where nρ is the unit normal
vector of the embedding surface of the three-brane. In the case at hand we have
1
nρ = p
(0, −z 0 (σ), 0, 0, 1, ~05 )
1 + z 0 (σ)2
(3.61)
in the coordinates (t, r, θ, φ, xi ) used in (3.16). To see this note that the tangent vector
to the 3-brane is V ρ = √ 1 0 2 (0, 1, 0, 0, z 0 (σ), ~05 ), so that nρ V ρ = 0. Using the second
1+z (σ)
fundamental tensor computed in (3.21) we then find after some algebra
1
z0
2 F0 4 sinh2 α
F0 F 0
2
00
02
K=
z
+
2
=
−
sin2 ζ ,
(1
+
z
)
=
−
σ
F F
σ
σ F 1 + 4 sinh2 α
(1 + z 02 )3/2
(3.62)
where F0 ≡ F (σ0 ). To examine this further, we note that for the main branch we have
sinh2 α ≥ 1/2 so that the the third factor in (3.62) is of order one. We have furthermore
checked that the extrinsic curvature (3.62) reaches its maximum Kmax in the region σ ∼ σ0
close to the end of the throat. We thus get
Kmax ∼ −
1
sin2 ζ .
σ0
(3.63)
Hence the second condition in (3.59) requires to be in the regime where rc (σ0 ) σ0 / sin2 ζ.
However, this is already ensured by the stronger condition rc (σ0 ) σ0 coming from the
first condition in (3.59). Thus, in conclusion, it is sufficient to impose the condition
rc (σ0 ) σ0
(3.64)
to ensure the validity of the probe approximation. To get a better understanding of the
√
condition (3.64), we now consider it in the regime where
σ0 / κ is very small. Using the
√
definition of κ in (3.24) the condition becomes σ03 kgs `s . It is interesting to note that
the N -dependence has canceled out in this condition. It is intuitively clear that the larger
the number of F-strings, the greater the minimum radius should be in order to neglect
backreaction. Moreover, for sufficiently weak string coupling it is always possible to be in
the correct range.
3.3
Separation between branes and anti-branes in
wormhole solution
3.3.1
Brane-antibrane wormhole solution
In Section 3.2 we found a configuration where N coincident thermally excited D3-branes
have an electric flux such that the equipotential surfaces of the electric field are on twospheres. The shape of the configuration is given by the function z(σ) in Eq. (3.46). As σ
ranges from ∞ to σ0 the bulk coordinate z(σ) ranges from zero to z(σ0 ) where the tangent
of the brane is orthogonal to the tangent of the brane at σ = ∞. Thus, we have k electric
flux lines all pointing towards a center, and as σ decreases the density of the electric flux on
66
3.3 Separation between branes and anti-branes in wormhole solution
the equipotential two-spheres increases. However, rather than having a central singularity
as in the linear theory of Maxwell electrodynamics, the non-linear nature of the DBI theory
on the D3-brane gives instead a bending of the brane preventing us from reaching the point
where the singularity should have been. The spherically symmetric electric flux thus causes
the bending of the D3-brane in the bulk creating a throat on the brane ending at r = σ0
and z = z(σ0 ) = ∆/2 as illustrated in Figures 1.1 and 1.2. As stated in Section 3.2 we
should distinguish between solutions z(σ0 ) = ∞ and z(σ0 ) < ∞. For z(σ0 ) = ∞ we have
an infinite spike. Instead, for z(σ0 ) < ∞ the solution as it is violates the conservation of
D3-brane and F-string charge. To remedy this one can attach a mirror of the solution,
reflected in x1 around x1 = z(σ0 ) = ∆/2, as illustrated in Figure 1.4. This we call a
brane-antibrane wormhole solution. Another possibility, that we explore in Section 3.5, is
to attach k non-extremal F-strings to the throat.
In this section we shall see that it is not possible to find finite-temperature solutions
with z(σ0 ) = ∞, i.e. the infinite spike, unlike in the zero temperature case (see Section
1.6). Thus, this section is focused on the brane-antibrane wormhole solution.
In more detail, the brane-antibrane wormhole solution has N coincident D3-branes
extending to infinity at x1 = 0, and N anti D3-branes (i.e. oppositely charged) extending
to infinity at x1 = ∆ = 2z(σ0 ). Then centered around r = 0 we have a wormhole where for
each constant x1 slice you have a two-sphere where a flux of k F-strings is going through
in the positive x1 direction. At x1 = z(σ0 ) the two-sphere has the minimal radius r = σ0 .
The solution has altogether four parameters: The D3-brane charge N , the F-string charge
k, the temperature T and the minimal radius σ0 . It is important to note here that the
extensive quantities given in (3.47)-(3.50) should be multiplied by a factor of 2 since we
add a mirror of the solution.
In this section we consider the separation distance ∆ = 2z(σ0 ) between the N D3branes and N anti D3-branes for a given brane-antibrane wormhole configuration defined
by the four parameters N , k, T and σ0 . We consider only the branch (3.42) connected to
the extremal solution. Thus, we have from (3.46)
− 12
Z ∞
F (σ 0 )2
0
∆=2
dσ
−1
,
(3.65)
F (σ0 )2
σ0
with F (σ) given by (3.31), (3.42) and (3.40). From considering (3.31), (3.42) and (3.40)
we see that F (σ) depends on N , k and T only through the variables κ and T̄ defined
in (3.24) and (3.37). Therefore, the separation distance is a function with dependence
∆ = ∆(T̄ , σ0 , κ). However, changing κ corresponds to a uniform scaling of the system.
Indeed, it is easy to show that ∆(T̄ , σ0 , κ) has the scaling property
σ
√
0
∆(T̄ , σ0 , κ) = κ ∆ T̄ , √ , 1 .
(3.66)
κ
Thus, it is enough to find ∆ for κ = 1. In detail, given ∆ for a certain σ0 and T̄ with
√
√
κ = 1 the general κ configuration is found by rescaling ∆ → κ∆ and σ0 → κσ0 while
keeping T̄ fixed.
3.3.2
Diagrams for separation distance ∆ versus minimal radius σ0
In the following we examine the behavior of the separation distance ∆(T̄ , σ0 , κ = 1) ≡
∆(T̄ , σ0 ). We set κ = 1 in the rest of this section since, as argued above, one can always
reinstate κ by a trivial rescaling.
67
Chapter 3. Heating up the BIon
We shall examine the
lytical methods. We first
four different values of T̄
case of zero temperature
Note that 0 ≤ T̄ ≤ 1.
separation distance ∆(T̄ , σ0 ) by using both numerical and anaevaluate ∆(T̄ , σ0 ) numerically. The behavior of ∆ versus σ0 for
is shown in Figures 3.1 and 3.2, where we compare it with the
(Figure1.5). We have chosen the values T̄ = 0.05, 0.4, 0.7, 0.8.
D
D
30
10
25
8
20
15
6
10
4
5
0.0
0.5
1.0
1.5
2.0
2.5
Σ0
0.0
0.5
1.0
1.5
2.0
Σ0
Figure 3.1: On the figures the solid red line is ∆ versus σ0 either for T̄ = 0.05 (left
figure) or T̄ = 0.4 (right figure) while the blue dashed line corresponds to T̄ = 0. We
have set κ = 1.
D
D
10
20
8
15
6
10
4
5
0.0
0.5
1.0
1.5
2.0
Σ0
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Σ0
Figure 3.2: On the figures the solid red line is ∆ versus σ0 either for T̄ = 0.7 (left figure)
or T̄ = 0.8 (right figure) while the blue dashed line corresponds to T̄ = 0. We have set
κ = 1.
By looking at the curves in Figures 3.1 and 3.2 we can see that there are new interesting
features with respect to the zero temperature case displayed in Figure 1.5. We first notice
that σ0 is bounded from below. Indeed, we found in Section 3.2 the bound (3.39)
σ0 ≥ σmin ≡
T̄ 2
.
(1 − T̄ 8 )1/4
(3.67)
This is in contrast to the zero temperature case where one can take σ0 → 0 corresponding
to the infinite spike solution. For most values of T̄ we further have the feature that when
increasing σ0 from σmin then ∆ increases until it reaches a local maximum denoted by
∆max . Increasing σ0 further ∆ decreases until it reaches a local minimum that we denote
68
3.3 Separation between branes and anti-branes in wormhole solution
∆min (for most values of T̄ this is also the global minimum). Increasing σ0 further ∆
increases monotonically and follows increasingly closely the zero temperature value of ∆
as a function of σ0 . In Figure 3.3 we have displayed the behavior of ∆ at σ0 = σmin , ∆max
and ∆min for all values of T̄ . We see from this that there is a critical value of T̄ given
by T̄b ' 0.8 beyond which ∆max and ∆min ceases to exist. Note also that just before T̄
reaches T̄b one has that ∆ at σ0 = σmin is smaller than ∆min , unlike for lower values of T̄
where ∆min is the global minimum.
D
16
14
Dmax
12
DΣ0=Σmin
10
8
6
4
2
0.0
Dmin
0.2
0.4
0.6
0.8
1.0
T
Figure 3.3: ∆max (blue curve), ∆min (red curve), ∆ at σmin (black curve) as a function
of the temperature T̄ for κ = 1.
We can conclude from the above that the there is no direct analogue of the infinite
spike solution with σ0 = 0 and z(σ0 ) = ∞ for non-zero temperature. As one can see from
(3.67) σ0 can not reach zero for non-zero temperature. Moreover, for T̄ ≤ T̄b the local
maximum ∆max is always finite, as one can see from Figure 3.3. Thus, there are no finite
values of σ0 for which ∆ is infinite. However, in Section 3.5 we will examine a different
way to find an analogue of the infinite spike solution at finite temperature.
The plots in Figures 3.1, 3.2 and 3.3 illustrate a number of interesting features:
• We observe that for any temperature, for sufficiently large σ0 , the finite temperature
curve is increasingly close to the corresponding extremal curve as σ0 increases. In
particular, in accordance with expectations, as the temperature is lowered a larger
part of each of the two curves is close to each other. However, for any non-zero
temperature T̄ < T̄b there is always the new branch connected to σmin going up to
∆max .
• As the temperature approaches its maximum value (T̄ → 1) the thin throat branch
diminishes, and the part of the curve that remains coincides increasingly with that
of the extremal one.
• Because of the appearance of a maximum ∆max and minimum ∆min for temperatures
in the range 0 < T̄ < T̄b , there exist values of the brane separation ∆ for which there
are three possible phases.
69
Chapter 3. Heating up the BIon
3.3.3
Analytical results
We now consider what we can say analytically about the ∆ versus σ0 behavior either in the
small temperature regime or the large σ0 regime using the analysis of Section 3.2.3 for the
branch connected to the extremal configuration6 and the results listed in Appendix B.2.
We begin by considering the ∆ versus σ0 behavior for small temperatures T̄ 1. We
first consider the minimum ∆min of the ∆(σ0 ) curve (for small temperatures this is a global
minimum). The small temperature expansion for z 0 (σ) in (3.54) can be used to compute the
small temperature expansion of the brane separation (3.65) in the corresponding wormhole
configuration. The result is given in Appendix B.2, from which we quote
∆ = ∆0 + T̄ 4 ∆1 + O(T̄ 8 ) .
(3.68)
Here the leading term ∆0 is the extremal result (1.93) (with κ = 1) and ∆1 is given in
(B.11). Using now (3.68) we find that the minimum is found at
σ0 = 1 + x T̄ 4 + O(T̄ 8 )
√
3
2
Γ − 41 Γ 81
6Γ − 14 Γ 43 Γ
1
√
x≡ √ +
+
5πΓ 81
3 6 1024 23/4 3π 5/2
13
8
.
(3.69)
∼ −0.103
Note that the leading result σ0 = 1 reproduces the one found in Section 1.6.2. The
corresponding value of the minimum separation distance is
√
√
2 2π Γ 45
2πΓ 58
4
+ O(T̄ 8 ) ,
∆min =
− T̄ √
(3.70)
Γ 34
3 3 Γ 18
showing that for small temperatures the minimum is slightly lower than for T = 0.
We turn now to the local maximum ∆max for small temperatures. Using the expansion
(B.10) of ∆ in powers of T̄ 4 up to and including the T̄ 12 term we find that the σ0 value at
which the local maximum occurs is
σ0 = a1 T̄ 2/3 + a2 T̄ 10/3 + O(T̄ 18/3 ) ,
(3.71)
with a1 and a2 given numerically by a1 ' 0.693 and a2 ' −0.00243.7 Note that the leading
scaling σ0 ∝ T̄ 2/3 in (3.71) for the position of the maximum is consistent with the fact
that it should be in between σmin , which goes like T̄ 2 to leading order, and the σ0 value
in (3.69) corresponding to minimum ∆min which goes like T̄ 0 to leading order. The local
maximum corresponding to (3.71) is
∆max '
b1
,
T̄ 2/3
(3.72)
where b1 ' 3.28 as computed using the expansion (B.10) to order T̄ 12 . Note furthermore
that the contributions to each term in the expansion of σ0 in (3.71) come from each term
of the expansion of ∆ in (B.10). This is the reason why we expand ∆ up to T̄ 12 instead of
6
Corresponding analytical results can be obtained for the branch connected to the neutral 3-brane,
using Section B.1, but these will not be discussed in this chapter.
7
To order T̄ 4 in the expansion (B.10) we find a1 ' 0.707 and a2 ' 0.0908. Going to order T̄ 12 one
instead finds a1 ' 0.693 and a2 ' −0.00243.
70
3.3 Separation between branes and anti-branes in wormhole solution
considering only the first two terms (i.e. up to the order T̄ 4 ) as we did in the determination
of the minimum.
We can also study what happens for σ0 large. Here we can use (3.57) and in an
expansion for small temperature we can compute the first correction, yielding
√
√
πΓ 54
κ2
4 1
8
∆σ0 →∞ =
2σ0 + 3 1 − T̄
3 + O(T̄ ) .
(3.73)
6
σ0
Γ 34
We see that for large σ0 the leading behavior of ∆ is linear in σ0 as in the extremal case
(see Eq. (1.96)).
3.3.4
Equilibrium and stability of the brane-antibrane wormhole configuration
The brane-antibrane wormhole configuration at finite temperature described in this section
is considered to be a static configuration in the background of hot flat space. We consider
in this subsection briefly the validity of this with regard to what regime we should be in
and with regard to the stability of our configuration.
Solving the extrinsic EOM (3.22), or more generally (1.106), means that one has found
an equilibrium configuration of the long-range dynamics of the system of thermal D3-branes
with electric flux. However, this does not say anything about whether this equilibrium is
stable or not. One type of instabilities that can occur are due to local fluctuations of
the black D3-F1 brane bound state. Investigating the existence of such an instability
boils down to considering whether the black brane is subject to the Gregory-Laflamme
instability. As found in [35] for long wave-lengths this is equivalent to whether the speed
of sound is imaginary which again is equivalent to the local thermodynamic stability of
the D3-F1 brane bound state. Since we are considering the branch (3.42) connected to
the extremal D3-F1 brane we have local thermodynamic stability and hence there is no
Gregory-Laflamme instability.
Another type of instability one can consider is whether our solution is an unstable
saddle-point of the action. We consider this question in detail in Section 3.4.2 where
we analyze it by computing a regularized free energy for the brane-antibrane wormhole
configuration at finite temperature.
Finally, one should consider the accuracy of the extrinsic EOM (1.106) in order to argue
in which regime one can claim that a static solution exists. The extrinsic EOM (1.106) is
subject to corrections coming from the backreaction that the brane has on its surrounding
geometry. Considering the brane-antibrane wormhole configuration it is evident that the
leading source of the backreaction is the attraction between the N D3-branes and the N
anti-D3-branes, separated by the distance ∆. Since we are in the branch (3.42) connected
to the extremal D3-F1 brane the force between the brane is of the same order as that
found in the zero-temperature system. Therefore, the considerations in Section 1.6.2 on
the extremal brane-antibrane-wormhole solution applies also in this case, and we find that
the characteristic time-scale over which the N D3-branes and N anti-D3-branes are seen
to attract each other is
3
t
1
∆
∼√
.
(3.74)
`s
gs N `s
Thus, for smaller time-scales than this we can effectively regard the brane-antibrane wormhole configuration at finite temperature as being static. Note here that gs N 1 in order
71
Chapter 3. Heating up the BIon
for the supergravity approximation of the D3-F1 probe branes to be valid, thus since we as
a minimum should require t `s we get the necessary condition ∆/`s (gs N )1/6 for the
existence of an intermediate regime in which the configuration can be regarded as static.
3.4
Comparison of phases in canonical ensemble
In this section we study the physics of the three different phases in the brane-antibrane
wormhole configuration found in Section 3.3. We define the free energy of the system
and use this to compare the three phases in order to see which is the thermodynamically
preferred one. We consider what happens when changing the temperature or the separation
distance in the system. Furthermore, we compare the free energy to the energy of the phases
in the zero-temperature case of the BIon solution [23,24], that we reviewed in Section 1.6.
Finally, we provide some heuristic considerations about the dynamics of the system away
from equilibrium.
3.4.1
Choice of ensemble and measurement of the free energy
In Section 3.3 we found that the (σ0 , ∆) diagram has important qualitative differences
with the (σ0 , ∆) diagram of the BIon solution of the Born-Infeld theory. Notably, for
temperatures not too large there exists three distinct phases, i.e. three possible values of
σ0 , for a range of separation distances ∆. It is therefore a natural question which of the
three phases are preferred thermodynamically. Before venturing into such a comparison, we
should briefly ponder on which thermodynamic ensemble one should consider, i.e. which
quantities to keep fixed in the comparison. We imagine here that the two systems of
N D3-branes and N anti-D3-branes are infinitely extended, at least in comparison with
the range of values of σ0 that we will consider. Two obvious quantities to keep fixed are
thus the D3-brane charge (which can be measured at each point of the D3-branes) and
the temperature T . This is simply because it would take an infinite amount of energy
to change the D3-brane charge or the temperature.8 With respect to the temperature
we can say that the infinite extension of the branes means that the branes, up to some
distance away from the wormhole, correspond to a heat bath for the part of the brane
system with the wormhole with k units of F-string flux that connects the two infinitely
extended branes. Therefore we are obviously in the canonical ensemble. In addition, also
the number of F-strings k should be fixed. This is because if we consider a sufficiently
large value of σ so that we are far away from the part with the wormhole then we can still
measure the k F-strings by making an integral over the two-sphere with radius σ. From the
charge conservation along the F-string worldvolume direction we then get that k should
be kept fixed in our thermodynamical ensemble. Finally, it is also natural to keep fixed ∆,
the separation between the D3-branes and anti-D3-branes, since that is also something we
can measure far away from the part of the system where the branes are connected with a
wormhole.
In summary, we work in an ensemble with T , N , k and ∆ kept fixed. The first law
of thermodynamics (3.51) generalizes to dM = T dS + µD3 dN + µF1 dk + f d∆ when we
allow for variations of ∆ where f is the force between the D3-brane systems. We see
from this that the correct free energy for our ensemble is F = M − T S with the variation
8
One can of course change the temperature locally but we are here concerned with systems in global
thermodynamical equilibrium.
72
3.4 Comparison of phases in canonical ensemble
dF = −SdT + µD3 dN + µF1 dk + f d∆. Thus, we can write F(T, N, k, ∆). However, as
described in Section 3.3, we can have up to three distinct phases given (T, N, k, ∆) which
we can label by σ0 . Thus, we write below F(T, N, k, ∆; σ0 ) where σ0 is understood only
as a label that points to which branch we are on, and should not be understood as a
thermodynamic variable. From (3.30), (3.46) and (3.31) we find
2 Z ∞
4TD3
F (σ)2
p
F(T, N, k, ∆; σ0 ) =
dσ
.
(3.75)
πT 4 σ0
F (σ)2 − F (σ0 )2
The goal is now to compute (3.75) for the various phases for different temperatures and
brane separations.
In our application we shall keep N and k strictly fixed. Thus, as such, there is no
reason to consider them as variables. As remarked in Section 3.3.1, F (σ) only depends
on N , k and T through the variables κ and T̄ . Furthermore, changing κ only amounts
to a uniform rescaling of the system. Indeed, it is easy to show that finding a solution
with a given T̄ , σ0 , ∆ and F for κ = 1 the general κ configuration is found by rescaling
√
√
σ0 → κσ0 , ∆ → κ∆ and F → κ3/2 F while keeping T̄ fixed. Thus, we choose to set
κ = 1 in the rest of this section since one can always reinstate it by a trivial rescaling and
since we are not interested in varying κ. In accordance with this, we shall consider the
variation of F with respect to the rescaled temperature T̄ rather than T . With this we
can write (3.75) as
Z
9πN TD3 ∞
F (T̄ , σ)2
F(T̄ , ∆; σ̂0 ) = √
dσ p
,
(3.76)
3T̄ 4 σ0
F (T̄ , σ)2 − F (T̄ , σ0 )2
where F (T̄ , σ) is defined as the function F (σ) given by (3.31), (3.42) and (3.40) for κ = 1
and a given T̄ . For our purposes below we furthermore choose to set N TD3 to one, i.e. we
will measure the free energy in units of N TD3 . We can thus write
Z ∞
F(T̄ , ∆; σ0 ) =
dσ h(T̄ , σ, σ0 )
σ0
h(T̄ , σ, σ0 ) ≡ √
9πF (T̄ , σ)2
p
.
3T̄ 4 F (T̄ , σ)2 − F (T̄ , σ0 )2
(3.77)
√
Consider now the integrand h(T̄ , σ, σ0 ) in (3.77). Using that in the regime of large σ/ κ
the function F (σ) behaves as (3.55) we see that for large σ
h(T̄ , σ, σ0 ) =
9πσ 2 g(T̄ )
√
+ O(σ −2 ) .
4
3T̄
(3.78)
Thus the integral over σ in (3.77) is clearly divergent, which is expected since the system
of branes is infinitely extended along the D3-brane worldvolume directions. However, we
can get rid of this divergence consistently as follows. First we choose to only keep fixed T̄
and instead consider all possible values of ∆. In this way we can think of the free energy
F(T̄ , ∆(σ0 ); σ0 ) as a function of T̄ and σ0 . Our goal is now to compute F(T̄ , ∆(σ0 ); σ0 )
for a large range of σ0 values with σ0 ≥ σmin (T̄ ) = (T̄ −8 − 1)−1/4 . Pick now a σ0 = σcut
outside this range. We then consider the difference between the free energy at σ0 and at
σcut . This can be written as
δF(T̄ , ∆(σ0 ); σ0 ) ≡ F(T̄ , ∆(σ0 ); σ0 ) − F(T̄ , ∆(σcut ); σcut )
Z σcut
Z ∞
h
i
=
dσh(T̄ , σ, σ0 ) +
dσ h(T̄ , σ, σ0 ) − h(T̄ , σ, σcut ) .
σ0
σcut
73
(3.79)
Chapter 3. Heating up the BIon
Since the divergent part of the integrand for large σ is independent of σ0 the divergent part
cancels out in the second integral of the RHS of Eq. (3.79). Moreover, since the correction
to the leading part for large σ in (3.78) goes like σ −2 the second integral of the RHS of
Eq. (3.79) is convergent. Therefore, δF(T̄ , ∆(σ0 ); σ0 ) as defined in (3.79) is well-defined
for a given σcut . We can furthermore infer that the dependence on σcut only enters as an
additive constant. Indeed, if we consider two values σ0 = a1 , a2 with a1 < a2 < σcut we
find that δF(T̄ , ∆(a1 ); a1 ) − δF(T̄ , ∆(a2 ); a2 ) does not depend on σcut . We thus see that
we can use (3.79) to measure the free energy since we only need the relative measure of
the free energy between the possible branches.
3.4.2
Comparison of phases
We now compare the free energy of the three distinct phases as found in Section 3.3 for
temperatures not too large.
We first consider the free energy for a fixed temperature. For definiteness, we consider
the behavior of the free energy for the temperature T̄ = 0.4. As one can see from Figure
3.3 we expect the qualitative features to be the same for the whole range of temperatures
T̄ from 0 to T̄b ' 0.8. The (σ0 , ∆) diagram for T̄ = 0.4 is displayed in Figures 3.1 and
3.2. In Figure 3.4 we display δF(T̄ , ∆(σ0 ); σ0 ) as a δF versus ∆ diagram for T̄ = 0.4
(with σcut chosen so that the upper branch starts with zero free energy). We see from
these diagrams that in the range of separation distances ∆ from ∆min ' 3.7 to ∆max ' 6.3
the thermodynamically favored branch, i.e. the branch with least free energy, is given by
the branch in Figures 3.1 and 3.2 that goes between ∆min to ∆max with d∆/dσ0 < 0.
Instead when ∆ ≥ ∆max there is only one available branch. However, as discussed below
we believe this is an unstable saddlepoint. Qualitatively, this is the behavior of the phases
for temperatures 0 < T̄ ≤ T̄b ' 0.8. For T̄ > T̄b we have at most one available phase for a
given ∆.
F
0
-10
Turning Point
-20
-30
-40
-50
-60
3.5
4.0
4.5
5.0
5.5
6.0
6.5
D
Figure 3.4: The free energy δF versus ∆ for T̄ = 0.4 and κ = 1.
Consider instead a fixed separation distance ∆ = ∆0 between the two systems of branes.
Increase now the temperature slowly from zero temperature. Then the thermodynamically
dominant phase is the phase for which d∆/dσ0 < 0, i.e. the phase that goes from ∆min
74
3.4 Comparison of phases in canonical ensemble
to ∆max . Above the critical temperature T̄ = T̄c for which ∆0 = ∆max (T̄ ) the phase
with d∆/dσ0 < 0 does not exist anymore and the only available phase is the one with
d∆/dσ0 > 0 that starts at ∆min . Note that we used here that ∆max (T̄ ) is a monotonically
decreasing function of T̄ , see Figure 3.3. However, we believe this phase is an unstable
saddlepoint so the system should decay before reaching T̄c towards another end state. This
will be discussed further below in Section 3.4.4.
From the above considerations of the free energy we note that it has variation dF =
−SdT + f d∆ using here that we do not allow for variations of N and k. Consider the
derivative9
∂F
f=
,
(3.80)
∂∆ T̄
f is the force between the infinitely extended branes. Since it is a brane-anti-brane system
the force is always positive f > 0. This we can in fact observe in the F versus ∆ diagram of
Figure 3.4. Furthermore, since the force is continuous as we follow the curve this explains
the behavior that the phases meet in cusps with zero angle.
3.4.3
Energy in the extremal case
It is interesting to compare the finite temperature behavior of the free energy with that of
the energy in the extremal case. Also in the extremal case the energy given in Eq. (1.88) is
divergent and requires a regularization. We can adopt the same regularization used above
for the free energy and define a regularized energy as
δE(∆(σ0 ); σ0 ) ≡ E(∆(σ0 ); σ0 ) − E(∆(σcut ); σcut )
Z ∞
Z σcut
h
i
dσh(σ, σ0 ) +
dσ h(σ, σ0 ) − h(σ, σcut ) ,
=
σ0
(3.81)
σcut
where now h(σ, σ0 ) is given by the energy density dH/dσ in the extremal case written in
Eq.(1.88)
σ 4 + κ2
h(σ, σ0 ) = 8πTD3 p
.
(3.82)
σ 4 − σ04
In this case the expression for the energy as a function of ∆ can be written exactly by
eliminating σ0 in the solution (1.87) using Eq.(1.95). Since there are two solutions for σ02
in (1.95) these generate two branches in the energy as a function of ∆, one corresponds to
the “thin throat” branch, and it is energetically favored, and the other that corresponds to
the “thick throat” branch. This is represented in Figure 3.5.
Comparing with the free energy diagram of Figure 3.4 we see that the branch with
highest energy/free energy behaves the same whether the temperature is turned on or not.
This branch ends in both cases at ∆min . However, the qualitative difference comes in for
the remaining branches. For the zero temperature case in Figure 3.5 the second branch
with lower energy extends all the way to infinite separation. Instead, as seen in Figure 3.4,
for non-zero temperature there is a turning point at ∆max . If we imagine taking the zero
temperature limit of the thermal case depicted in Figure 3.4 what happens qualitatively is
that the turning point at ∆max is pushed towards infinity thus leaving only the two infinite
branches meeting at ∆min . This is in accordance with the fact that we get the total energy
9
Note that since F and δF only differ by an additive constant it does not matter whether we use F or
δF in the Eq. (3.80).
75
Chapter 3. Heating up the BIon
E
0
-10
-20
-30
-40
-50
-60
3.5
4.0
4.5
5.0
5.5
6.0
6.5
D
Figure 3.5: The energy δE versus ∆ for κ = 1 and with TD3 = 1.
of the configuration by taking the zero temperature limit of F = M − T S. This is also
what is shown in Eq. (3.58).
3.4.4
Heuristic picture away from equilibrium
We propose here a heuristic picture of the dynamics of the phases. In the above, we
have considered static configurations in thermal and mechanical equilibrium. However,
it is interesting to ask how the different phases we have found behave when allowing for
dynamics.
We consider here what happens for a fixed temperature T̄ = 0.4 and for different choices
of values of the separation distance ∆.10 Thus, we impose the boundary conditions on the
system in the form of the separation distance ∆ and temperature T̄ , as measured sufficiently
far away from the wormhole. The equilibrium solutions for T̄ = 0.4 are displayed in Figures
3.1 and 3.2 with free energy diagram given in Figure 3.4. We now fix ∆ = 5. This means
we have three available equilibrium configurations, corresponding to three different values
of the minimal radius σ0 . We illustrate this in the left part of Figure 3.6.
Now, one can imagine going away from equilibrium to consider configurations with
different values of σ0 but still ∆ = 5, just as one would be able to consider a harmonic
oscillator away from its equilibrium configuration. This amounts to having a potential for
the system for a range of σ0 values. We sketched such a possible potential V for ∆ = 5
and T̄ = 0.4 in the right part of Figure 3.6. Note that the values of the potential V at
the equilibrium points are taken to be the values of the free energy as displayed in Figure
3.4 since the free energy is related to the total action of the solution. We see now that
in this potential we have two local maxima and one local minimum. The local minimum
is, obviously, the phase with the least free energy. Thus, we expect that this phase is
stable to small perturbations. Instead for the other two phases, corresponding to the two
local maxima, a small perturbation could move the system increasingly further away from
equilibrium.
10
One could instead consider only one value of ∆ and different choices of T̄ . However, the heuristic
picture would amount to the same.
76
3.5 Thermal spike and correspondence with non-extremal string
�
V
10
-10
8
-20
6
-30
4
-40
0.0
0.5
1.0
1.5
2.0
Σ0
-50
Figure 3.6: Left side: Free energy for T̄ = 0.4 and ∆ = 5. Right side: Heuristic
depiction of potential for T̄ = 0.4 and ∆ = 5.
For the large σ0 phase, what can happen is that having a perturbation that makes σ0
smaller would tend to take the system towards the stable equilibrium solution. So, one
would presumably end up in the stable configuration. Instead, making a perturbation that
would tend to increase σ0 would result in a run away type of instability. We believe this
should be in the form of a time-dependent solution where the radius of the wormhole keeps
increasing and thus the brane-antibrane system will disappear. Thus, one could think of
this as a brane-antibrane annihilation process. This is presumably related to open string
tachyon condensation of the brane-antibrane system [104].
For the small σ0 phase, a perturbation that would increase σ0 would presumably end
up in the stable phase. Instead, a perturbation that would tend to decrease σ0 should be
such that it makes the wormhole more and more thin. We speculate that this process could
end up in annihilating the F-string flux from the branes and the end point would thus be
the system of infinitely extended flat branes and anti-branes, but without the wormhole.
Finally, one could contemplate what happens for other values of ∆. Taking ∆ = 4
would remove the local maximum for small σ0 as one can see from Figures 3.1 and 3.2.
Instead taking ∆ = 7 both the local maximum for small σ0 and the local minimum would
disappear. Thus, one is left with an unstable phase.
3.5
Thermal spike and correspondence with
non-extremal string
In this section we explore the question of whether there is a configuration of k coincident
F-strings ending on N coincident D3-branes at non-zero temperature. Unlike for the zero
temperature case, where this configuration for N = 1 is described by the infinite spike
BIon solution (see Section 1.6), our non-zero temperature analogue of the BIon solution
does not allow for an infinite spike (see Section 3.3). We begin this section with some
general considerations, and then we propose how the configuration in question is made
by matching up a non-extremal black string solution with the throat solution found in
Section 3.2. This involves finding a regime with small temperatures in which we can ignore
the presence of the D3-branes in the D3-F1 system. In this regime, the D3-F1 brane
bound state behaves very similar to a fundamental non-extremal string. We show this by
77
Chapter 3. Heating up the BIon
computing the mass and entropy densities of the D3-F1 brane bound state and comparing
it with the corresponding quantities for the non-extremal fundamental string.
3.5.1
General considerations
In Section 1.6 we reviewed the infinite spike solution of the DBI theory
z(σ) =
κ
,
σ
(3.83)
first found in [23]. The interpretation of this solution is that of k coincident straight Fstrings ending on a single D3-brane. This brane intersection happens at a right angle,
ensuring the spherical symmetry around the string. Why is this interpretation correct?
Besides that one can measure that the effective tension of the spike has the right magnitude,
the other physical requirement is that sufficiently far away from the D3-brane we should
not be able to see the effect of the D3-brane on the configuration anymore, and any
physical measurements should be consistent with having k F-strings. Naively, this could
seem impossible since for any value of z the radius of the throat of the spike is finite and
we could thus send in a very small observer inside the throat to see that there still is a
D3-brane charge present. However, we are saved by the fact that the DBI theory is not
an exact theory. It is an effective theory where one integrates out the open string scale.
Thus, if we are sufficiently far away, the radius of the spike is of the same magnitude as
the open string scale, and we cannot see “inside the spike” anymore. Therefore, at this
point the spike solution is indistinguishable from k coincident F-strings. Or, said in a
different way, since the infinite spike solution (3.83) is not valid anymore from this point
on we should match the DBI theory solution with k coincident F-strings at that point. If
k is sufficiently large, this could involve matching up the DBI solution with a supergravity
solution of k coincident F-strings. Note that at the correspondence point where we match
the DBI solution and the F-string supergravity solution we go beyond the validity of both
solutions. However, one can extrapolate the DBI solution beyond its validity because of
supersymmetry.
Turning now to the thermal configuration found in Section 3.2 we found that the
minimal radius σ0 is bounded from below (see Eq. (3.39)) and that z(σ0 ) ≤ ∆max /2 when
we are not in the “thick throat” branch that starts at ∆min and continues with increasing
σ0 and ∆ such that ∆ ∝ σ0 for large σ0 (see Section 3.3). Obviously this means that
we do not have any immediate generalization of the infinite spike solution (3.83) in the
non-zero temperature case since we do not have z(σ0 ) = ∞ away from the “thick throat”
branch. However, in analogy with the zero temperature case, we shall argue below that it
is possible to match the blackfold type of solution of Section 3.2, in a certain regime, with
a supergravity solution of k coincident F-strings at non-zero temperature, and thereby to
construct a configuration that can be interpreted as k F-strings ending on N D3-branes.
Just as in the zero temperature case, this involves matching up the two different solutions,
the blackfold solution and the F-string supergravity solution, at a correspondence point
where we have to go beyond the validity of both solutions. We argue for the matching
here by using the blackfold solution beyond its validity to show that its physical behavior
is very close to that of k non-extremal F-strings, and the slight numerical differences can
be explained by the fact that we do not have supersymmetry to protect us. We illustrate
the matching of the blackfold solution and non-extremal black F-string solution at the
correspondence point in Figure 3.7.
78
3.5 Thermal spike and correspondence with non-extremal string
z direction
Correspondence point
6
Non-extremal black F-strings
D3-F1 blackfold configuration
Figure 3.7: Illustration of the matching of the D3-F1 blackfold configuration and the
non-extremal black F-strings at the correspondence point.
3.5.2
Correspondence point for matching of throat to F-strings
In Section 3.2 we found a solution using the blackfold approach for a spherically symmetric
configuration of N D3-branes with a throat supported by an F-string flux such that the
F-string charge measured over each spherical surface is k. The throat ends in a minimal
two-sphere of radius σ0 . In this section we find a regime of the solution of Section 3.2 in
which it behaves like k non-extremal F-strings at the end of the throat, i.e. at σ = σ0 .11
This enables us to match the D3-F1 blackfold configuration to k non-extremal black Fstrings. As we explain below, this means that the end of the throat σ = σ0 serves as the
correspondence point between the two different solution, as illustrated in Figure 3.7.
Since we are aiming to generalize the extremal infinite spike solution of [23] to non-zero
temperature, as discussed above in Section 3.5.1, we approach this problem in the low
temperature limit in which we are close to extremality. We begin therefore this section by
examining the thermodynamics of the supergravity solution of k non-extremal F-strings at
low temperature, and subsequently find a corresponding regime of the non-extremal D3-F1
solution of Section 3.2.
Non-extremal black F-strings at low temperature
The supergravity solution for k coincident non-extremal black F-strings lying along the z
direction is
ds2 = H −1 (−f dt2 + dz 2 ) + f −1 dr2 + r2 dΩ27
e2φ = H −1 , B0z = H −1 − 1 , H = 1 +
r06 sinh2 ᾱ
r6
, f = 1 − 06 ,
6
r
r
(3.84)
here written in the string frame. This supergravity solution is a good description of k
non-extremal F-strings for k 1 and gs2 k 1.12 From this the mass density along the z
11
In order to compare our findings with the results for a non extremal fundamental string we work in
this section in terms of the quantities (T, N, k), TF1 and TD3 instead of κ and T̄ .
12
That gs2 k 1 follows from demanding that the curvature length scale (GTF1 k)1/6 ∼ (gs2 k)1/6 `s for the
supergravity solution of k coincident extremal F-strings is larger than the string scale.
79
Chapter 3. Heating up the BIon
direction can be found using Ref. [95]
13
2 (1 + 6 cosh2 ᾱ)
4 (cosh2 ᾱ − 1)
35 TD3
312 TD3
dMF1
2
=
,
k
=
,
2 T 12 cosh10 ᾱ
dz
27 π 3 T 6 cosh6 ᾱ
212 π 6 TF1
(3.85)
where we eliminated r0 in favor of the temperature T and the second equation follows from
charge quantization. Combining the two equations one can eliminate ᾱ and write the mass
density in terms of k, T , TD3 and TF1 . Note that in solving the second equation of (3.85)
for ᾱ we choose the branch connected to the extremal solution. For small temperatures
one can expand the mass density as follows
2 k2 π3 T 6
dMF1
16 (TF1 k π)3/2 T 3 40TF1
9
+
= TF1 k +
+
O
T
.
2
dz
81TD3
729TD3
(3.86)
This is the mass density of k coincident F-strings for low temperatures so that the F-strings
are close to extremality as found in the regime k 1. We see the leading part as expected
is given by k times the extremal string tension TF1 . We also read from this expression that
being near extremality means that we are in the regime
√
TF1 k T 3
1.
(3.87)
TD3
In terms of the illustration in Figure 3.7 we can say that (3.86) is the mass density in the
z direction for the non-extremal black F-strings in the upper part of the drawing, away
from the correspondence point.
Finding matching regime in the thermal D3-F1 configuration
Having found the low temperature behavior (3.86) of the mass density of k F-strings at
low temperature near extremality we now look for a regime of the thermal D3-F1 blackfold
configuration of Section 3.2 in which its mass density is similar at the correspondence point
at the end of the throat σ = σ0 . With such a regime in hand, we can match the D3-F1
blackfold configuration and the non-extremal F-string supergravity solution, as explained
above in Section 3.5.1.
Using (3.28) along with the D3-F1 blackfold configuration given by (3.46) and (3.42)
we can compute the mass density14
2T 2 F (σ) 2 4 cosh2 α + 1
dM
= D3
σ
.
dz
πT 4 F (σ0 )
cosh4 α
(3.88)
where, as in Section 1.6 for the extremal case, we divided by the derivative of the solution,
z 0 (σ), to construct the mass density with respect to the transverse direction z(σ) to the
D3-brane. We now expand this in the regime T̄ 1 both since we want to match the
result to the small temperature expansion of the non-extremal F-strings (3.86) but also
since T̄ = 1 is the maximal temperature for the D3-brane without F-string flux and thus
a regime in which the D3-branes are suppressed with respect to the F-strings would have
to be far from T̄ = 1. Using (3.54) written in terms of (T, N, k), TF1 and TD3 we obtain
13
We write Eq. (3.85) in terms of TD3 rather than the 10D Newtons constant G. This is done using the
2
relation 16πG = (2π)7 `8s gs2 = (2π)/TD3
.
14
Note that we have a factor two less in the mass since here we do not include the mirror of the solution.
80
3.5 Thermal spike and correspondence with non-extremal string
a perturbative expansion of the mass density (3.88) in powers of the temperature T for
T̄ 1
2 k2
3 k3
p
3
3πTF1
7π 2 TF1
dM 2
4
2+
=
T
k
1
+
x
(1+x
)T
+
(1+x2 ) 2 T 8 +O(T 12 ) , (3.89)
F1
2
2
4
4
dz σ=σ0
32TD3 σ0
512TD3 σ0
where we defined the quantity
x≡
4πTD3 σ02 N
.
TF1 k
(3.90)
Note that from (3.89) it is apparent that we need a stronger requirement than T̄ 1 for
the correction terms in the expansion (3.89) to be small, namely that
T̄ 4 √
x
.
1 + x2
(3.91)
We now demand that we are in a regime in which the mass density (3.89) is dominated
by the presence of the k F-strings to (and including) order T 8 . This basically means
that any term with N dependence should be suppressed in comparison with the terms
with only k dependence. For √
the leading order term in (3.89) this clearly means that
x 1. Since we recognize TF1 k 1 + x2 as the 1/2 BPS mass density formula for a D3-F1
brane bound state we see that x 1 physically is the requirement that the F-strings
dominate in the extremal bound state. Turning to the next order in the temperature, the
T 4 term in (3.89), we see that the requirement of the dominance of the F-strings means
2 k 2 T 4 /(T 2 σ 2 ). We can write this condition as x3 T̄ 4 . We see that this
TF1 k x2 TF1
D3 0
condition implies the T 0 condition x 1 since our starting point is that T̄ 1. Going on
3 k 3 /(T 4 σ 4 ) dominates over both x2 T k and
to the order T 8 we should require that T 8 TF1
F1
D3 0
2
4
2
2
2
x T TF1 k /(TD3 σ0 ). While the latter reduces to the condition x3 T̄ 4 the first one gives
instead the condition x2 T̄ 4 and we see that this condition implies both the T 0 order
condition x 1 and the T 4 order condition x3 T̄ 4 . If we now consider the condition
(3.91) on the expansion of (3.89) we note that for x 1 this becomes T̄ 4 x which is
consistent with the x2 T̄ 4 condition. Thus, in summary, demanding that the physics of
the k F-strings should dominate over the N D3-branes at the end of the throat σ = σ0 to
(and including) order T 8 means that we should be in a regime with
x2 T̄ 4 x .
(3.92)
If one should include higher order corrections in the temperature expansion one would get
stronger requirements on x. Assuming (3.92) the mass density (3.89) now becomes
2 k2
3 k3
3πTF1
7π 2 TF1
dM 4
8
12
=
T
k
+
T
+
(3.93)
F1
2 σ2
4 σ 4 T + O(T ) ,
dz σ=σ0
32TD3
512TD3
0
0
Comparing this to the mass density for the F-strings (3.86) we see that in order for the
thermal D3-F1 configuration at σ = σ0 to behave like k F-strings near extremality we
should take σ0 to have the following dependence on the temperature
√
σ0 =
TF1 k
TD3
12
√
√
TF1 k 3
6
T s0 + s1
T + O(T ) ,
TD3
(3.94)
where s0 and s1 are numerical coefficients which can be determined by requiring that the
T 3 and T 6 terms in Eqs. (3.86) and (3.93) agree. We find the precise coefficients below.
81
Chapter 3. Heating up the BIon
Note that from the regime (3.92) we have T̄ 4 /x 1 which becomes the condition (3.87)
when we insert (3.94).
The condition (3.92) can be written equivalently as T̄ 4 x T̄ 2 . Since from (3.90)
√
we see that x ∼ σ02 /κ we can also write it as T̄ 2 σ0 / κ T̄ . This shows that we are
√
well above σmin but also below σ0 / κ ∼ T̄ 2/3 for which ∆ ∼ ∆max . If we insert (3.94) in
√
σ0 / κ T̄ this condition can be written as N TD3 /(kTF1 ) T 2 . This is consistent with
(3.87) when N 3 /k 2 gs .
Validity of the probe approximation and the correspondence point
Having found the regime (3.92) with σ0 given by (3.94) we have identified a regime in
which the D3-F1 blackfold configuration of Section 3.2 at the end of the throat σ = σ0
behaves as k non-extremal F-strings near extremality. As illustrated in Figure 3.7, the end
of the throat σ = σ0 is what we call the correspondence point. In our case we take this to
signify that it is a point in which it is possible to match the two descriptions although the
validity of the two descriptions does not overlap. One should therefore extrapolate the two
descriptions in order to match them. Above we did this by finding the regime in which the
D3-F1 blackfold description of the system looks like the non-extremal F-string description
at the correspondence point.
That the validity of the approximation of the D3-F1 blackfold configuration breaks
down when we are in the regime (3.92), (3.94) is seen as follows. In Section 3.2.4 it was
found that the probe approximation is valid provided rc σ0 where rc is the charge radius
of the D3-F1 brane bound state. Note that from (3.92) we have that x 1 which means
√
2 .
that σ0 κ. Using this one finds that the condition rc σ0 reduces to kTF1 σ06 TD3
Inserting (3.94) we then find that the blackfold approximation requires
√
kTF1 3
T 1,
TD3
(3.95)
which is seen to be exactly the opposite of the requirement (3.87). Thus, clearly we are
beyond the validity of the blackfold description. However, this is in fact consistent with
what we would like to do. If we for a moment imagine that we could reach a regime in
which both the D3-F1 blackfold description and the non-extremal F-string description are
valid, we would run in to problems. Indeed, an observer smaller than the size of the throat
at σ = σ0 would be able to see that there is a hole in the middle of the string and that
there still is a D3-brane charge present. Instead, that the blackfold regime breaks down
means that the backreaction to the system becomes strong before we reach σ = σ0 and
this can thus close off the hole in the middle of the throat and thereby also cancel out the
D3-brane charge.
Of course, the fact that we go far beyond the validity of the blackfold description also
means that it is limited what we can conclude from our approach. However, at the same
time the breakdown of the blackfold description happening precisely in the regime in which
we get a behavior similar to the F-strings is perhaps the clearest indication that our match
between the D3-F1 blackfold and the non-extremal F-strings is possible.
To find the exact configuration in which k non-extremal F-strings dissolve into N nonextremal D3-branes would be a very difficult problem. One could try to add order by order
corrections on the blackfold side, taking into account the backreaction, and at the same
time add corrections to the F-string side, although it is not clear at present how one should
82
3.5 Thermal spike and correspondence with non-extremal string
do that. Note here that even in the extremal case such a configuration has not been found,
thus as a first approach one should start with that.
Extrapolating the D3-F1 configuration
Finally, we end this section with seeing how far we can take the extrapolation of the D3-F1
blackfold configuration in making it look like k non-extremal F-strings. Note that a more
accurate calculation should take into account that one should extrapolate both descriptions
instead of only the D3-F1 blackfold description. However, we do not know how the F-string
side changes as we get closer to the D3-branes.
We already remarked above that the mass density (3.93) can be matched to (3.86) with
the precise coefficients up
including) order T 6 by fixing
√ to (and
√ √
√ s0 and s1 in (3.94). We
4
find the numbers s0 = 9 3/(16 2 π) and s1 = 43π 5/4 /(1728 6). This can be continued
to higher orders as well. However, fixing dM/dz as function of k and T does not fully
characterize the thermodynamics of the string. This requires matching another quantity
as well. We consider here the entropy density dS/dz. For k non-extremal F-strings we find
2 k2 π3 T 5
dSF1
8 (TF1 kπ)3/2 T 2 16TF1
=
+
+ O T8 ,
2
dz
27TD3
243TD3
(3.96)
in the regime (3.87). One can now play the same game as for dM/dz and fix s0 and s1
in (3.94) so that the D3-F1 blackfold configuration at σ = σ0 reproduces dSF1 /dz. As
one could expect, the values of s0 and s1 that one needs to match dM/dz and dS/dz are
different. Thus, one cannot simultaneously match the two quantities exactly. However,
this would also be highly surprising since we are beyond the validity of the blackfold
approximation, as discussed above. Nevertheless, it is interesting to compare how different
the first correction to the extremal configuration is since for the extremal configuration we
have supersymmetry and all the quantities thus match exactly. Computing s0 for the two
quantities we find
s0 | dM ' 0.69 , s0 | dS ' 0.65 .
(3.97)
dz
dz
We see that they are about 6% different. This is encouraging, as it basically means that we
could reproduce the F-string thermodynamics within 6% accuracy which is a rather good
accuracy given that we are extrapolating the blackfold description beyond its validity.
83
Chapter 4
Thermal string probes in AdS and
finite temperature Wilson loops
The thermal generalization of the BIon solution considered in the previous chapter allowed
to introduce our new method to describe thermal brane probes. Of course the same method
can be applied to other more complex and interesting configurations. In particular it would
be worth using it in the context of the AdS/CFT correspondence. In fact there are many
interesting systems whose holographic duals involve branes probing the finite temperature
AdS5 × S 5 background. Up to now the Euclidean DBI probe method was used to describe
these branes, but as we argued in Section 1.7.2 this is not always so accurate. Our new
method seems instead to be more suited to be used in the AdS/CFT correspondence at
finite temperature.
In this chapter we consider the simplest application of the new thermal brane probe
method to a relevant system in the finite temperature AdS/CFT duality, namely a thermal
string probe embedded in the AdS black hole (so finite temperature) background (1.67)(1.68). To be more specific we will consider such a system in order to give an holographic
description of Wilson loops in the finite temperature N = 4 SU (Nc ) super Yang-Mills
theory [45].
4.1
Finite temperature Wilson loops: standard
method and new method
Wilson loops are an important class of non-local observables in the study of strongly
coupled gauge theories. Their expectation values provide non-trivial information on confinement, quark screening and phase transitions and they can in particular be used to
compute the potential for a quark-antiquark pair.
In the AdS/CFT correspondence a Wilson loop is dual to a fundamental string probe
with its worldsheet extending into the bulk of Anti-de Sitter space and ending at the
location of the loop on the boundary of AdS [28, 29]. At zero temperature, the classical
Nambu-Goto (NG) action provides an effective description of the F-string probe.1 However,
for finite temperature gauge theory the F-string is probing a finite temperature space-time
such as an AdS black hole. In this case, the classical NG action does not take into account
the thermal excitations of the string that arise since one can locally regard a static string
probe as being in a heat bath.
1
This has led to an impressive weak-strong coupling interpolation of circular Wilson loops [105–107].
84
4.1 Finite temperature Wilson loops: standard method and new method
C
AdS boundary
string worldsheet
Figure 4.1: Holographic description of a Wilson loop: The string worldsheet extends
into the AdS bulk and its boundary coincide with the loop C lying in the AdS boundary.
The arguments that support this claim are the same presented in Section 1.7.2 for the
Euclidean DBI probe method, so we do not repeat them. Of course they also holds for
the NG action since the latter can be viewed as the DBI action for a 1-brane without any
U (1) field strength turned on.
Hence for an AdS black hole the classical NG action becomes an increasingly inaccurate
description as one approaches the event horizon since by the Tolman law [108] the local
temperature for a static string probe is redshifted towards infinity. Thus, one can possibly
miss not just quantitative corrections but also qualitative effects if one uses the classical
NG action.
One way to find an accurate description of an F-string probe in a finite temperature
background would be to quantize the NG action and in this way include the thermalization
of the string DOFs. Indeed, in [109] the leading quadratic correction was taken into account
for a string probing an AdS black hole. However, to find the equivalent of the NG action
for finite temperature string probes one should find an effective action that includes all
orders in the temperature [46, 47] (see also the review [110]).
In this chapter we apply and discuss a new approach to find the effective action for
an F-string probe in a finite temperature background. The new approach to string probes
parallels the one to the finite-temperature D-brane probes discussed in Chapter 3 to generalize the BIon solution to finite temperature. Indeed, in analogy to what we did for the
thermal D-branes, we shall use the supergravity (SUGRA) solution for a non-extremal Fstring to describe a finite temperature Wilson loop by employing the blackfold methods2 .
Using this approach one finds an effective description for an F-string probe at finite temperature in which the thermal excitations of the string are fully taken into account. Since
the SUGRA F-string requires having many coincident F-strings the finite-temperature Fstring probe corresponds to a Wilson loop in the k̃-symmetric product of the fundamental
representation where k̃ is the number of F-strings.3 While the classical NG action is a valid
description of a string probe at zero temperature and for k̃ = 1, gs 1 the new approach
is valid for both finite and zero temperature and in the regime 1 k̃ Nc , gs2 k̃ 1.
To illustrate the consequences of the new holographic description of finite temperature
Wilson loops we consider the specific case of rectangular Wilson loops in finite-temperature
N = 4 super Yang-Mills theory with gauge group SU (Nc ) in the large Nc limit [45]. The
expectation value of this Wilson loop gives the potential for a Q-Q̄ pair in the gauge theory,
2
Applications of the blackfold method in (A)dS backgrounds were already considered in Refs. [111,112].
We choose the “uncomfortable” notation of putting a tilde over the symbol k in order to distinguish it
from the one used in the previous Chapter.
3
85
Chapter 4. Thermal string probes in AdS and finite temperature Wilson loops
where Q corresponds to the symmetric representation of k̃ quarks. This is furthermore
related to the tension of k̃-strings.4 On the holographically dual side, the gauge theory in
question is described as a black hole in AdS5 . This is in the Poincaré patch corresponding
to the fact that the superconformal gauge theory is deconfined. The Q-Q̄ pair is dual to k̃
coincident probe F-strings attached to the locations of the two particles at the boundary
of AdS5 .
One important general motivation of studying quark-gluon plasma physics using the
AdS/CFT correspondence is that N = 4 SYM theory at finite temperature shares many
similarities with QCD, with the advantage that the former is much easier to study. Consequently, the study of Wilson loops at finite temperature and their application to the
physics of quark-antiquark pairs, has provided an impetus to study dynamical issues, such
as the energy loss of a quark moving through a strongly coupled plasma (see e.g. [114,115]).
In many instances such studies provide a very good approximation to the physics of QCD
including Debye screening in strongly coupled non-abelian plasma. For example, motivated by the observation that at weak coupling N = 4 SYM theory approximates very well
the physics of QCD if one compares the two theories for coinciding values of the Debye
mass [116–118], in [119] it was examined to which extent the similarity between the two
theories holds at strong coupling.5
Our analysis reveals various new results and we highlight the main points here. Denoting the distance between the Q-Q̄ pair by L, we find in the small LT expansion of the free
energy for the rectangular Wilson loop a new correction term to the Coulomb potential, as
compared to the first correction observed using the extremal F-string probe. In particular,
for sufficiently small temperatures the former is the leading correction to the Coulomb
force potential (see Eq. (4.35)). We also study the rectangular Wilson loop for finite values
of LT . As expected, we find that there is a phase transition to a phase with two Polyakov
loops, one for each charge, corresponding to the Debye screening of the charges. However,
we find an order 1/Nc correction to the onset of the Debye screening relative to what an
extremal F-string probe gives. In addition to these results we perform a careful analysis of
the validity of the probe approximation and show that it breaks down close to the event
horizon.
4.2
Thermal F-string probe
We saw that the blackfold approach presented in Chapter 2 is a general framework that
enables one to describe black branes in the probe approximation. The probe approximation
means that the thickness of the brane is much smaller than any of the length scales in the
background or in the embedding of the brane (apart from the length scale corresponding
to the local temperature). In the case of a thermal F-string probe we employ the SUGRA
solution of k̃ coincident black F-strings in type IIB SUGRA in 10D Minkowski space. This
solution has the string frame metric [59]
1
1
ds2 = H − 2 (−f dt2 + dx21 ) + H 2 (f −1 dr2 + r2 dΩ27 ) ,
(4.1)
4
The tension of k̃-strings has been studied in QCD both analytically as well as on the lattice (see
e.g. [113]).
5
For the numerous interesting applications, recent developments and other works, we refer the reader
to the review [18] which explains and emphasizes the important role of using AdS/CFT techniques to
interpret the physics of QCD and the phenomenology of heavy ion collisions.
86
4.2 Thermal F-string probe
where
r06
r06 sinh2 α
,
H
=
1
+
r6
r6
and with dilaton field φ and two-form Ramond-Ramond gauge fields C(2) given by
f =1−
e2φ = H , C01 = (H −1 − 1) coth α .
(4.2)
(4.3)
From this we can compute the energy-momentum tensor [95]
T00 = Ar06 (7 + 6 sinh2 α) , T11 = −Ar06 (1 + 6 sinh2 α) ,
(4.4)
with A = Ω7 /(16πG) and Ω7 is the volume of the unit 7-sphere S 7 . The temperature, the
entropy density and the charge are
Ts =
3
, s = 4πAr07 cosh α , k̃TF1 = 6Ar06 cosh α sinh α ,
2πr0 cosh α
(4.5)
with TF1 = 1/(2π`2s ). As shown in Sections 2.5.3 and 2.6.2 the action principle for a
stationary blackfold uses the free energy as the action since the first law of thermodynamics
is equivalent to the EOMs. For a SUGRA F-string probe in a general background with
redshift factor R0 one finds the free energy
Z
Z
F = dV(1) R0 (T00 − Ts s) = A dV(1) R0 r06 (1 + 6 sinh2 α) .
(4.6)
Note that for this free energy the action principle requires holding the charge k̃ fixed during
a variation.
The background we want to probe is the AdS black hole in the Poincaré patch times a
five-sphere. This is because we want to study the rectangular Wilson loop in N = 4 SYM
on S 1 × R3 , the S 1 meaning that it is at finite temperature. The metric of this background
is
ds2 =
R2
z4
2
2
2
2
−1 2
2
2
(−f
dt
+
dx
+
dy
+
dy
+
f
dz
)
+
R
dΩ
,
f
(z)
=
1
−
,
1
2
5
z2
z04
(4.7)
where R is the AdS radius, the boundary of AdS is at z = 0 and the event horizon is at
z = z0 . The background spacetime (4.7) is the same as the one presented in Section 1.5.1
in Eqs. (1.67)-(1.68), with z = R/r and z0 = R/r0 . The temperature of the black hole
as measured by an asymptotic observer is T = 1/(πz0 ). Note that the five-form RamondRamond flux will not play any role in the following. We consider now the following choice
for the embedding of the F-string probe
t = τ ≡ σ 0 , z = σ ≡ σ 1 , x = x(σ) ,
(4.8)
since we want to study a static string probe that extends between the point charge Q at the
location (x, y1 , y2 ) = (0, 0, 0) and the point charge Q̄ at the location (x, y1 , y2 ) = (L, 0, 0)
on the boundary of AdS at z = 0 (see Figure 4.2).
Furthermore, the string is located at a point on the five-sphere which on the gauge
theory side means that we are breaking the R-symmetry. The induced metric and redshift
factor for this embedding are
γab dσ a dσ b =
i
Rp
R2 h
2
−1
02
2
−f
dτ
+
f
+
x
dσ
, R0 (σ) =
f (σ)
2
σ
σ
87
(4.9)
Chapter 4. Thermal string probes in AdS and finite temperature Wilson loops
yi
Q
Q̄
0
L
x
AdS boundary
z
z0
Figure 4.2: Illustration of the string setup.
It is important to note that the redshift factor R0 induces a local temperature T = T /R0
which is the temperature that the static string probe locally is subject to, T being the
global temperature of the background space-time as measured by an asymptotic observer.
Thus imposing
3
T
Ts =
=
(4.10)
2πr0 cosh α
R0
ensures that the probe is in thermal equilibrium with the background. Taking everything
into account the free energy for the thermal F-string probe in the background (4.7) with
the ansatz for the embedding (4.8) can be obtained by plugging the redshift factor R0 (4.9)
and r0 in terms of T , according to Eq. (4.10), into (4.6) yielding
F =A
3
2πT
6 Z
dσ
p
R8
1 + 6 sinh2 α(σ)
. (4.11)
1 + f (σ)x0 (σ)2 G(σ) , G(σ) ≡ 8 f (σ)3
σ
cosh6 α(σ)
Varying with respect to x(σ) this gives the EOM
f (σ)x0 (σ)
!0
p
G(σ)
1 + f (σ)x0 (σ)2
= 0.
(4.12)
For the most part we will be considering solutions with x0 > 0 and the boundary conditions
x(0) = 0 and x0 → ∞ for σ → σa . In this case one can write the general solution as
0
x (σ) =
− 12
f (σ)2 G(σ)2
− f (σ)
.
f (σa )G(σa )2
(4.13)
The full solution which goes from (x, z) = (0, 0) to (x, z) = (L/2, σa ) and back to (x, z) =
(L, 0) is found by mirroring the solution (4.13) around x = L/2. The distance L between
the charge Q and the charge Q̄ is then
Z
σa
L=2
dσ
0
− 12
f (σ)2 G(σ)2
− f (σ)
.
f (σa )G(σa )2
(4.14)
For clarity we introduce the dimensionless parameter σ̂ = πT σ. In terms of this we find
2
LT =
π
Z
σ̂a
dσ̂
0
− 12
f (σ̂)2 H(σ̂)2
− f (σ̂)
,
f (σ̂a )H(σ̂a )2
88
(4.15)
4.2 Thermal F-string probe
with f (σ̂) = 1 − σ̂ 4 , σ̂a = πT σa ,
H(σ̂) =
f (σ̂)3 1 + 6 sinh2 α(σ̂)
25 k̃TF1
f (σ̂)3 sinh α(σ̂)
,
κ̃
≡
=
.
σ̂ 8
37 AR6
σ̂ 6 cosh5 α(σ̂)
cosh6 α(σ̂)
(4.16)
Thus, we see that LT depends only on the dimensionless quantities κ̃ and σ̂a . Note that
the equation for κ̃ (4.16), since it is derived from the equation for the SUGRA F-string
charge (4.5), enforces the charge conservation and can be used to find α(σ̂). In terms of
the gauge theory variables k̃, λ and Nc , κ̃ can be written
√
27 k̃ λ
κ̃ = 6 2 ,
(4.17)
3 Nc
using the AdS/CFT dictionary for the AdS radius R4 = λ`4s and the string coupling
4πgs = λ/Nc with λ being the ’t Hooft coupling of N = 4 SYM theory.
In addition to the above solution of the EOM (4.12) we consider in the following also
another type of solution, namely that of an F-string extending from a point at the boundary
straight down towards the horizon. This solution trivially solves (4.12) with x0 (σ) = 0. In
detail we consider the two solutions extending from the boundary at either (z, x) = (0, 0)
or (z, x) = (0, L), i.e. at the position of the Q or the Q̄ point charge.
4.2.1
Critical distance
From the equation for κ̃ in (4.16) and the fact that sinh α/ cosh5 α is bounded from above
with maximal value 24 /55/2 we see that for a given κ̃ the equation can only be satisfied
provided σ̂ ≤ σ̂c with σ̂c given by
s
55/3 2 55/6 1
σ̂c2 = 1 + 14/3 κ̃ 3 − 7/3 κ̃ 3 .
(4.18)
2
2
We see from this that σ̂c < 1 which means that one reaches the critical distance σ̂c before
reaching the horizon. The physical origin for this critical distance is that we require the Fstring probe to be in thermal equilibrium with the background (4.7). The critical distance
then arises because the SUGRA F-string has a maximal temperature for a given k̃ and
since the Tolman law means that the local temperature goes to infinity as one approaches
the black hole. This is a qualitatively new effect which means that the probe description
breaks down beyond the critical distance σ̂c . We analyze in detail the validity of the probe
approximation below.
4.2.2
Probe approximation
It is important to consider the validity of the probe approximation for the SUGRA F-string.
In the context of the blackfold approach, the probe approximation essentially means that
we should be able to piece the string probe together out of small pieces of SUGRA F-strings
in hot flat ten-dimensional space-time. For this to work, the local length scale of the string
probe, i.e. the “thickness” of the string, should be much smaller than the length scale of
each of the pieces of F-string in hot flat space. Since we want to consider the branch of
the SUGRA F-string connected to the extremal F-string the thickness of the F-string is
the charge radius rc = r0 (cosh α sinh α)1/6 , as one can see following considerations similar
89
Chapter 4. Thermal string probes in AdS and finite temperature Wilson loops
to the ones presented in Section 3.2.4 for the BIon system. Using the formulas above, one
finds easily that rc ∝ κ̃1/6 R. Looking at the AdS black hole background (4.7) we find that
R and z0 ∝ 1/T are the two length scales in the metric. Since we need to require that
the sizes of the pieces of F-string should be smaller than the length scales of the metric we
need that rc R and rc 1/T . This gives the conditions
1
κ̃ 1 , RT κ̃− 6 .
(4.19)
Given that κ̃ 1, the condition RT κ̃−1/6 is easily fulfilled as it just gives a rather
weak upper bound on how high the asymptotic temperature T can be. As an immediate
consequence of the condition κ̃ 1 we see that the critical distance σ̂c given in Eq. (4.18)
- for which the probe reaches the maximal temperature and beyond which the probe
description in terms of a SUGRA F-string description breaks down - is very close to the
horizon: 1 − σ̂c ∝ κ̃1/3 for small κ̃.
As stated earlier, the regime of validity of the SUGRA F-string is 1 k̃ Nc and
2
λ k̃ Nc2 . Instead, the probe approximation requires κ̃ 1. We see that this is consistent
with the regime of validity of the SUGRA F-string provided that λ Nc2 . This translates
to gs Nc which is trivially satisfied since we assume weak string coupling gs 1.
The conditions (4.19) are not sufficient to ensure the validity of the probe approximation. Indeed, our analysis of the length scales of the background (4.7) is only valid away
from the near horizon region z0 − σ z0 . In the near horizon region z0 − σ z0 further
analysis is required. A relevant quantity to consider is the local temperature T = T /R0 .
In order for the probe approximation to be valid we need that the local temperature varies
over sufficiently large length scales such that we can regard the probe locally as a SUGRA
F-string in hot flat space of temperature T . Thus, we need in particular that
Ξ≡
r T0
p c
1,
T 1 + x0 (σ)2
(4.20)
i.e. that the variation of the local temperature is small over the length scale of the F-string
probe. For the straight string solution in which x0 = 0 we compute
rc T 0
rc
2rc (πT σ)4
= √ +
.
T
R f
Rf 3/2
(4.21)
As a check we can see that for σ z0 the condition rc T 0 /T 1 reduces to rc R.
Considering instead the near horizon region z0 − σ z0 we find in the κ̃ → 0 limit
that rc T 0 /T 1 requires z0 − σ κ̃1/9 z0 . Thus, when we reach z0 − σ ∼ κ̃1/9 z0 , the
probe approximation breaks down. In particular we notice that for σ ' σc the probe
approximation is not valid, indeed rc T 0 /T ∼ κ̃−1/3 .
Let us see now which limitation the condition (4.20) imposes for the connected string
configuration. In the near horizon region, assuming that x0 1, we can write
√
1
σ̂a − σ̂
6
Ξ ∼ κ̃
.
(4.22)
(1 − σ̂)3/2
Taking σ̂a ' σ̂c we find, in the small κ̃ limit, that Ξ ∼ κ̃1/6 /(1 − σ̂). The condition Ξ 1
is then equivalent to z0 − σ κ̃1/6 z0 which is a less severe bound with respect to the
one found for the straight string solution. However in both cases the probe approximation
90
4.3 Physics of the rectangular Wilson loop
breaks down before we reach the critical distance σ = σc where the local temperature
reaches the maximal possible temperature of the SUGRA F-string.
Finally, we should also consider the extrinsic curvature of the solution (4.13). Since we
already took into account the variation of the background, the easiest way to analyze the
extrinsic curvature is to neglect the derivatives of the metric. Doing this we can write the
length scale of the extrinsic curvature as
Lext (σ) =
R(1 + f x0 2 )3/2
.
σf |x00 |
(4.23)
It is not hard to see that the extrinsic curvature is maximal at the turning point σ = σa
where we find
0
f (σa ) 2G0 (σa ) −1
2R
.
(4.24)
Lext (σa ) = p
+
G(σa ) σa f (σa ) f (σa )
Analyzing this for σ z0 we find Lext (σa ) ∼ R which means that the probe approximation
is valid as long as rc R. Considering instead σ ' σc we find Lext ∼ rc , thus the
probe approximation breaks down at this point, hence we should require z0 − σ κ̃1/3 z0 .
However, this is already guaranteed by the stronger condition z0 − σ κ̃1/9 z0 for the
straight string solution or z0 − σ κ̃1/6 z0 for the connected string solution which we
found above.
4.2.3
Small κ̃ expansion of solution
Since by (4.19) we need to require κ̃ 1 it is useful to find the solution (4.13) to the EOM
in this limit. We first record that α(σ̂) from (4.16) is
√
3
(1 − σ̂ 4 ) 2
1
5 σ̂ 3 κ̃
√
cosh α =
− −
+ O(κ̃) .
4 32 (1 − σ̂ 4 ) 23
σ̂ 3 κ̃
2
(4.25)
Using this in (4.13) we find

3
√ σ̂ 7 1−σ̂4 2 − σ̂ 3 σ̂ 4
a
κ̃ a 1−σ̂a4


p
√
+ O(κ̃) .(4.26)
1 −
4
4
4
4
4
4
3
1 − σ̂ (σ̂a − σ̂ )
(1 − σ̂ )(σ̂a − σ̂ )

x0 = πT
dx
dσ̂
=
σ̂ 2
p
1 − σ̂a4
We use these formulas below to analyze both the small κ̃ limit as well as the small LT
limit.
4.3
Physics of the rectangular Wilson loop
We now record our results for a Q-Q̄ pair in N = 4 SYM at temperature T using the new
F-string probe method described above. Our setup is that we have a point charge Q in
the k̃-symmetric product of the fundamental representation located at (z, x) = (0, 0) and
the opposite charge Q̄ in the k̃-symmetric product of the anti-fundamental representation
located at (z, x) = (0, L). Between the pair of charges we have k̃ coincident F-strings
probing the AdS black hole background (4.7). We have illustrated this in Figure 4.3.
91
Chapter 4. Thermal string probes in AdS and finite temperature Wilson loops
0
0
L
Q
Q̄
x
0
boundary
0
L
Q
Q̄
x
boundary
σa
σc
critical distance
σc
critical distance
z0
horizon
z0
horizon
∞
∞
z
z
Figure 4.3: Left side: F-strings stretched between the Q-Q̄ pair corresponding to a
rectangular Wilson loop. Right side: Two straight strings stretching from the two charges
to the event horizon corresponding to two Polyakov loops.
4.3.1
Regularized free energy
From (4.11) we can write the free energy of the string extended between the Q and Q̄
charges as
Z σ̂a
q
√
dσ̂
1
F = λk̃T
(1 − X) 1 + f x0 2 , X ≡ 1 − tanh α −
.
(4.27)
2
σ̂
6 cosh α sinh α
0
We see that this is divergent near σ = 0 corresponding to an ultraviolet divergence in
the gauge theory. The interpretation of this is that (4.27) is the “bare” free energy of
the rectangular Wilson loop. We thus need a prescription to compute the regularized free
energy. The physically most appealing one is to compare the free energy of the rectangular
Wilson loop to that of two Polyakov loops, one for each of the point charges Q and Q̄.
In the gauge theory a Polyakov loop is a Wilson line along the thermal circle. For
the charge Q (Q̄) the Polyakov loop is in the k̃-symmetric product of the fundamental
representation (anti-fundamental representation). Seen from the string side, the Polyakov
loops corresponds to two strings stretching from the two charges in a direct line towards
the event horizon with x = 0 and x = L (as shown in Figure 4.3). This is trivially a
solution of the EOM (4.12) since x0 = 0.
The prescription is thus to subtract the “bare” free energy of the two Polyakov loops
since the ultraviolet divergences are the same in this case. To do this in a controlled way,
we introduce an infrared cutoff at z = σcut near the event horizon. Note that σcut ≤ σc .
The “bare” free energy we should subtract from (4.27) is
Z σ̂cut
√
dσ̂
Fsub = λk̃T
(1 − X) .
(4.28)
σ̂ 2
0
The resulting difference ∆F = F −Fsub between the free energies of the rectangular Wilson
loop and the two Polyakov loops takes the form
∆F = Floop − 2Fcharge ,
92
(4.29)
4.3 Physics of the rectangular Wilson loop
with
Floop (T, L, k̃, λ) =
√
λk̃T
1
− +
σ̂a
Z
σ̂a
0
!
p
(1 − X) 1 + f x0 2 − 1
dσ̂
,
σ̂ 2
1√
λk̃T
Fcharge (T, k̃, λ, σcut ) = −
2
1
σ̂cut
Z
+
0
σ̂cut
X
dσ̂ 2 ,
σ̂
(4.30)
(4.31)
where Floop is interpreted as the regularized free energy of the rectangular Wilson loop
and Fcharge the regularized free energy for each of the Polyakov loops, respectively. Notice
that we used the dependence of σa (L, T ) and σcut to separate the contributions for the two
types of free energies. We see that substituting σa with σcut in (4.30), putting x0 = 0 and
dividing with two gives (4.31), which means that the two free energies (4.30) and (4.31)
can be said to be regularized in the same way.
With respect to the Polyakov loop free energy Fcharge we can only use it as the regularized free energy if we can remove the infrared cutoff. This can be done in the κ̃ → 0
limit where σ̂c → 1. Letting σcut asymptote to σc we find the free energy
Fcharge (T, k̃, λ) = −
1√
λk̃T .
2
(4.32)
Note in particular that by numerical analysis we find that
Z
lim
κ̃→0 0
σ̂c
dσ̂
X(σ̂)
= 0.
σ̂ 2
(4.33)
We notice that the free energy (4.32) does not depend on how σcut asymptotes to σc .
This is encouraging also in view of the fact that the probe approximation is not valid
close to σc . While (4.32) is found in the κ̃ → 0 limit we claim that it is also valid for
small non-zero κ̃. Indeed, from
√ (4.31) we see that the free energy in general should be
of the form Fcharge = −a(κ̃) λk̃T /2 where a(κ̃) is a constant
that depends on κ̃. Using
√
dFcharge = −Scharge dT we find the entropy Scharge = a(κ̃) λk̃/2. Thus, a dependence on
κ̃ in a(κ̃) would mean that the entropy as computed from an extremal F-string probe,
corresponding to the κ̃ = 0 case, and our thermal F-string probe should be different when
applied at zero temperature T = 0, which obviously does not make sense. Moreover, one
would expect an entropy that is extensive in the number of F-strings k̃ at zero temperature,
which also would require that a(κ̃) has no dependence on κ̃. Hence we conclude that (4.32)
is also valid for κ̃ > 0. This means that when using ∆F = Floop −2Fcharge below to compare
the free energies of the rectangular Wilson loop with that of the two Polyakov loop we use
(4.30) to find the free energy of the rectangular Wilson loop and (4.32) for the free energy
of the Polyakov loop.
4.3.2
Free energy of rectangular Wilson loop for small LT
We begin by exploring the length L between the Q and Q̄ charges for small σ̂a = πT σa
using the general result (4.26) for κ̃ 1. We find
!
√
√
√
2 2π
2π
1 √ 4
2 2π 5
7
LT =
κ̃σ̂a −
(4.34)
1 2 σ̂a +
1 2 − 6
1 2 σ̂a + O(σ̂a ) .
Γ( 4 )
3Γ( 4 )
5Γ( 4 )
93
Chapter 4. Thermal string probes in AdS and finite temperature Wilson loops
We see that small LT is equivalent to small σ̂a . The leading term is the same as for an
extremal F-string probing the zero temperature AdS background.
√ For the higher order
terms we notice here the appearance of the term proportional to κ̃σ̂a4 in the expansion.
In general the expansion of LT has terms of the form κ̃m/2 σ̂a3m+4n+1 . The terms with
κ̃ mark an important departure from the results obtained using the extremal F-string to
probe the AdS black hole background (4.7), which here would correspond to setting κ̃ = 0,
thus ignoring the thermal excitations of the F-string.
Considering now the regularized free energy of the rectangular Wilson loop (4.30) we
find using (4.25), (4.26) and (4.34) the expansion
!
√
Γ( 41 )4 √
3Γ( 41 )4
λk̃
4π 2
3
4
Floop = −
+
κ̃(LT ) +
(LT ) + · · ·
(4.35)
L
96
160
Γ( 14 )4
for LT 1. Clearly, the leading term is the well-known Coulomb force potential found
by probing AdS5 × S 5 in the Poincaré patch with an extremal F-string [28]. Considering
the higher order terms in the free energy (4.35) we see that for κ̃ = 0 we regain the results
found previously in the literature by using an extremal F-string probe
in the AdS black hole
√
background (4.7). Thus, the higher order term suppressed as κ̃(LT )3 compared to the
leading term in (4.35) is a new term that appears as consequence of including the thermal
excitations of the F-string probe, and demanding that the F-string probe is in thermal
equilibrium with the background.
We can now compare this to the term suppressed as
√
(LT )4 in (4.35). Since κ̃ ∝ k̃ λ/Nc2 we see that the new term is the dominant correction
to the Coulomb force potential provided
p
k̃λ1/4
LT .
(4.36)
Nc
Thus, we see that for sufficiently small temperatures the leading correction to the Coulomb
force potential can only be seen by including the thermal excitations of the F-string probe.
This is a very striking consequence of our new thermal F-string probe technique which
means that one misses important physical effects by ignoring the thermal excitations of the
F-string probe and merely using the extremal F-string as probe of a thermal background.
4.3.3
Finite LT and Debye screening of charges
We begin by describing the space of solutions for our static F-string probe extended between
Q and Q̄ as obtained from the general solution (4.13) of the EOM (4.12) (see Figure 4.3
for an illustration of this configuration). Given a value of the charge parameter κ̃, the
asymptotically measured temperature T and the distance L between Q and Q̄ one can
examine whether (4.13) gives rise to a solution. As we now describe, there can be either
zero, one or two solutions available given values of κ̃ and LT . For a particular solution,
one has a particular value of σa which measures the closest proximity of the F-string probe
to the black hole horizon in the background (4.7), namely at the point (z, x) = (σa , L/2).
Fixing the value of κ̃, a convenient way to examine the available solutions is to turn things
around and ask for the available solutions given instead T and σa , or, more precisely, the
dimensionless quantity σ̂a = πT σa , which measures the location of the closest proximity of
the F-string probe in a rescaled coordinate system where the F-string touching the event
horizon would correspond to σ̂a = 1. Given σ̂a , LT is found from (4.15). One can now
94
4.3 Physics of the rectangular Wilson loop
LT
0.25
0.20
0.15
0.10
0.05
σ̂a
0.2
0.4
0.6
0.8
1.0
Figure 4.4: LT as a function of σ̂a for various values of κ̃: κ̃ = 0.01 (blue line), κ̃ = 0.001
(red line), κ̃ = 0.0001 (green line). For comparison we also plot the result obtained using
the extremal probe (dashed line) [46, 47] (see Eq. (4.37)).
plot LT as a function of σ̂a for various values of the charge parameter κ̃. This is done
numerically in Figure 4.4.
From Figure 4.4 we see that for a given κ̃ there exists solutions in the range from σ̂a = 0
to σ̂a = σ̂c (κ̃). Here σ̂c (κ̃) is given by (4.18) and is the critical distance from the event
horizon for which the F-string probe reaches the maximal temperature (note that for small
κ̃ the probe approximation breaks down before reaching σ̂c (κ̃)). For a given κ̃ we write
(LT )c as the value of LT for the solution with σ̂a = σ̂c (κ̃). We notice that the quantity
LT always has a maximum value (LT )max for a given value of κ̃. We denote the value of
σ̂a for which this maximum is attained by σ̂0,max .
If we consider what happens for given values of κ̃ and LT , we see that for LT < (LT )c
we find one solution. Instead in the range (LT )c ≤ LT < (LT )max we have two branches of
solutions. Then for LT = (LT )max we reach the maximal possible value of LT for a given
value of κ̃. Thus, for LT > (LT )max we do not have any available solution corresponding
to the F-string probe stretching between Q and Q̄.
For comparison we have also plotted in Figure 4.4 the result for LT obtained using the
extremal F-string probe. This corresponds to κ̃ = 0, as can be seen from (4.25) and (4.26),
and integrating (4.26) in that case one finds
√
2 2π p
1 3 5 4
4
LT |κ̃=0 =
.
(4.37)
, ; ; σ̂
2 σ̂a 1 − σ̂a 2 F1
2 4 4 a
Γ 14
This matches with the result found in Refs. [46, 47] using the extremal F-string probe in
the AdS Poincaré black hole background. LT also exhibits a maximum in this case: Its
value is Lmax T ' 0.277 and it is reached for σ̂a ' 0.85.
The thermal F-string probe induces both qualitative and quantitative differences compared to the extremal F-string probe. The qualitative difference consists in that, while for
95
Chapter 4. Thermal string probes in AdS and finite temperature Wilson loops
the extremal probe one has two solutions available for given LT , for the thermal F-string
probe there is only one solution when LT is sufficiently small. This is clear from Figure
4.4. In detail, we see that there is only one solution available for LT < (LT )c . Taking into
account the validity of the probe approximation the values of LT for which there are two
solutions are even more restricted. We also notice a clear quantitative difference in the
dependence of LT on σ̂a as well as the value
√ of (LT )max . In particular, notice from (4.26)
that the value of (LT )max receives a O( κ̃) correction for small κ̃, which is thus a 1/Nc
effect that is missed by the less accurate extremal F-string probe.
We now turn to the free energy of the Wilson loop and its comparison to that of two
Polyakov loops. As described above, the latter configuration corresponds to two straight
strings stretching towards the horizon from the charge Q and Q̄ respectively, and exists
for all values of LT . We therefore want to determine which of the phases is preferred by
computing the difference in their free energies ∆F. As we found above, ∆F = Floop −
2Fcharge with Floop given by (4.30) and Fcharge by (4.32). In case ∆F is less than zero the
Wilson loop is thermodynamically preferred in the canonical ensemble. Using the exact
solution (4.13) as well as (4.16)
we have plotted in Figures 4.5 and 4.6 for various values
√
of κ̃ the quantity L∆F/(k̃ λ) as a function of LT and σ̂0 respectively.
L∆F
√ 0.05
k̃ λ
0.05
0.10
0.15
0.20
0.25
-0.05
-0.10
-0.15
-0.20
√
Figure 4.5: The quantity L∆F/(k̃ λ), with ∆F the free energy difference, as a function
of LT for various values of κ̃: κ̃ = 0.01 (blue line), κ̃ = 0.001 (red line), κ̃ = 0.0001 (green
line). For comparison we also plot the result obtained using the extremal probe (dashed
line) [46, 47] (see Eq. (4.38)).
96
LT
4.3 Physics of the rectangular Wilson loop
L∆F
√ 0.05
k̃ λ
σ̂a
0.2
0.4
0.6
0.8
1.0
-0.05
-0.10
-0.15
-0.20
√
Figure 4.6: The quantity L∆F/(k̃ λ), with ∆F the free energy difference, as a function
of σ̂a for various values of κ̃: κ̃ = 0.01 (blue line), κ̃ = 0.001 (red line), κ̃ = 0.0001 (green
line). For comparison we also plot the result obtained using the extremal probe (dashed
line) [46, 47] (see Eq. (4.38)).
As in Fig.4.4, we have also plotted the corresponding results obtained using the extremal
F-string probe. This can be found again by substituting the κ̃ = 0 limit of (4.25) into the
expressions (4.30), (4.32), yielding for the free energy difference (4.29)
s
"
!#
Z σ̂a
√
1
σ̂ 4 − 1
1
2
∆F|κ̃=0 = λkT − + 1 +
dσ̂
−1
σ̂a
σ̂a
σ̂
σ̂ 4 − σ̂a4
0
"
√ 3/2
#
√
2π
1 − σ̂04
1 3 1 4
,
(4.38)
= λkT 1 −
, ; ; σ̂
2
2 F1
2 4 4 0
σ̂0 Γ 41
This agrees with expression of the energy of the solution found in Refs. [46, 47].
Focussing first on Figure 4.5, we see that the intersection of the curves with the horizontal axis determines the onset, denoted by (LT )|∆F =0 , where the thermodynamically
preferred configuration is that of two straight strings as illustrated in Figure 4.3 corresponding to two Polyakov loops in the gauge theory. The corresponding phase transition,
also observed for the extremal probe, can be interpreted as Debye screening of the Q-Q̄
pair. One finds that, as one moves to (small) non-zero values of κ̃, the onset moves to
higher values of LT . We thus see that using the thermal F-string probe we find that the
pair is less easily screened compared to what one can see with the less accurate extremal
F-string probe, and that this effect moreover becomes stronger as κ̃ increases. Note also
that one never reaches (LT )max as the quarks are screened before reaching this distance.
Finally, we note that the intersection of the curves with the vertical axis corresponds to
the leading Coulomb potential term given in (4.35).
Another aspect that one can see from Figure 4.5 is the free energy comparison for
the two phases of Wilson loops that are present for certain values of LT . The cusp of
the swallow tail is where one reaches (LT )max , with the lower part of the swallow tail
corresponding to 0 < σ̂a < σ̂0,max , and the upper part corresponding to σ̂0,max ≤ σ̂a ≤ σ̂c .
97
Chapter 4. Thermal string probes in AdS and finite temperature Wilson loops
The features mentioned in regard to Figure 4.5 can alternatively be seen from Figure
4.6, now as function of σ̂a . Denoting σ̂a |∆F =0 as the onset for quark screening, one may
check in this connection that σ̂a |∆F =0 < σ̂0,max < σ̂c is valid for any value of κ̃.
We thus conclude that, with respect to the free energy and in particular quark screening,
the thermal F-string probe sees quantitative effects that the less accurate extremal F-string
probe misses. In particular, for the onset of quark screening we find for small κ̃
√
(LT )|∆F =0 ' 0.240038 + 0.0379706 κ̃ ,
(4.39)
√
so a κ̃ correction term to the result obtained previously using the extremal F-string
probe.
√
Thus for a finite value of LT , the quantitative difference mainly consists in having κ̃
corrections to thepextremal probe results. From the point of view of the gauge theory this
corresponds to a k̃λ1/4 /Nc correction to the previously obtained results for the potential
between the charges, as well as to the critical value (LT )|∆F =0 of LT where the charges
becomes screened.
Finally, we note that the thermal effects observed in the system studied in this chapter,
bear some resemblance to those found in Chapter 3 for the thermal BIon. Indeed, also
in that case we found that the finite temperature system behaves qualitatively different
compared to its zero-temperature counterpart. In particular, for a given separation between
the D-brane and anti-D-brane there are either one or three phases available, while at zero
temperature there are two phases. Furthermore, analysis of the free energy of the finite
temperature generalization of the BIon shows a similar swallow tail structure as found
above.
98
Chapter 5
Thermal DBI action at weak and
strong coupling
In Chapters 3 and 4 we have presented and applied a new method to describe thermal
brane probes based on the blackfold approach. This method is valid in the strongly coupled
regime, gs N 1, gs being the string coupling and N the number of coinciding D-branes,
since it describes the thermal branes through the non-extremal black branes solution of
supergravity. As discussed in details, in this regime the D-branes backreact locally but at
a much smaller length scale than that of the background which means one can still regard
them as probe branes with an effective action.
It should be noted however that the regime gs N 1, necessary in order for this
approach to be sensible, is the opposite with respect to the one usually considered for
brane probes. Indeed the DBI action, which has been used in the literature to deal with
D-brane probes (both at zero and at finite temperature), is valid only at weak coupling,
gs 1 and N = 1, i.e. when the D-branes are not sufficiently heavy to backreact on the
background geometry. For this reason, in order to have a more consistent relation with the
wide amount of results obtained with the standard DBI approach, it would be important
to find a way to describe thermal brane probes, without neglecting the thermal excitations
on the brane while being in the same regime of validity of the DBI. This is essentially the
aim of the discussion we face here.
This chapter is devoted to the study of the effective action for D-branes in thermal
backgrounds [48]. We call this action the thermal DBI action. We consider firstly the
weakly coupled regime computing the one-loop effective action for Dp-branes with electric and magnetic fields. This provides the first correction to the DBI action for small
temperatures.
In the second part of the chapter, for the case of D3-branes, we compare the results
obtained at weak coupling with the corresponding ones at strong coupling, which can be
derived using the supergravity description for the D-branes.
5.1
One-loop finite temperature DBI action
We compute the one-loop correction to the DBI action in order to find a thermal effective
action for a non-extremal D-brane for small temperatures. We achieve this exploiting the
background field method. We expand the (supersymmetric) DBI action for a Dp-brane
around a classical “background” field configuration corresponding to constant electric and
magnetic fields on the worldvolume of the brane. According to the finite temperature
99
Chapter 5. Thermal DBI action at weak and strong coupling
field theory formalism we compute the one-loop free energy which provides the first finitetemperature correction to the DBI action.
We shall assume a static spacetime background and a static embedding for the Dpbrane in the background, with σ 0 = X 0 , so that g0µ = 0 for µ 6= 0, γ0i = 0, i = 1, ..., p
and γ00 = g00 . Moreover, we do not turn on the dilaton field, the Ramond-Ramond
fluxes or the Kalb-Ramond field. With the above assumptions we should be able to write
the general thermal DBI action in terms of the following variables: The red-shift factor
√
√
R0 = −g00 = −γ00 , the global temperature T , the space-like part of the worldvolume
metric γij , the electric field F0i and the “magnetic field” Fij .
If we introduce the locally flat frame as
γab = ea ā eb b̄ ηāb̄
(5.1)
a b
and define
p Fāb̄ = eā eb̄ Fab , then we have that the Dp-brane action should only depend on
R0 , T , det(γij ) and Fāb̄ . Thus, one can find the action by combining the case of general
Fab with γab = ηab and the case of general γij and R0 with Fab = 0.1 For this reason we
perform the computation considering a flat background and a flat induced metric.
5.1.1
D3 with electric field
The DBI action for a D3-brane embedded in 10-dimensional flat space-time in the static
gauge, including fermionic DOFs [83], reads
Z
ISD3 = −TD3
d4 σ
q
− det ηab + ∂a X I ∂b X I − 2ψ̄Γa ∂b ψ + 2π`2s Fab .
(5.2)
The integral is taken over the worldvolume coordinates σ a = (t, x, y, z), a, b, ... = 0, 1, 2, 3,
the coordinates X µ (σ a ) define the embedding and Fab = ∂a Ab − ∂b Aa is the field strength
of the U (1) gauge field Aa living on the brane. ψ is a 10-dimensional Majorana-Weyl spinor
and Γa the 10-dimensional gamma matrices. We work in the static gauge X a = σ a and so
X I , with I = 4, ..., 9, are the target space coordinates orthogonal to the brane.
We expand the fields around a classical configuration
(q)
Aa = A(cl)
a + Aa ,
I
I
X I = X(cl)
+ X(q)
,
ψ = ψ(cl) + ψ(q) .
(5.3)
(cl)
I
We take X(cl)
and ψ(cl) to be constant and Aa so as to obtain a constant electric field
E1 along one of the direction on the brane, say x. Thus the corresponding classical field
strength is


0 E1 0 0

1 
(cl)
(cl)
−E1 0 0 0 .
Fab = ∂a Ab − ∂b A(cl)
(5.4)
a =
0 0 0
2π`2s  0
0
0 0 0
1
This is valid if one neglects all the derivative terms, which is the same approximation regime of the
DBI action.
100
5.1 One-loop finite temperature DBI action
We can now expand the DBI action (5.2) as ID3 ' I0 + I2,A + I2,X + I2,ψ + ... where
h
i
I
, ψ(cl) ,
I0 = ID3 A(cl) , X(cl)
1 δ 2 ID3 (q) (q)
I2,A =
F F ,
2 δFab δFcd (cl) ab cd
δ 2 ID3
1
∂a X I ∂b X J ,
I2,X =
(q)
(q)
I
J
2 δ∂a X δ∂b X (cl)
1
δ 2 ISD3 I2,ψ = ψ̄(q)
∂a ψ(q)
2
δ ψ̄δ∂a ψ (cl)
(5.5)
and we discarded the linear terms which in fact vanish because of the EOMs.
I2,A written in terms of the components of the electric and magnetic fields E (q) and
B (q) associated to the quantum field A(q) is2
Z
I2,A ∼
h
2 2
i
2
2
2
2
2
dt d3 r −E(q)x
+ B(q)x
1 − E12 + B(q)y
+ B(q)z
− E(q)y
− E(q)z
1 − E12 ,
(5.6)
analogously we have
Z
I2,X ∼
2
I
I
dt d3 r X(q)
−∂0 + ∂12 + 1 − E12 ∂22 + ∂32 X(q)
,
(5.7)
and
Z
I2,ψ ∼
dt d3 r ψ̄(q) [− (Γ0 + Γ1 E1 ) ∂0 + (Γ1 + Γ0 E1 ) ∂1
+ 1 − E12 (Γ2 ∂2 + Γ3 ∂3 ) ψ(q)
(5.8)
In order to study the finite temperature correction to the DBI action we have to perform
a Wick rotation t → iτ and compute the partition function Z defined by
Z
E
E
E
δh
(q)
I
Z = DAa DX(q) Dψ̄(q) Dψ(q) δ(h) det
e−I2,A −I2,X −I2,ψ ,
(5.9)
δχ
(q)
I and antitaking care to impose periodic boundary conditions for the fields Aa and X(q)
periodic boundary conditions for the spinor fields ψ(q) in the τ direction with period equal
to the inverse temperature β =
1/T . h is a functional of the gauge field and its derivatives
and it is taken to be zero h A(q) , ∂A(q) = 0 in order to fix the gauge; χ is the gauge
(q)
0(q)
(q)
transformation parameter Aa → Aa = Aa + ∂a χ.
From Eqs. (5.6), (5.7) and (5.8) we see that the partition function decouples in the
following way
Z
Z
Z
E
E
E
δh
−I2,ψ
−I2,A
−I2,X
(q)
I
Z=
DAa δ(h) det
e
DX(q) e
Dψ̄(q) Dψ(q) e
= ZA ZX Zψ ,
δχ
(5.10)
and therefore we can treat the various contributions separately.
2
In the expressions for I2,A , I2,X and I2,ψ we discard the overall constant factors since they are irrelevant.
101
Chapter 5. Thermal DBI action at weak and strong coupling
(q)
Let us start considering the gauge field partition function. Choosing h = A3
(q)
(q)
(q)
A 3 → A3 + ∂ 3 χ
h = A3 = 0 ,
⇒
δh
= ∂3
δχ
(5.11)
(q)
E written in term of the gauge potential A
the Euclidean action I2,A
a becomes
E
I2,A
Z
∼
β
Z
dτ
d3 rAT MA ,
(5.12)
0
(q)
(q)
(q) T
where A = A0 , A1 , A2
and the matrix M is

−∂12 − ∂22 + ∂32

M=
∂0 ∂1
1 − E12
−∂02 − ∂22 + ∂32
2
∂0 ∂2 1 − E1
∂0 ∂2 1 − E12
∂0 ∂1
1 − E12
2
∂1 ∂2 1 − E1


∂1 ∂2 1 − E12
.
2
2
2
2
2
− 1 − E1 ∂0 + ∂1 + ∂3 1 − E1
(5.13)
The partition function is simply given by
ZA = (det ∂3 ) (det M)−1/2 .
(5.14)
The determinant of M can be computed by substituting each entry in the matrix by
the corresponding eigenvalue and then by taking the infinite product over all the possible
eigenvalues, namely
det M =
∞
Y Y
k n=−∞
2
2πn
2πn
2
k1 + k22 + k32 1 − E12
−
k
−
k
1
−
E
1
2
1
β
β
2
2πn
2πn
2
2
2
2
−
k
+
k
+
k
1
−
E
−k
k
1
−
E
1
1 2
2
3
1
1
β
β
!
2
2πn
2πn
2
2
2
2
2
2
k2 1 − E1
−k1 k2 1 − E1
1 − E1
+ k1 + k3 1 − E1 −
β
β
"
#
2
2
∞
Y Y
2πn
=
k32
+ f (k) ,
f (k) = k12 + 1 − E12 k22 + k32 .
(5.15)
β
n=−∞
k
The factor k32 simplifies when one takes into account the determinant of ∂3 coming from
the gauge fixing
Y
k3
#−1
"
∞
Y Y
2πn 2
det ∂3
k3
"
#=
=
+ f (k)
.
ZA = √
∞
β
Y Y
det M
2πn 2
k n=−∞
k3
+ f (k)
β
k n=−∞
(5.16)
Computing the infinite product over n exploiting the ζ-function regularization technique
one obtains for the free energy relative to the gauge DOFs
Z
√
2V3
d3 k 1 p
1
−β f (k)
GA = − log ZA =
β
f
(k)
+
log
1
−
e
.
(5.17)
β
β
(2π)3 2
102
5.1 One-loop finite temperature DBI action
Note that in the case of zero background electric field (E1 = 0) this exactly gives the
expected result, i.e. the free energy of a gas of photons.
Using Eq. (5.7), the scalar contribution to the partition function reads
ZX
∞
2
−3 Y Y
2
2
2
2
= det ∂0 + ∂1 + 1 − E1 ∂2 + ∂3
=
"
k n=−∞
2πn
β
#−3
2
+ f (k)
, (5.18)
3.
which is indeed equal to the cube of the gauge field partition function (5.16), ZX = ZA
Thus the scalar free energy is GX = 3GA .
For the fermonic contribution to the partition function one gets
Zψ =
∞
Y Y
k n=−∞
"
π(2n + 1)
β
#4
2
+ f (k)
.
(5.19)
The infinite product over n can be computed again by means of the ζ-function regularization technique yielding the free energy for the fermionic fields
8V3
Gψ = −
β
Z
√
d3 k 1 p
−β f (k)
β f (k) + log 1 + e
.
(2π)3 2
Putting the three contributions GA , GX and Gψ together the term 21 β
we get that the total free energy of the system is
8V3
G1 = GA + GX + Gψ =
β
Z
(5.20)
p
f (k) cancels and
i
√
√
d3 k h −β f (k)
−β f (k)
−
log
1
+
e
(5.21)
log
1
−
e
(2π)3
The integral can be computed expanding the sum of the logarithms as
log(1 − x) − log(1 + x) = −2
∞
X
x2n+1
,
2n + 1
(5.22)
n=0
p
so that in cylindric coordinates (q = k22 + k32 ) the integration is straightforward and
yields
q
h
i
2 + 1 − E 2 q2
Z ∞
Z ∞
∞
exp
−(2n
+
1)β
k
X
1
1
8V3
G1 = 2
dk1
dq q
π β 0
2n + 1
0
n=0
Z ∞
∞
X
8V3
1
1 −nβk1
=− 2 2
dk1 (2n + 1)k1 +
e
(5.23)
β
π β 1 − E12 n=0 (2n + 1)3 0
∞
X
16V3
1
T 4 V3
1
T 4 π 2 V3
,
=− 2 4
=
−
ζ
4,
=
−
H
2
2
4
2
π β 1 − E1 n=0 (2n + 1)
π 2 1 − E1
6 1 − E12
P
−s and we used
in which ζH is the Hurwitz zeta function defined as ζH (s, a) = ∞
n=0 (n + a)
ζH (4, 1/2) = π 4 /6.
G1 provides the first finite temperature correction to the DBI action. Generalizing this
√
to a non-flat induced metric and introducing the red-shift factor R0 = −γ00 we can
103
Chapter 5. Thermal DBI action at weak and strong coupling
straightforwardly construct the thermal DBI action for a D3-brane with an electric field
up to the order T 4 in a small temperature expansion:
Z
ID3 = −TD3
Z
= −TD3
"
q
q
2
d σR0 det (γij ) 1 − E1 1 −
#
T 4π2
4
6R04 TD3 1 − E12
"
p
d σ − det(γab + 2π`2s Fab ) 1 −
3/2 + . . .
T 4 π 2 γ 3/2
4
6R04 TD3 [− det(γab + 2π`2s Fab )]3/2
#
+ ...
,
(5.24)
where γ = − det γab .
5.1.2
D-branes with electric and magnetic fields
We now repeat the calculation for a D3-brane with both electric and magnetic fields. Then
we extend the result to a generic Dp-brane.
D3-brane
We start from the supersymmetric DBI action (5.2) for a D3-brane and again we expand
the fields around a classical configuration according to the Eqs. (5.6), (5.7) and (5.8) but
this time the classical gauge field is such that

(cl)
(cl)
Fab = ∂a Ab
− ∂b A(cl)
a

0
E1
0
0
1 
0
B3
0
−E1
.
=

2
0 −B3
0 B1 
2π`s
0
0 −B1 0
(5.25)
This is the most general form for a constant gauge field on the brane: We choose the frame
such that the electric field is directed along the x axis and the magnetic field lies in the xz
plane
E = (E1 , 0, 0) , B = (B1 , 0, B3 ) .
(5.26)
Following the same steps as before we get for the one-loop free energy
T 4 π 2 V3 1 − E 2 + B 2 − (E · B)2
T 4 π 2 V3 1 − E12 1 + B12 + B32
=
−
G1 = −
2
2
6
6
1 − E12
1 − E2
#
"
T 4 π 2 V3 1 + B 2
(E × B)2
=−
+
2 .
6
1 − E2
1 − E2
(5.27)
The thermal corrected DBI action is therefore
"
#
p
Z
4 π 2 γ 3/2 − det(γ + 2π`2 F )
p
T
ab
s ab
ID3 = −TD3 d4 σ − det(γab + 2π`2s Fab ) 1 −
+ ... .
6R04 TD3 [− det(γab + 2π`2s Fab )]2B=0
(5.28)
104
5.1 One-loop finite temperature DBI action
Generalization to Dp-branes
Now we extend the result just found for D3-branes to the case of generic Dp-branes. The
super-DBI action has the same form of Eq. (5.2)
Z
q
ISDp = −TDp dp+1 σ − det ηab + ∂a X I ∂b X I − 2ψ̄Γa ∂b ψ + 2π`2s Fab .
(5.29)
where now the worldvolume indices a, b, ... run from 0 to p, so that we have p − 1 gauge
DOFs and the number of scalar field X I is 9 − p. Thus we always have 8 overall bosonic
DOFs which exactly matches the number of fermionic DOFs. Following exactly the same
procedure as before we expand this action around a classical field configuration. We pick
the classical scalar and fermionic fields to be constant and the classical gauge field corresponding to a constant electromagnetic field whose most general form, exploiting the
rotational invariance, can be written as


0 E1
E1 0 b1





−b1 0
1 

(cl)
(cl)
(cl)
Fab = ∂a Ab − ∂b Aa =
(5.30)

.
..

2π`2s 
.



0
bp−1 
−bp−1 0
The one-loop free energy of Dp-branes turns out to be
p
− det(η + 2π`2s F (cl) ) η 2
Vp T p+1 Γ (p) ζ p + 1, 12
G1,Dp = −
p
p+1 .
22(p−2) π 2 Γ p2
− det(η + 2π`2s F (cl) )bi =0 2
The thermal DBI action up to the T p+1 order can be thus written as
Z
p
IDp = − TDp dp σ − det(γab + 2π`2s Fab ) ×


p


− det(γ + 2π`2s F ) γ 2
T p+1 Γ (p) ζ p + 1, 12
1−
+
.
.
.
.
p
p+1


22(p−2) π 2 Γ p
2
Rp+1 − det(γ + 2π`2 F (cl) )
2
s
0
(5.31)
(5.32)
bi =0
It is interesting also to derive thermal correction to the EM tensor. Recalling that the EM
tensor is defined as
2 ∂IDp
ab
TDp
≡√
,
(5.33)
γ ∂γab
using (5.32) one can easily obtain the leading finite temperature correction to the EM
tensor for the Dp-brane
p−1
16Vp T p+1 Γ (p) ζ p + 1, 12
− det(γ + 2π`2s F ) γ 2
ab
T1,Dp = −
p
p+1
4p π 2 Γ p2
2
2
γ00 det(γ + 2π`s F )
bi =0
n
(5.34)
−1 ab
−1 ba
−1 ab
2
2
γ + 2π`s F
+ γ + 2π`s F
+p γ
−1 ab
−1 ba i p − 1 a0 b0
p+1h
2
2
−
γ + 2π`s F |bi =0
+ γ + 2π`s F |bi =0
−
δ δ
.
2
γ00
105
Chapter 5. Thermal DBI action at weak and strong coupling
5.2
DBI action from supergravity
The aim of this section is to compare the weakly coupled results obtained before with
the corresponding ones at strong coupling for the specific case of D3-branes. We start
by considering the case of D3-branes with an electric field. We recover the Gibbs free
energy which is the right quantity to compare with the one loop free energy computed at
weak coupling. We also show that a more convenient way to compare weakly and strongly
coupled results consists in taking into account the entropy instead of the Gibbs free energy.
Then we extend the comparison to D3-branes with both electric and magnetic fields.
We do this by considering separately the cases of parallel and orthogonal electric and
magnetic fields.
5.2.1
D3-brane with electric field
From the supergravity view the system of D3-branes with electric field corresponds to the
F1-D3 brane bound state, which we already encountered in Section 3.1. We recall the
string frame metric of this solution
√
1 H −1 2
2
2
2
2
2
ds = √
−f dt + dx1 + D(dx2 + dx3 ) + √
f dr + r2 dΩ25 ,
(5.35)
DH
D
with
r04 sinh2 α
r04
,
f
=
1
−
, D−1 = cos2 θ + sin2 θH −1 .
(5.36)
r4
r4
The dilaton, the NS-NS Kalb-Ramond two-form and the R-R gauge potentials are
H =1+
e2φ = D−1 , B01 = sin θ(H −1 − 1) coth α ,
(5.37)
C23 = tan θ(H −1 D − 1) , C0123 = cos θD(H −1 − 1) coth α .
The D3-brane current is
2
J = 2π 2 TD3
cos θr04 cosh α sinh α dt ∧ dx1 ∧ dx2 ∧ dx3 ,
(5.38)
giving the charge quantization condition for the D3-brane charge
N = 2π 2 TD3 cos θr04 cosh α sinh α .
(5.39)
From the Kalb-Ramond field Bµν we read off the corresponding world-volume field
strength3
F01 = TF1 sin θ tanh α .
(5.40)
The temperature is given by
T =
1
.
πr0 cosh α
3
(5.41)
Strictly speaking one should work with B(2) in a gauge where it is zero at the horizon, and then the
value of B(2) is the one that is read off at infinity. Since B(2) is zero at infinity this means that effectively
speaking the value of B(2) is read off as minus the value of B(2) at the horizon. Note also that B(2) +2πls2 F(2)
is the gauge invariant quantity thus one divide with 2πls2 to get F(2) .
106
5.2 DBI action from supergravity
The mass and the entropy are
M=
π2 2
T V3 r04 (5 + 4 sinh2 α) ,
2 D3
2
S = 2π 3 TD3
V3 r05 cosh α
(5.42)
from which we can compute the Helmholtz free energy F = M − T S:
F=
π2 2
T V r4 (1 + 4 sinh2 α) .
2 D3 (3) 0
(5.43)
Note that the Helmholtz free energy is not the right quantity to compare with the
effective action computed at weak coupling. Indeed F depends on T , N and k:
dF = −SdT + ωdN + µdk ,
(5.44)
where µ is essentially the electric field
Z
µ=
dV(1) F01 ,
and k denotes the number of F-strings
Z
2 2
kTF1 = 2π TD3 dV(2) sin θr04 cosh α sinh α .
(5.45)
(5.46)
However we are looking for a quantity that depends on the electric field (and not on k).
Thus we need to perform a Legendre transform on the Helmholtz free energy by subtracting
to the latter the quantity µk
2
µk = 2π 2 TD3
V(3) sin2 θr04 sinh2 α .
(5.47)
In this way we obtain the Gibbs Free energy G = M − T S − µk
G=
π2 2
T V r4 (1 + 4 cos2 θ sinh2 α) ,
2 D3 (3) 0
(5.48)
and of course it turns out that
dG = −SdT + ωdN − kdµ .
(5.49)
Thus the quantity to compare with the DBI action is the Gibbs free energy.4
The Gibbs free energy G is a function of the three parameters θ, r0 and α. Using the
above formulas one can rewrite it as a function of T , N and E1 . As an intermediate step,
we find


q
1
.
q
G = N TD3 V(3)  tanh2 α − E12 +
(5.50)
2
2
2
4 cosh α tanh α − E1
where we defined
E1 ≡
F01
.
TF1
4
(5.51)
We ignored here the issue that strictly speaking one should integrate the thermodynamic conjugate of
k along the x1 direction.
107
Chapter 5. Thermal DBI action at weak and strong coupling
We find
√
sinh α
2 3 4 π2N T 4
T̄ ≡
= cos θ
.
9
2TD3
cosh3 α
(5.52)
4 8
T̄ cosh6 α − (1 − E12 ) cosh2 α + 1 = 0 .
27
(5.53)
using this we get
Define the variables
x ≡ (1 − E12 ) cosh2 α , cos δ ≡
T̄ 4
.
(1 − E12 )3/2
(5.54)
With this we can write (5.53) as
4 cos2 δ 3
x − x + 1 = 0.
27
(5.55)
Notice however that it is not obvious from the above equations that T̄ 4 ≤ (1 − E12 )3/2 .
Thus, the question is whether this comes out of (5.53) in the sense that one can only find
valid solutions for T̄ 4 ≤ (1 − E12 )3/2 . To test this we solved (5.55) for the different values
of cos δ with the results
cos δ = 0.99
cos δ = 1
cos δ = 1.01
Roots: − 3.0269 , 1.3897 , 1.6372
Roots: − 3 , 1.5 , 1.5
Roots: − 2.9736 , 1.4868 ± 0.1212i
(5.56)
This show that for cos δ > 1 none of the three branches give valid solutions as they are
either complex or negative. Hence we can conclude that cos δ ≤ 1 and we infer the bound
T̄ 4 ≤ (1 − E12 )3/2 .
(5.57)
Thus, the presence of the electrical field lowers the maximal temperature of the brane.
But note also that for a given temperature this inequality means that the electrical field
is bounded from above
q
E1 ≤
8
1 − T̄ 3 .
(5.58)
Between the three branches of solutions of (5.55) the correct branch, when being connected
to the extremal case, is
√
3 cos 3δ + 3 sin 3δ
x=
.
(5.59)
2
cos δ
In terms of x we can write the Gibbs free energy as

r
q
1
1
.
G = N TD3 V(3) 1 − E12  1 − + q
x 4x 1 − 1
x
(5.60)
For small temperatures δ is close to π/2 and we find the expansion
√
√
3 3
1
3
2
x=
− −
cos δ −
cos2 δ + O(cos3 δ) .
2 cos δ 2
12
27
108
(5.61)
5.2 DBI action from supergravity
This allows to obtain G for small T̄
"
q
G = N TD3 V(3)
1 − E12
T̄ 4
T̄ 8
1− √
−
···
6 3(1 − E12 )3/2 54(1 − E12 )3
#
,
that we can rewrite in terms of the temperature T as
q
π4N 2
π2N
T4
T8
2
−
G = N TD3 V(3) 1 − E1 1 −
2 (1 − E 2 )3 + · · · .
8TD3 (1 − E12 )3/2 32TD3
1
(5.62)
(5.63)
Eq. (5.63) is the small temperature expansion of the Gibbs free energy at strong coupling. Let us denote the latter as Gstrong , to distinguish with the one computed at weak
coupling (5.23), accordingly denoted by G1,weak . Comparing the first finite temperature
correction G1,strong (the term ∼ T 4 ), taking N = 1, to G1,weak , given in Eq. (5.23), we find
the remarkable result
G1,strong
3
= .
(5.64)
G1,weak
4
Of course if we set the electric field E1 to zero this is nothing but the well known result
found in [86, 87] (that we already reviewed in Section 1.5.1). Nevertheless there are no a
priori reasons for which one could expect that the famous 3/4 factor extends also to the
case of D3-branes with electric field. This result is therefore very interesting, but, as we
will shortly see, we can even go further.
Comparison of the entropies
Instead of comparing the free energies at weak and strong coupling one can compare the
entropies. This has the following advantages:
1. The entropy as a quantity is independent of the ensembles unlike the free energy.
2. One needs only the leading order behavior of cosh α and sin θ for small T .
We now compute the entropy to leading order for small T at strong coupling. We begin
with the general result from the supergravity solution
2 5
S = 2π 3 V3 TD3
r0 cosh α =
We have
2 V
2TD3
3
.
π 2 T 5 cosh4 α
(5.65)
π2N T 4
sinh α
= cos θ
.
2TD3
cosh3 α
(5.66)
π4N 2T 8
6
2
2 cos2 θ cosh α − cosh α + 1 = 0 .
4TD3
(5.67)
This gives
For T → 0 we have that α → ∞ hence to leading order we can neglect the 1 in the above
third order equation for cosh2 α. Accordingly to leading order for small T we find
cosh2 α '
2TD3 cos θ
.
π2N T 4
109
(5.68)
Chapter 5. Thermal DBI action at weak and strong coupling
Plugging this into (5.65) we find
Sstrong '
π2
T3
V3 N 2 2 .
2
cos θ
(5.69)
From (5.40) and (5.51) we see that to leading order E1 = sin θ for small T hence
S1,strong =
T3
π2
.
V3 N 2
2
1 − E12
(5.70)
to leading order for small T .
This (with N = 1) should be compared to weak coupling entropy, which can be computed from G1,weak , given in Eq. (5.23), as
S1,weak = −
∂G1,weak
∂T
=
E1
2π 2
T3
.
V3
3
1 − E12
(5.71)
Thus, comparing S1,strong and S1,weak , immediately we recover the 3/4 factor between
the strongly and weakly coupled results.
5.2.2
D3-brane with parallel electric and magnetic fields
Now we take into account D3-branes with parallel electric and magnetic fields. The supergravity solution corresponding to this system is the (F1,D1)-D3 brane bound state, with
parallel F1 and D1 fluxes. This solution can be derived starting from the F1-D3 bound
state given in Eq. (5.35)-(5.36), by means of T-duality transformations and rotations.
So, let us remind the reader the Buscher rules for T-duality along a direction x (see
Ref. [80])
Bxµ
gxµ
1
e2φ
, g̃xµ =
, B̃xµ =
, e2φ̃ =
,
gxx
gxx
gxx
gxx
i
i
1 h
1 h
+
− gxµ gxν + Bxµ Bxν , B̃µν = Bµν +
Bxµ gxν − gxµ Bxν ,
gxx
gxx
g̃xx =
g̃µν = gµν
(5.72)
with µ, ν 6= x.
As we said, we start from the F1-D3 brane solution (5.35)-(5.36). We first perform a
T-duality along x2 . Since g2µ = δ2,µ g22 and B2µ = 0 this gives
1
g̃22
1
1
H2
e2φ
H2
=
= 1 , e2φ̃ =
= 3 ,
g22
g22
D2
D2
(5.73)
with all other components of gµν and Bµν unchanged. The R-R fields are
C(1) = tan θ(DH −1 − 1)dx3 , C(3) = cos θD(H −1 − 1) coth α dt ∧ dx1 ∧ dx3 .
(5.74)
We then rotate along the x2 x3 -plane
x2
x3
=
cos ϕ − sin ϕ
sin ϕ cos ϕ
110
x̂2
x̂3
,
(5.75)
5.2 DBI action from supergravity
so that the metric becomes
i
1 h
H cos2 ϕ + D sin2 ϕ ,
DH
i
1 h
ĝ33 = g̃22 sin2 ϕ + g̃33 cos2 ϕ = √
H sin2 ϕ + D cos2 ϕ ,
DH
1
ĝ23 = cos ϕ sin ϕ(−g̃22 + g̃33 ) = √
cos ϕ sin ϕ(D − H) ,
DH
ĝ22 = g̃22 cos2 ϕ + g̃33 sin2 ϕ = √
(5.76)
while the other components are left unchanged, and the R-R forms are given by
C(1) = tan θ(DH −1 − 1)(sin ϕdx̂2 + cos ϕdx̂3 ) ,
C(3) = cos θD(H
−1
(5.77)
− 1) coth α dt ∧ dx̂1 ∧ (sin ϕdx̂2 + cos ϕdx̂3 ) .
We now perform a T-duality along the x2 direction again. Using that B̂2µ = 0 this gives
the new metric
√
1
(ĝ23 )2
DH
g22 =
,
g
=
ĝ
,
g
=
ĝ
−
=
,
(5.78)
11
11
33
33
ĝ22
ĝ22
H cos2 ϕ + D sin2 ϕ
the new dilaton
e2φ =
e2φ̂
H
,
=
ĝ22
D(H cos2 ϕ + D sin2 ϕ)
(5.79)
the new Kalb-Ramond field
B01 = sin θ(H −1 − 1) coth α ,
B23 =
0
g23
cos ϕ sin ϕ(D − H)
,
=
0
g22
H cos2 ϕ + D sin2 ϕ
(5.80)
and the R-R forms
C(0) = tan θ sin ϕ(DH −1 − 1) ,
C(2) = cos θ sin ϕD(H −1 − 1) coth α dt ∧ dx1 + tan θ cos ϕ(DH −1 − 1)dx2 ∧ dx3 ,
C(4) = cos ϕ cos θD(H −1 − 1) coth α dt ∧ dx1 ∧ dx2 ∧ dx3 .
(5.81)
The above is the supergravity solution for the (F1,D1)-D3 brane bound state.
From the four-form C(4) we can derive the D3-brane current
2
J = 2π 2 TD3
cos ϕ cos θr04 cosh α sinh α dt ∧ dx1 ∧ dx2 ∧ dx3 ,
(5.82)
which gives the charge quantization condition
N = 2π 2 TD3 cos ϕ cos θr04 cosh α sinh α .
(5.83)
We read off the electric and magnetic field from the Kalb-Ramond two-form, obtaining
−1
−1
E = TF1
F01 = sin θ tanh α , B = TF1
F23 =
111
tan ϕ cos2 θ sinh2 α
.
cosh2 α cos2 θ + sin2 θ + tan2 ϕ
(5.84)
Chapter 5. Thermal DBI action at weak and strong coupling
To leading order for small T we find
E = sin θ ,
B = tan ϕ .
(5.85)
The temperature and the entropy of the solution are
T =
2T 2 V3
1
2 5
, S = 2π 3 V3 TD3
r0 cosh α = 2 5 D3 4 .
πr0 cosh α
π T cosh α
(5.86)
Thus the charge quantization constraint (5.83) becomes
sinh α
π2N T 4
= cos ϕ cos θ
,
2TD3
cosh3 α
(5.87)
π4N 2T 8
cosh6 α − cosh2 α + 1 = 0 .
2
2
2
4TD3 cos ϕ cos θ
(5.88)
which squared yields
In the T → 0 limit, since α → ∞, we can neglect the 1 in the above equation for cosh2 α.
Hence to leading order for small T we find
cosh2 α '
2TD3 cos ϕ cos θ
.
π2N T 4
(5.89)
Plugging this into (5.86) and using (5.85) we find
S1,strong =
2
T3
π2
π2
2 31 + B
V3 N 2 2
=
V
N
T
.
3
2
cos ϕ cos2 θ
2
1 − E2
(5.90)
Finally, this (with N = 1) can be compared to the weak coupling entropy, which can
be obtained as the derivative of G1,weak given in (5.27) with respect to the temperature,
after setting E × B to zero
S1,weak = −
∂G1,weak
∂T
=
E,B
2π 2
1 + B2
V3 T 3
.
3
1 − E2
(5.91)
Thus, also in presence of parallel electric and magnetic field one obtains the ratio
S1,strong /S1,weak = 3/4.
5.2.3
D3-brane with orthogonal electric and magnetic fields
At this point one is tempted to think that the 3/4 factor between strong and weak coupling
also holds in the more general case of non parallel E and B. Let us check this guess. Since
we have just checked the result for parallel E and B, it is enough to consider the case
of orthogonal E and B. This corresponds at strong coupling to the configuration of a
(F 1 ⊥ D1) k D3, which however has never been considered in the literature. For this
reason we have to construct this new supergravity solution.
112
5.2 DBI action from supergravity
Constructing a new supergravity solution
We now explicitly show how the new supergravity solution of (F1 ⊥ D1) k D3 can be
obtained from the non extremal D3-brane solution using T-dualities, rotations and boosts.
So, let us consider a black D3-brane solution
√
1
ds2 = √ [−f dt2 + dx21 + dx22 + dx23 ] + H[f −1 dr2 + r2 dΩ25 ] ,
(5.92)
H
e2φ = 1 , B(2) = 0 , C(2) = 0 , C(4) = (H −1 − 1) coth α dt ∧ dx1 ∧ dx2 ∧ dx3 , (5.93)
r04 sinh2 α
r04
,
f
=
1
−
,
r4
r4
We T-dualize along x3 to obtain a smeared black D2-brane
√
1
ds2 = √ [−f dt2 + dx21 + dx22 ] + H[dx23 + f −1 dr2 + r2 dΩ25 ] ,
H
√
e2φ = H , B(2) = 0 , C(3) = (H −1 − 1) coth α dt ∧ dx1 ∧ dx2 .
H =1+
(5.94)
(5.95)
(5.96)
x2 x3 -plane
We then rotate along the
x2
x3
=
cos ϕ − sin ϕ
sin ϕ cos ϕ
x̂2
x̂3
.
(5.97)
The components of the metric that change under such a rotation are the following
i
1 h
ĝ22 = g22 cos2 ϕ + g33 sin2 ϕ = √
cos2 ϕ + H sin2 ϕ ,
H
i
1 h 2
2
2
sin ϕ + H cos2 ϕ ,
ĝ33 = g22 sin ϕ + g33 cos ϕ = √
(5.98)
H
1
ĝ23 = cos ϕ sin ϕ(−g22 + g33 ) = √ cos ϕ sin ϕ(H − 1) ,
H
and the three-form becomes
C(3) = cos ϕ(H −1 − 1) coth α dt ∧ dx1 ∧ dx̂2 − sin ϕ(H −1 − 1) coth α dt ∧ dx1 ∧ dx̂3 . (5.99)
We subsequently perform a boost in the x̂3 direction
t̃
t
cosh η sinh η
=
,
x̃3
x̂3
sinh η cosh η
(5.100)
so that
√
1
f
H cos2 ϕ + √ sin2 ϕ sinh2 η ,
g̃tt = gtt cosh2 η + ĝ33 sinh2 η = − √ cosh2 η +
H
H
√
f
1
2
2
2
2
2
g̃33 = gtt sinh η + ĝ33 cosh η = − √ sinh η +
H cos ϕ + √ sin ϕ cosh2 η ,
H
H
H −1
1
g̃t3 = cosh η sinh η(gtt + ĝ33 ) = √
cosh η sinh η(cos2 ϕ +
),
sinh2 α
H
sinh η
g̃t2 = sinh η ĝ23 = √
cos ϕ sin ϕ(H − 1) ,
H
cosh η
g̃23 = cosh η ĝ23 = √
cos ϕ sin ϕ(H − 1) ,
H
(5.101)
113
Chapter 5. Thermal DBI action at weak and strong coupling
and
h
C(3) =(H −1 − 1) coth α cos ϕ cosh η dt̃ ∧ dx1 ∧ dx2
i
+ cos ϕ sinh η dx1 ∧ dx2 ∧ dx̃3 − sin ϕ dt̃ ∧ dx1 ∧ dx̃3 .
(5.102)
Note that in Eq. (5.101) we used that f = 1 − (H − 1)/ sinh2 α.
Finally, we T-dualize along the x̃3 direction, obtaining for the Kalb-Ramond form
1
2ϕ+
(H
−
1)
cosh
η
sinh
η
cos
g̃3t
sinh2 α
,
Bt3 = −
=
2
2
2
g̃33
f sinh η − cosh η H cos ϕ + sin2 ϕ
(5.103)
(H − 1) cosh η cos ϕ sin ϕ
g̃23
=
B23 = −
g̃33
f sinh2 η − cosh2 η H cos2 ϕ + sin2 ϕ
and for the metric
(g̃t3 )2
ĝtt ĝ33
=
,
g̃33
g̃33
(g̃23 )2
,
g22 = g̃22 −
g̃33
gtt = g̃tt −
gt2 = g̃t2 −
g33
g̃t3 g̃23
,
g̃33
(5.104)
1
=
.
g̃33
One has the following T-duality transformations for R-R-fields [120] for the case in which
Bµν = 0 before the T-duality
Cµν = C̃µν3 , Cµνρ3 = C̃µνρ −
1
g̃ C̃
.
g̃33 3[µ νρ]3
(5.105)
Thus for the R-R two-form potential we find
h
i
C(2) = (H −1 − 1) coth α − sin ϕ dt ∧ dx1 + cos ϕ sinh η dx1 ∧ dx2 ,
(5.106)
and for the R-R 4-form potential we have
Ct123 = C̃t12 −
1
(g̃3t C̃123 + g̃32 C̃t13 ) .
g̃33
(5.107)
This gives
C(4) = (H −1 − 1)D̃ coth α cos ϕ cosh ηdt ∧ dx1 ∧ dx2 ∧ dx3 ,
where
(H − 1) sinh2 η(cos2 ϕ +
D̃ = 1 −
1
)
sinh2 α
− sin2 ϕ
cosh2 η(H cos2 ϕ + sin2 ϕ) − f sinh2 η
(5.108)
.
(5.109)
Entropy for small temperatures
With the above procedure we have obtained the supergravity solution of (F1 ⊥ D1) k
D3. The quantity that we need is the entropy in the small T limit. Firstly we need the
D3-brane charge current which can be read off from the leading term of Ct123 for r → ∞.
Since D̃ → 1 we find
2 4
J = 2π 2 TD3
r0 cos ϕ cosh η cosh α sinh α dt ∧ dx1 ∧ dx2 ∧ dx3 ,
114
(5.110)
5.2 DBI action from supergravity
giving the charge quantization condition
N = 2π 2 TD3 r04 cos ϕ cosh η cosh α sinh α .
(5.111)
From the Kalb-Ramond two-form we obtain the electric and magnetic fields E and B
−1
−1
E = TF1
F03 = tanh η , B = TF1
F23 =
cos ϕ sin ϕ sinh2 α
.
cosh η(sin2 ϕ + cosh2 α cos2 ϕ)
(5.112)
To leading order for T → 0 these are given by
E = tanh η ,
so that
1 − E2 + B2 =
B=
1
,
cosh η cos2 ϕ
2
tan ϕ
,
cosh η
1
= cosh2 η ,
1 − E2
(5.113)
(5.114)
which we combine to
cosh2 η
1 − E2 + B2
=
.
(1 − E 2 )2
cos2 ϕ
The temperature and entropy of this supergravity solution are
T =
(5.115)
2 V
2TD3
1
3
2 5
. (5.116)
, S = 2π 3 V3 TD3
r0 cosh α cosh η = 2 5
πr0 cosh α cosh η
π T cosh4 α cosh4 η
Plugging r0 expressed in terms of T according to Eq. (5.116) into the charge quantization condition (5.111) we get
π2N T 4
cos ϕ sinh α
=
,
2TD3
cosh3 η cosh3 α
(5.117)
which can be rewritten in the following form
π 4 N 2 T 8 cosh6 η
cosh6 α − cosh2 α + 1 = 0 ,
2 cos2 ϕ
4TD3
(5.118)
thus giving an equation for cosh α. As usual, T → 0 corresponds to α → ∞ and so to
leading order one can neglect the 1 in (5.118). Hence to leading order for T → 0 we find
cosh2 α '
2TD3 cos ϕ
2
π N T 4 cosh3 η
.
(5.119)
This can be plugged into the expression of the entropy given in Eq. (5.116) yielding the
leading term of the entropy for small temperature
S1,strong =
2
2
π2
cosh2 η
π2
2 31 − E + B
V3 N 2 T 3
=
V
N
T
.
3
2
cos2 ϕ
2
(1 − E 2 )2
(5.120)
The corresponding entropy at weak coupling can be derived from the Gibbs free energy
(5.27) (setting E · B to zero) as follows
∂G1,weak
2π 2
1 − E2 + B2
S1,weak = −
=
V3 T 3
.
(5.121)
∂T
3
(1 − E 2 )2
E,B
The comparison between the strongly and weakly coupled entropies (with N = 1) yields
also in this case to the result S1,strong /S1,weak = 3/4. So we can conclude that the 3/4
factor between the strongly and weakly coupled regime holds in the most general case of
D3-branes with both electric and magnetic fields.
115
Conclusions
In this thesis we have presented a new approach to brane probes in thermal backgrounds,
which, conversely to the commonly employed ones, takes into account the thermal excitations of the branes DOFs induced by the temperature of the background. In fact, the
standard approaches used so far, treat such probes as if the temperature of the background does not affect their physics. For example, a D-brane probing a finite temperature
background is usually studied through the DBI action, which means that the D-brane is
considered as extremal even if it is embedded in a hot background. As discussed in Section 1.7, one can thus show that this procedure is not accurate. This motivates our search
for a new approach in which the probe is instead considered as a thermal object.
Specifically, we have proposed two methods to achieve a consistent description of thermal branes. The first one [36, 37, 45], uses the recently developed blackfold approach
[35, 38–44] (presented in Chapter 2). The second one, valid only for D-branes, consists in
deriving a thermally corrected version of the DBI action [48].
In this work we mainly focused on the blackfold-based method. It is worth emphasizing that while the blackfold approach was originally conceived and applied in connection
with the approximate analytic construction of novel black hole solutions of Einstein gravity
and supergravity in five and more dimensions, the results of this thesis illustrate that it
has a far broader range of applicability. In particular, the applications to new stationary
blackfold solutions considered at first have mainly focused on black holes with compact
horizons, and many new possible horizon topologies have been found. However, the approach is perfectly suited as well to describe the bending of black branes into other types
of geometries. Consequently, it is the appropriate starting point to describe the dynamics
of branes probing thermal backgrounds.
In Chapter 3 we used this method to examine what happens to the BIon solution
[23, 24] when one switches on the temperature [36, 37]. Since the (zero temperature) BIon
corresponds in the supergravity picture to an extremal D3-F1 probe brane system curved
in a flat space background [121] it is natural to base the thermal generalization on a nonextremal D3-F1 probe brane system curved in hot flat space. The equilibrium conditions
for such a configuration can then be computed from the blackfold equations. As shown in
Section 1.7, the latter equations are in fact the natural non-extremal generalization of the
DBI EOMs, providing an alternate (heuristic) derivation of the firmly established blackfold
method.
According to the blackfold prescription, we have presented a thermodynamic action,
given by the free energy, which yields the extrinsic blackfold equations describing the
thermal generalization of the BIon. This action reduces to the DBI Hamiltonian in the
116
Conclusions
zero temperature limit, and may hence be regarded as the finite temperature/closed string
analogue of the corresponding DBI Hamiltonian. As seen more generally in the blackfold
approach, the thermodynamic origin of the action implies that the (mechanical) extrinsic
blackfold EOMs are equivalent to requiring the first law of thermodynamics.
From the action we were able to obtain the explicit solution (3.46) for the slope of
the embedding function describing the brane profile of a thermal D3-brane configuration
with electric flux ending in a throat. We showed that there are two branches of solutions,
one connected to the extremal configuration and the other connected to the neutral black
3-brane configuration. In most of our analysis we focused on the former branch. We have
discussed the resulting finite temperature wormhole configuration of N D3-branes and
parallel anti-D3-branes separated by a distance ∆ and connected by a wormhole with Fstring charge. We found that the finite temperature system behaves qualitatively different
than its zero-temperature counterpart. In particular, for a given separation between the
D-branes and anti-D-branes, while at zero temperature there are two phases, at finite
temperature there are either one or three phases available.
Moreover, from our results of Section 3.3 it seems that for small temperature and large
enough σ0 the non-extremal BIon solution is well-approximated by the extremal BIon
solution.5 We take this to mean that in this range using the (abelianized U (1)N ) DBI
action as a probe of hot flat space could be a good approximation to our thermal D-brane
probe. It also illustrates that there are certain regimes where the two methods give different
results which means that using our new method can change the results both quantitatively
and qualitatively for certain regimes (e.g. the “thin throat” branch of the extremal solution
becomes two branches with a maximum ∆).
In Section 3.4 we discussed the thermodynamics of the three branches of solutions
found. This was done by comparing the free energy for the branches in the canonical
ensemble. In Section 3.5 we then solved the apparent non-existence of a thermal generalization of the infinite spike solution. For this we consider a different type of generalization
by matching the supergravity solution of k non-extremal strings to our thermal D3-brane
configuration with electric flux ending in a throat.
In Chapter 3 we focused exclusively on the thermal generalization for the BIon in the
case of D3-branes with electric flux. The construction however can be readily generalized
for Dp-branes by starting with the EM tensor of the non-extremal Dp-F1 brane system.
Branes probing thermal background have been used in several important applications
in the context of the AdS/CFT correspondence. This is one of the main motivations which
make the new method for the description of thermal brane probe so interesting.
Notice that our method is not confined to D-branes but can be used more generally for
all types of brane probes in thermal backgrounds. Indeed it is as well suited for M-brane
probes in M-theory, or F-string probes and NS5-brane probes in string theory. For example
in [122, 123] the method has been used to study the M2-M5 version of the BIon system.
In Chapter 4 we applied our new method to the holographic study of finite temperature Wilson loops. This has been the first application of the method in the AdS/CFT
correspondence framework. In the holographic dictionary a Wilson loop is dual to a Fstring probe ending on the loop which is located at the boundary of the AdS space. In
particular we studied a rectangular Wilson loop in finite temperature N = 4 SYM with
5
Recall that σ0 denotes the minimum wormhole radius.
117
Conclusions
gauge group SU (Nc ). This system has already been considered in the literature using an
extremal F-string probe in the AdS black hole background described by the NG action.
However using the NG action for such a string probe is inaccurate just like using the DBI
action for a D-brane probe in a thermal background. According to our method the string
probes are instead described through the non-extremal F-string solution of supergravity
and are demanded to be in equilibrium with the background. Note that the supergravity
solution, in order to exist, requires having k̃ coincident F-strings, with k̃ large. In the
dual boundary picture the k̃ coincident F-strings correspond to a charge Q which is the
symmetric representation of k̃ quarks. Thus the Wilson loop we considered is in the k̃symmetric product of the fundamental representation. From the expectation value of this
Wilson loop operator one can easily extract the potential between a Q-Q̄ pair, separated
by a distance L.
Analogously to the hot BIon case, the free energy of the supergravity F-string provides
the thermodynamic action yielding to the correct blackfold EOMs. The dual configuration
corresponding to the rectangular Wilson loop is given by a string joining the two charges.
Imposing the suitable boundary conditions for such a configuration we obtained the solution
of the EOMs, as given in Eq. (4.13).
We noted that the EOMs trivially admit also a solution describing a straight string.
From this solution one can construct an alternative bulk configuration for the Q-Q̄ pair
formed by two disjoint such straight strings starting from each charge and pointing toward
the horizon. The latter configuration corresponds to two Polyakov loops.
The results of our study confirm that in using the thermal F-string probe there are
both qualitative and quantitative effects that the less accurate extremal F-string probe
misses. In particular we showed that our results reproduce
the previous results found in
√
the literature [46, 47] in the strict limit in which κ̃ ∼ k̃ λ/Nc2 is taken to be zero. For all
the quantities considered we found corrections in κ̃ relative to what an extremal F-string
probe gives.
One of the novel effects is the new term in the free energy for the rectangular Wilson
loop, in the small LT regime, as displayed
p in Eq. (4.35). Relative to the leading Coulomb
potential, this term is proportional to k̃λ1/4 (LT )3 /Nc and for sufficiently small temperatures it is actually the dominant finite-temperature correction to the Coulomb potential.
It would be very interesting to examine whether such a correction also appears at weak ’t
Hooft coupling in N = 4 SYM at finite temperature.
In Section 4.3, studying the thermodynamics of the system, we found that it undergoes
a phase transition when the quantity ∆F changes sign. ∆F is indeed defined as the
difference between the free energy of the Wilson loop and that of two Polyakov loops (see
Eq. (4.29)). For small LT the Wilson loop configuration is thermodynamically favored,
since ∆F is negative. For larger value of LT the quantity ∆F becomes positive indicating
a transition to the phase with two Polyakov loops. This phase transition is naturally
interpreted as the Debye screening of the charges. Also for the onset of the Debye screening
we found corrections in κ̃ relative to the known result obtained using the extremal probe.
In our analysis we furthermore carefully examined the conditions of validity of the
probe approximation pointing out that it actually breaks down close to the event horizon.
It is important to mention that just as we have considered the thermal backreaction
effects in the worldvolume of the string, one can also consider the backreaction on the
background (4.7) due to the thermal radiation in the bulk. Unlike the effects we found,
which are of order 1/Nc , the bulk backreaction is of order 1/Nc2 , since it is governed by
118
Conclusions
the gravitational coupling, and is therefore subleading.
An inherent feature of the blackfold-based method to treat thermal brane probes is
that it only works in the strongly coupled regime, i.e. where one can trust the black
brane solution of supergravity through which one describes the brane. For a system of
N coincident D-branes the method is valid when gs N 1. It is worth noting that in
the standard approach commonly used in the literature, the D-brane probes are instead
considered in the opposite regime, namely N = 1 and gs 1, which is indeed the regime
of validity of the DBI.
In Chapter 5 we showed that it is possible to have a consistent description of thermal
D-brane probes, which, conversely to the blackfold one, is valid in the weakly coupled
regime. This results in a new approach to study D-brane probes in thermal backgrounds
which is more accurate than the DBI, since it takes into account the thermal excitations of
the internal DOFs of the brane, and, at the same time, works in the same regime of validity
of the DBI. This alternative approach consists in describing the brane through a thermal
version of the DBI action, which can be derived perturbatively for small temperature, by
computing its quantum corrections. We referred to this action as the “thermal DBI action
at weak coupling”. In Chapter 5 we computed the one-loop effective action of the DBI for
a generic Dp-brane with electric and magnetic fields, thus obtaining the leading correction
for small temperature to the DBI action [48].
For the specific case of D3-branes we also compared the thermal DBI action at weak
coupling with the corresponding one at strong coupling. The latter can be obtained using
the blackfold approach, by computing the Gibbs free energy. Remarkably we found that the
famous 3/4 factor between strongly and weakly coupled regimes for the D3-brane without
electric and magnetic fields actually extends to the full non-linear DBI regime. It would
be certainly worth analyzing more carefully the inherent origin of this 3/4 factor.
As showed extensively throughout the thesis, the novel approach to brane probes in
thermal backgrounds, taking into account the thermal excitations of the probe and demanding thermal equilibrium between the probe and the background, can lead to new
effects that one misses using an extremal probe. For this reason it would be important to
use this method to reconsider the several interesting studies in which an extremal brane
was used to probe a thermal background. The newly developed weakly coupled thermal
DBI action, being more strictly connected with the standard DBI approach, seems to be
particularly suited to find whether and to what extent the results found in the literature
can be improved.
To be more specific, possible applications regard the holographic study of phenomenologically valuable properties of the strongly coupled QCD. For instance, one can consider
relevant D-branes setups such as the D3/D7 systems, (see [18] and references therein), the
D4/D6 systems [7] or the Sakai-Sugimoto model [8, 9].6
Another important application concerns the thermal generalizations of the Wilson loop,
the Wilson-Polyakov loop, in high-dimensional representations considered in [32, 33], by
studying D3/D3 (symmetric representation) or D3/D5 (antisymmetric representation) systems. It would be interesting to see if using our method for the thermal D-brane probes
one can manage to solve the discrepancies between gauge theory and gravity results found
for the symmetric representation.
6
Generically, with Dp/Dq system we mean Nf Dq-brane probes in a backgound generated by a stack
of Nc coincident Dp-branes, with Nf Nc .
119
Appendix A
Geometry of embedded submanifolds
In this appendix we collect some relevant definitions and results on the geometry of submanifold embeddings.1
A.1
Extrinsic curvature
Consider the submanifold W whose embedded is specified by X µ (σ a ). The induced metric
γab on the submanifold is the pull-back of the spacetime metric onto W
γab = gµν ∂a X µ ∂b X ν ,
(A.1)
and the first fundamental tensor of the surface is
hµν = γ ab ∂a X µ ∂b X ν .
(A.2)
It can be easily checked that hµ ν actually is the projector onto directions tangent to W,
indeed
hµ ν ∂a X ν = ∂a X µ ,
hµ ν hν ρ = hµ ρ .
(A.3)
The orthogonal projection tensor ⊥µν
⊥µν ∂a X µ = 0 ,
⊥µ ν ⊥ν ρ = ⊥µ ρ .
(A.4)
can be obtained by the following decomposition of the metric
gµν = hµν + ⊥µν ,
(A.5)
Any background tensor tµ1 µ2 ... ν1 ν2 ... can be pulled-back onto a worldvolume tensor
ta1 a2 ... b1 b2 ... using the pull-back map ∂a X µ as follows
ta1 a2 ... b1 b2 ... = ∂a1 X µ1 ∂a2 X µ2 · · · ∂ b1 Xν1 ∂ b2 Xν2 · · · tµ1 µ2 ... ν1 ν2 ... ,
(A.6)
∂ b Xν = γ bc hνρ ∂c X ρ .
(A.7)
where
The extrinsic curvature tensor is defined as
Kµν ρ = hλ µ hσ ν ∇λ hρ σ = −hλ µ hσ ν ∇λ ⊥ρ σ .
1
(A.8)
This appendix derives from [35]. A more detailed treatment of the subject can be found in [124].
120
A.1 Extrinsic curvature
Using the projector property (A.3) one can straightforwardly show that the first two indices
are tangential to the worldvolume while the last is orthogonal
⊥µ ν Kσν ρ = ⊥σ ν Kνµ ρ = hν ρ Kσµ ν = 0 .
(A.9)
The trace of the extrinsic curvature gives the mean curvature vector
K ρ = hµν Kµν ρ = ∇µ hµρ .
(A.10)
The extrinsic curvature tensor can be rewritten as
Kµν ρ = hν σ ∇µ hσ ρ .
(A.11)
where ∇µ is the tangential covariant derivative
∇µ = hµ ν ∇ν .
(A.12)
Applying the tangential derivative on the second formula in (A.3) one obtains
2Kµ(νρ) = ∇µ hνρ = −∇µ ⊥νρ .
(A.13)
For any vector v tangent to W we have [124]
v µ v ν Kµν ρ = −v µ v ν ∇ν ⊥µ ρ = −v ν ∇ν (v µ ⊥µ ρ ) + ⊥ρ µ v ν ∇ν v µ
= ⊥ρ µ v̇ µ ,
(A.14)
where
v̇ µ = v ν ∇ν v µ .
(A.15)
Denoting by N any vector orthogonal to W we instead have
Nρ Kµν ρ = Nρ hν σ ∇µ hσ ρ = −hν ρ ∇µ Nρ .
(A.16)
The first two indices of the extrinsic curvature tensor are symmetric
K[µν] ρ = 0
(A.17)
as follows from the integrability of the subspaces orthogonal to ⊥µν .2
Note that even if tµ1 µ2 ... ν1 ν2 ... is a background tensor with indices parallel to W, in
general ∇µ tµ1 µ2 ... ν1 ν2 ... has both parallel and orthogonal components. The parallel projection along all indices is related to the worldvolume covariant derivative Da ta1 a2 ... b1 b2 ... as
in (A.6). This can be shown by using the equation that relates the
coefficients
connection
for the background metric and the worldvolume metric, Γρµν and bac respectively,
∂a X µ ∂b X ν hσ ρ Γρµν = ∂c X σ bac − hσ ρ ∂a ∂b X ρ .
(A.18)
In particular, the divergences of tensors are related as
hν1 µ1 · · · ∇ρ tρµ1 ... = ∂a1 X ν1 · · · Dc tca1 ... .
(A.19)
Such relations allow to dispense with the use of worldvolume coordinate tensors and derivatives in most formal manipulations. However, worldvolume coordinates are very practical
2
A proof of this can be found in the appendix of [35].
121
Appendix A. Geometry of embedded submanifolds
for explicit calculations and also allow us to highlight the distinction between intrinsic
(parallel to W) and extrinsic (orthogonal to W) equations.
Let us now consider the divergence of a totally antisymmetric tensor J (such as a
current associated to a gauge form field) parallel to the worldvolume. It is easy to show
that
⊥ρ µ1 ∇µ J µµ1 ... = 0
(A.20)
holds as an identity. This implies that the conservation equation
∇µ J µµ1 ... = 0
(A.21)
is equivalent to the worldvolume conservation equation
Da J aa1 ... = 0 ,
(A.22)
i.e. the orthogonal component of the current conservation equation (A.21) does not yield
any additional equation. For instance, for a 1-form current one has
∇µ J µ = Da J a ,
(A.23)
and continuity of charge is only meaningful as an intrinsic equation. This is in contrast to
the conservation of the stress energy tensor, where, as we saw in Chapter 2, the orthogonal
component gives independent extrinsic equations (see Eq. (2.39)).
It is also worth to derive more explicit expressions for the pull-back of the extrinsic
curvature tensor onto W in terms of X µ (σ),
Kab ρ = ∂a X µ ∂b X ν Kµν ρ = −∂a X µ ∂b X ν ∇µ ⊥ν ρ .
(A.24)
The property (A.4) implies
0 = ∂b X ν ∇ν (⊥σ ρ ∂a X σ ) = −Kab ρ + ⊥σ ρ ∂b X ν ∇ν (∂a X σ ) .
(A.25)
Expanding the covariant derivative in the last term and using ∂b X ν ∂ν = ∂b we find
Kab ρ = ⊥σ ρ ∂a ∂b X σ + Γσµν ∂a X µ ∂b X ν ,
(A.26)
which is reminiscent of the expression for the acceleration (deviation from self-parallel
transport) of a curve — indeed (A.14) makes this even more explicit. An alternative
expression with this same feature can be obtained by performing some manipulations:
⊥σ ρ ∂a ∂b X σ = ∂a ∂b X ρ − hρ σ ∂a ∂b X σ
= ∂a ∂b X ρ − bac ∂c X ρ + ∂a X µ ∂b X ν hρ σ Γσµν
ρ
µ
ν ρ
= Da ∂b X + ∂a X ∂b X h
σ
σ Γµν
(A.27)
,
where in the second line we used (A.18). Inserting the last expression into (A.26) we find
Kab ρ = Da ∂b X ρ + Γρµν ∂a X µ ∂b X ν .
122
(A.28)
A.2 Variational calculus
A.2
Variational calculus
Consider a congruence of curves with tangent vector N , that intersect W orthogonally
N µ hµν = 0 ,
N µ ⊥µν = Nν ,
(A.29)
and Lie-drag W along these curves. The only condition we impose on the congruence is
that is has to be smooth in a finite neighbourhood of W, so this realizes arbitrary smooth
deformations of the worldvolume X µ → X µ + N µ .
The Lie derivative of hµν along N is
£N hµν = N ρ ∇ρ hµν + hρν ∇µ N ρ + hµρ ∇ν N ρ ,
(A.30)
so, multiplying by hµ λ hν σ we obtain
hµ λ hν σ £N hλσ = hµ λ hν σ N ρ ∇ρ hλσ + hρν ∇µ N ρ + hµρ ∇ν N ρ .
(A.31)
The first term in the RHS is zero, since
hµ λ hν σ N ρ ∇ρ hλσ = −hµ λ hν σ N ρ ∇ρ ⊥λσ
= hµ λ ⊥λσ N ρ ∇ρ hν σ = 0 ,
(A.32)
and thus, using (A.16), we can rewrite Eq. (A.31) as
1
Nρ Kµν ρ = − hµ λ hν σ £N hλσ .
2
(A.33)
Multiplying by hµν both sides of the above equation we get
p
1
1
Nρ K ρ = − hµν £N hµν = − p £N |h| ,
2
|h|
(A.34)
where h = det hµν . These equations are the generalization of the expressions for the
extrinsic curvature of a codimension-1 surface. In particular the last one can be used to
derive the equation for a minimal-volume submanifold:
Z p
p
Vol =
|h| ⇒ δN Vol = − |h| Nρ K ρ
(A.35)
W
i.e. for variations along an arbitrary orthogonal direction N , extremal volume implies
K ρ = 0. This is of course the variational principle for Nambu-Goto-Dirac branes.
Consider now a more general functional
Z p
I=
|h| Φ
(A.36)
W
where Φ is a worldvolume function. Then
p
p
δN I = £N
|h| Φ = |h| (−Nρ K ρ Φ + N ρ ∂ρ Φ) .
(A.37)
Since N is an arbitrary orthogonal vector we have
δN I = 0
⇔
K ρ =⊥ρµ ∂µ ln Φ .
123
(A.38)
Appendix B
Hot BIon
This appendix is connected to Chapter 3 where we discuss the thermal generalization of
the BIon solution.
B.1
Analysis of branch connected to neutral
configuration
In this section we discuss some relevant limits of the solution branch (3.43) which is connected to the neutral 3-brane.
We begin by considering the solution for small temperature. In this case, we find that
in the limit of small temperatures we have
cosh2 α = 1 +
4
16
cos2 δ +
cos4 δ + O(cos6 δ) ,
27
243
(B.1)
so that after some algebra
σ2
− z (σ) = p 0 4
σ 4 − σ0
0
8 κ2 8
1+
T̄
,
27 σ04
(B.2)
with the next correction being of order T̄ 16 (1 + κ2 /σ04 )2 . The corresponding brane separation for the wormhole configuration then becomes
√
2 πΓ(5/4)
8 κ2 8
∆=
σ0 1 +
T̄
.
Γ(3/4)
27 σ04
(B.3)
It is also interesting to consider here what happens when the D3-brane charge goes to
zero, i.e. we have a neutral 3-brane with N = 0. In some sense, this can be viewed as
the opposite limit of the extremal limit connected to the first branch, which we examined
above. From (3.24) we see that as N → 0 we have that κ → ∞, so that from (3.40) we
find
T̂ 4
cos δ(σ) ≡ 2 ,
(B.4)
σ
where we have defined
T̂ 4 = T̄ 4 κ =
9π kTF1 T 4
√
.
2
16 3 TD3
124
(B.5)
B.2 ∆ expansion for small temperature
Note that in the final expression the N -dependence has canceled out, after substituting the
definitions of κ in (3.24) and T̄ in (3.37). From (B.4) we immediately read off the lower
bound on σ (and hence σ0 ) for a given temperature
σmin = T̂ 2 .
(B.6)
Substituting now (B.4) in the the relevant solution (3.43) for cosh α and using this in (3.45)
we obtain
!
2
8
σ
8
T̂
− z 0 (σ) = p 0 4 1 +
.
(B.7)
27 σ04
σ 4 − σ0
From this we then find the corresponding brane separation as
√
2 πΓ(5/4)
8 T̄ 8
.
∆=
σ0 1 +
Γ(3/4)
27 σ04
B.2
(B.8)
∆ expansion for small temperature
Here we show explicitly the small temperature expansion of ∆, i.e. the separation between
the two systems of D3-branes. Using
Z
∞
∆=2
dσ
σ0
− 12
F (σ)2
−1
,
F (σ0 )2
(B.9)
along with (3.31) and (3.42) we can expand ∆ for small T̄ up to the order T̄ 12
∆ = ∆0 + T̄ 4 ∆1 + T̄ 8 ∆2 + T̄ 12 ∆3 + O(T̄ 16 ) ,
where ∆0 is given in equation (1.93) and ∆1 , ∆2 , ∆3 are computed to be
p
p
2 π3 Γ 45
∆1
1 5 1
1
4
2
4
4
√ = 7
y + 1 −y y + 1 2 F1 − , , − ; − 4 − 1 + y
2 4 4 y
κ
3y Γ − 14
√
(B.10)
(B.11)
p
3/2
πΓ 14
y4 + 1
1
1 1 5
2
4
8
4
5y y + 1
2 2y + 5y 2 F1 − , , − ; − 4
2 4 4 y
648y 13 Γ − 14
1 5 5
1
4
4
12
8
4
− y + 1 4y + 7 2 F1 − , , − ; − 4
− 31 y − 6y + 11y + 10
2 4 4 y
(B.12)
pπ 1 4
y +1 ∆
3Γ 4
16
12
8
4
√3 =
7
−42y
−
y
+
65y
+
145y
+
105
κ
22680y 19 Γ − 45
3/2
1 1 9
1
6
4
12
8
4
(B.13)
−6y y + 1
171y + 326y + 294y + 385 2 F1 − , , − ; − 4
2 4 4 y
5/2
1
1 5 9
2
4
12
8
4
+ 3y y + 1
342y + 386y + 504y + 385 2 F1 − , , − ; − 4
2 4 4 y
√
where we defined y ≡ σ0 / κ and 2 F1 (a, b, c; z) is the hypergeometric function.
∆
√2 =
κ
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