Download Higher Spin Theories and Holography

HIGHER SPIN GRAVITY
AND HOLOGRAPHY
Simone Giombi
XXI Conferenza Sigrav
Alessandria, September 15, 2014
Spin
• Elementary particles can carry spin (integer or half-
integer multiple of Planck constant
)
• Particles of arbitrarily high spin theoretically possible
(Dirac 1936, Wigner 1939)
“Low spin” particles
• Known elementary particles have small values of the spin
quantum number
Particles and fields
• In the formalism of relativistic quantum field theory,
these particles are described by fields
• m,n =0,1,2,3 are space-time indices, in the usual
formalism of special relativity
Higher spin fields
• The generalization to higher spin is natural
• For integer spin s, they can be described by symmetric
traceless rank-s tensors
• Generalizes the photon (4-vector) and graviton (rank 2
tensor)
• The problem of writing consistent relativistic equations
describing propagation of higher spin fields, and their
interactions, is a non-trivial one and has a long history
(Dirac ‘36, Wigner ’39, Fierz-Pauli ‘39, Rarita-Schwinger ‘41,
Bargmann-Wigner ‘48, Fronsdal ‘78…)
Massive vs Massless
• Massive and massless spinning particles have some
crucial differences
• Massive relativistic particles of spin s have 2s+1
polarization states
• On the other hand, massless particles (of s ≥ 1) have
only two polarizations: these are the helicity states h=
+s,-s (for any spin s)
Gauge symmetry
• The photon can be described by the electromagnetic
potential
. By itself, this would describe more than
two states
• The unphysical (longitudinal) states are decoupled
because of the gauge symmetry
where e is a scalar gauge parameter.
• For the graviton, one has the linearized gauge symmetry
which is related to general coordinate invariance of
Einstein’s gravity.
Higher spin gauge symmetries
• For massless higher spins of general s, one must have a
gauge symmetry with a spin s-1 gauge parameter
• This gauge symmetry ensures that there are only two
physical polarizations h=+s,-s.
• So massless fields with spin greater or equal to 1 are
gauge fields.
• The presence of the gauge symmetries makes it nontrivial to write down consistent, covariant wave equations
(and Lagrangians) for massless higher spins
Free massless higher spins
• Fronsdal (1978) obtained the gauge invariant Lagrangian
and equations of motion for free massless higher spins
propagating in flat spacetime.
• Fronsdal’s equations can be also generalized to the de
Sitter (dS) and anti-de Sitter (AdS) space-time: the
maximally symmetric solutions of Einstein’s equations
with positive and negative cosmological constant
• A surprising feature: it is not possible to put the massless
higher spins on a general curved space-time (AragoneDeser 1979) in a way consistent with the gauge
symmetry.
Interactions?
• The problem of writing consistent interactions of the
higher spin fields is a much more difficult one
• Gauge invariance severely constrains the possible
interactions
• In flat space-time, “No-Go” theorems (Coleman-Mandula)
appear to forbid interactions of higher spin fields.
Gauge symmetry would imply the existence of
conserved charges that are too constraining to allow a
non-trivial scattering matrix: just free fields would be
allowed.
“Yes-Go”: Vasiliev theory
• However, under the assumption of a non-zero
cosmological constant , Fradkin and Vasiliev (‘87)
explicitly constructed consistent cubic interaction vertices
of higher spin fields
• In a series of works (’90-’92), Vasiliev was then able to
construct a fully non-linear consistent theory of interacting
massless higher spin fields. Works in AdS or dS space.
No smooth flat space limit.
• Originally developed in 4-dimensional space-time, but
generalizations to arbitrary dimensions were later
constructed
Vasiliev theory
• Crucial feature of the theory: consistency with non-linear
gauge symmetries requires that the spectrum contains an
infinite tower of higher spin fields
• In the simplest version of the 4d theory, the spectrum
around the AdS vacuum solution is
• In particular, it contains the massless spin 2 field, which is
nothing but the graviton !
Vasiliev higher spin gravity
• So, Vasiliev theory may be viewed as a peculiar theory of
gravity which generalizes Einstein’s theory by including an
infinite set of massless fields of all spins.
• Classical equations of motion are complicated, but fully
known. Quantization is not yet well understood, but one
may hope that due to the infinite dimensional gauge
symmetry, it could perhaps define a fully consistent, finite
model of quantum gravity.
• Maybe describing a highly symmetric phase of space-time
(early universe?). Would need to understand how to break
the higher spin symmetries.
Vasiliev theory
• It involves a remarkable construction using ideas from
twistor theory and non-commutative field theory
Vasiliev equations
• The 4d Vasiliev equations
• Essentially, A encodes the metric and all other higher
spin fields, and B contains the scalar field and the
curvatures (Weyl tensors) of the higher spin (HS) fields.
• Equations have a vacuum solutions corresponding to
AdS (or dS) space-time.
• The equations for linearized fluctuations are the
standard equations for free scalar, graviton and
massless HS fields (Fronsdal equations).
The AdS/CFT correspondence
• The Vasiliev theory in anti-de Sitter space plays an
important role in the context of the so-called “AdS/CFT
correspondence” (Maldacena, Gubser-Klebanov-Polyakov, Witten ‘98)
• A remarkable conjecture relating theories of quantum
gravity in anti-de Sitter space (the maximally symmetric
space with negative curvature) to ordinary quantum
field theories in one lower dimension
AdS/CFT
• The AdS/CFT correspondence is an exact equivalence,
or duality, between quantum gravity in AdS and a
conformally invariant quantum field theory (CFT) that
can be thought of as living at the boundary of AdS
Holographic duality: quantum gravitational physics
“encoded” in the QFT at the boundary.
AdS/CFT duality
• While it was originally discovered in the context of
string theory, AdS/CFT might be more general. It
does not, in principle, require string theory.
• Any consistent quantum gravity theory in AdS should
have a CFT dual
AdS/CFT dictionary
• There is a general dictionary relating the physics in
the “bulk” (AdS) and “boundary”.
• Operators in the CFT are dual to fields in the AdS
theory
AdS/CFT dictionary
• Basic observables: correlation functions of local
operators in the CFT.
“Witten diagrams”
The above diagram shows for instance how the cubic
coupling of the graviton determines the 3-point function of
the stress energy tensor in the CFT.
• The coupling constant in the bulk is related to N,
rougly speaking the number of fields in CFT. Large
N corresponds to weak coupling on the gravity side.
Higher spin theories and AdS/CFT
• Vasiliev theory was constructed long before AdS/CFT
• Its existence may seem surprising, however from the
point of view of AdS/CFT one can see that it is
natural that these theories do exist
• If Vasiliev theory defines a consistent quantum gravity
theory in AdS, what is its CFT dual?
Free vector models
• Consider a simple free theory of N massless scalar fields
in 2+1 spacetime dimensions
•
is an N-component scalar field, transforming in the
fundamental of the global U(N) symmetry (we can also
consider O(N) version by taking real scalar fields).
Free vector models
• By virtue of being a free theory, one can see that it has an
infinite number of conserved currents of all spins
• In addition, there is a scalar operator
Higher spins and AdS/CFT
• By the AdS/CFT dictionary, conserved currents are dual
to gauge fields
• Conserved currents of all spins should be then dual to
massless higher spin fields in AdS.
• The scalar operator J0 should be dual to a massive
scalar field in the bulk with
• In our case, this gives m2 = -2/l2, which is precisely the
mass of the scalar field fixed by Vasiliev theory.
• The operator spectrum of the free vector model exactly
matches the spectrum of Vasiliev theory!
Higher spins and AdS/CFT
• The dual higher spin fields are necessarily interacting in
order to reproduce the non-vanishing correlation
functions of HS currents in the CFT
• The Newton’s constant of the AdS theory scales as
GN ~ 1/N (in units of AdS radius).
• The large N limit corresponds as usual to weak
interactions in the bulk.
Higher spins and AdS/CFT
• Thus, we come at the natural (but surprising)
conclusion that Vasiliev theory in AdS may be exactly
equivalent to a free theory in one less dimension.
• Both sides of the duality are in principle accessible
perturbatively in 1/N.
• May lead to a derivation of AdS/CFT in a simple setting.
• May also shed light on the nature of the more
complicated gauge/string dualities in the limit of weak
coupling (strings in highly curved space).
Not just free theories
• The conjecture that the O(N)/U(N) free vector model
is dual to the Vasiliev theory was made by Klebanov
and Polyakov (2002)
• A crucial observation they made is that one can in
fact also consider the well-known interacting WilsonFisher fixed point of the quartic scalar field theory
The interacting vector model
• The quartic scalar field theory
at low energies is described by a strongly interacting
conformal field theory. This is the Wilson-Fisher IR fixed
point.
• This fixed point is relevant for understanding the vicinity
of 2nd order phase transitions in various statistical
systems with O(N) symmetry
Interacting vector model
• At large N, it can be shown that the fixed point theory
still possesses higher spin currents that are only
slightly broken
• It turns out that the dual theory in AdS is the same
Vasiliev theory dual to the free vector model, with the
only difference that the bulk scalar field is assigned a
different boundary condition in AdS space (Klebanov,
Witten; Klebanov, Polyakov)
Tests of higher spin/vector model duality
• These dualities between higher spin gravity and
vector models have been tested successfully at the
level of 3-point correlation functions
SG, Yin ‘09
• Requires solving Vasiliev equations to second order
in perturbation theory around AdS. A non-trivial task.
Matching partition functions
• Recently, we performed a different kind of test, based on
comparing the bulk and boundary vacuum partition
functions (SG, Klebanov; SG, Klebanov, Safdi; SG, Klebanov, Tseytlin)
• An interesting observable in a CFT, is the Euclidean path
integral of the theory on a round sphere S3
• It has been suggested (Jafferis, Klebanov, Pufu, Safdi; Myers,
that F is a “measure of degrees of freedom” in a
2+1 dimensional CFT. It is related to the entanglement
entropy across a circle. Decreases under “RG flow”
(Casini, Huerta).
Sinha)
• In the free vector model CFT, this F can be easily
computed from one-loop functional determinants.
Matching partition functions
• On the gravity side, the quantity F should be computed by
the vacuum path integral of Vasiliev theory on the Euclidean
AdS (hyperbolic 4-space)
• A non-trivial 1/N expansion, in principle (Recall GN ~ 1/N).
• On the other hand, the dual free CFT has a trivial 1/N
expansion. Can we see vanishing of 1/N correction in the
bulk?
One-loop tests
• The calculation of the leading piece in Fbulk is currently out
of reach (classical action in AdS for Vasiliev theory is not
yet understood)
• But the one-loop piece is actually not too difficult to
compute
• We just need to know the kinetic operators of the
massless higher spin fields, which are fixed by gauge
symmetry, and compute certain functional determinants in
AdS. This can be done e.g. by heat kernel techniques.
• Consistency with the duality requires that the one-loop
correction (as well as all higher orders) should vanish
One-loop tests
• Determinants of quantum fields in even-dimensional
spacetime (AdS4 for us) can have UV logarithmic
divergences
• For instance, pure Einstein gravity in AdS has nonvanishing cdiv (Christensen, Duff ‘79)
• For Vasiliev theory to be a “UV complete”, finite, theory
the logarithmic divergence should vanish.
• Can this happen due to the sum over infinite number of
fields?
One-loop tests
• An explicit calculation yields the result
• How does the logarithmic divergence vanish?
• Sum over spins need to be regularized
• It vanishes if we use Riemann z-function
regularization identities
• This can be done a bit more rigorously, by
appropriately regularizing the functional determinants
(spectral zeta-functions) in a way consistent with
higher spin symmetry, obtaining the same result.
One-loop finiteness
• Thus, we are led to the conclusion that Vasiliev theory
is a theory of gravity free of UV divergences at the oneloop level
• Similar cancellations of one-loop UV divergences are
known to happen in extended supergravity theories in
AdS4 (with N>4 susy)
• Here the cancellation is in a purely bosonic theory, but it
requires an infinite number of fields.
One-loop test
• Having shown that the divergent part vanishes, one can
also compute the finite piece (with a bit more effort)
• The result is indeed that, after summing over all integer
spins, the finite part also vanishes cfinite=0
• This is precisely consistent with the AdS/CFT duality!
• For the interacting Wilson-Fisher theory, Vasiliev theory
gives the correct non-zero value predicted by field
theory (it is cfinite =-z(3)/(8p2))
• Higher dimensional versions of the duality can be
checked as well (SG, Klebanov, Safdi). Vasiliev theory is
free of divergences in any space-time dimension.
Conclusions
• Interacting theories of massless higher spin particles can
be constructed if the cosmological constant is non-zero.
• They involve infinite towers of fields of all spins, including
the s=2 graviton. They are, in particular, models of
gravity.
• The Vasiliev theory in AdS was conjectured to be exactly
dual to simple vector model CFT’s.
• Via AdS/CFT, Vasiliev theories provide exact
gravitational duals not only to free theories, but also to
interesting interacting theories such as the critical
(Wilson-Fisher) O(N) model, the Gross-Neveu model,
CPN model, theories involving Chern-Simons gauge
fields...
Conclusions
• Supersymmetry is not necessary (though one can
consider supersymmetric versions).
• Our one-loop calculations suggest that Vasiliev theory is
finite (free of logarithmic UV divergences). Might be true
to all loops. A finite model of quantum gravity?
• Much simpler than a full-fledged superstring theory.
• Interesting not only for holography, but also as a model
of quantum gravity by itself.
• Much to be understood: quantization (action principle?);
study exact solutions (black holes? Resolution of
singularities?); connection to string theory?
THANK YOU!