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Recent issues in brane theory and
the gauge/gravity correspondence
Four possible topics
1. Domain Walls
2. Shadow Multiplets and the AdS/CFT in D=3
3. Algebraic Geometry, Cones and Conformal Field
Theories in D=3
4. D3 branes and ALE manifolds
P.Fré
Bogolyubov Institute, Kiev
Dec. 2001
p-Brane Actions
The parameter D and the
harmonic function H(y)
Electric and magnetic p-branes
“Elementary”
Conformal branes and AdS space
AdS is a special case of
a Domain Wall
ELECTRIC BRANE
These two forms
are related by a
coordinate
transformation
Coordinate patches and the
conformal gauge
Conformal
brane
a=0
Randall Sundrum gravity trapping
Kaluza Klein expansion
in non compact space
These potentials have a Volcano shape that
allows the existence of normalizable zero
mode describing the graviton in D-1
dimensions. The continuum Kaluza Klein
spectrum contributes only a small
correction to the D-1 dimensional
Newton’s law
Randall Sundrum
Positivity of the Wall Tension
The “dual frame” of Boonstra,
Skenderis and Townsend
We learn that although the AdS x S8-p is not a solution of supergravity, we can
notheless compactify on the sphere S8-p, or other compact manifold X8-p !!!
“Near brane” factorization in the
dual frame
The transverse cone
(D-p-1) - Cone
p-brane
2
dsbrane
 dx  d
In some
sense

An X8-p
compact
manifold is
the base of the
transverse
cone C( X8-p )
2
dsCone
 dr 2  r 2ds X2
In D=10 the p-brane splits the space into a d=p+1 world volume and a transverse
cone C( X8-p ) that has the compact manifold X8-p as base.
Domain wall supergravity from
“sphere reduction”
The DW/QFT correspondence of
Boonstra Skenderis & Townsend
This raises some basic questions
and we have some partial answers:
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Which supergravity is it that accommodates the Domain
Wall solution after the “sphere” reduction?
It is a “gauged supergravity”
But which “gauging” ?
Typically a non compact one. It is compact for AdS
branes!
What are the possible gaugings?
These are classifiable and sometimes classified
How is the gauging determined and how does it reflect
microscopic string dynamics?
??? This is the research frontier!
The shadow of the graviton
multiplet can be a gravitino
multiplet, suggesting a
superHiggs mechanism but…...
Conjecture:
Shadow
SUGRAS
It seems
that there
exist more
general Nextended
SUGRAS
with
SU(N0)
symmetry
where
N0<N
M 123  eigenvalue s of
Laplacians on X 7
The spectrum
is
determined
by
eigenvalues
of Laplace
Beltrami
operators
Structure
of
M-theory
AdS4
UIR
relations
Differential Geometry of X7
knows all the relations implied by
SUSY in AdS4
There are
4 neutral
scalars
whose
vev ‘s
trigger
superHiggs
R symmetry
* = modulus in
N=4
N=3
This
multiplet
is
Universal
in N=3
KK
This suggests
consistent
truncation!
The value E0=3
realized in the
Kaluza Klein model
is reached only at the
boundary of moduli
space
in N=4 standard
SUGRA
Lines of constant E0
over the disk
Questions-Conclusions
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SU (1,1) USp(3,3)
M sh 

 W6
U (1)
U (3)
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Standard N=4 SUGRA realizes
SuperHiggs with E0<3
M-theory à la KK realizes
SuperHiggs with E0=3
This realization follows from a
general shadowing mechanism
Truncation to graviton + massive
gravitino should be consistent since
harmonics are constants
This hints to the existence of a new
N=4 shadow supergravity based on
the scalar manifold Msh and an FDA
describing superHiggs where E0=3 or
bigger is allowed.
Shadow SUGRAS exist also for other
N?
is shadowing true also in D=5?
We know the CFT interpretation of
the universal multiplet but there is
no time…….!
The list of supersymmetric
homogeneous 7-dimensional
cosets was determined in the
eighties.
We have a
particular interest
in this coset that
yields N=3
supersymmetry
and reveals the
intriguing relation
between
shadowing and
susy breaking
Type IIB Supergravity, D3 branes
and ALE manifolds
1.
2.
Based on recent work by:
M. Bertolini, G. Ferretti, (P. F.) M.
Trigiante, L. Campos, P. Salomonson
hep-th 0106186
M.Billo, (P.F.), C. Herman, L.
Modesto, I. Pesando (to appear)
Introduction
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There are interesting solutions of type IIB theory, named fractional D3
branes.
 The gauge duals are non-conformal N=2 gauge theories in d=4
Fractional branes are commonly viewed as 5-branes wrapped on a vanishing
cycle of transverse space
 Transverse space is R2 x R4/G.
We have found a supersymmetric (BPS) D3-brane solution where transverse
space is R2 x ALE
 In the orbifold limit we recover fractional branes
 The warp factor is determined by a harmonic equation on ALE
 In Eguchi Hanson case the harmonic equation reduces to a confluent
Heun equation
 Open questions on the boundary action and the gauge dual.
Type IIB Sugra
The D3 brane
couples to C[4]
whose field
strength
involves a
Chern Simons
of lower forms
Castellani & Pesando (1991) established geometric formulation
The SL(2,R)~SU(1,1) structure
The solvable Lie algebra
parametrization of the coset
naturally introduces the
dilaton and the RR 0-form
The FDA defining the Theory
Note
the
Chern
Simons
As for all supergravities the algebraic structure is encoded in an FDA.
(D’Auria, Fré, (1982), Castellani, P. Van Nieuwenhuizen, K. Pilch (1982)
The rheonomic
parametrizations~susy rules
Fermionic
Components
are
expressed
in terms of
space-time
components
Our task: the equations to be
solved
Note that these
equations are written in
flat indices and appear
as constraints on the
curvature components
Fractional Branes= D3 branes on
orbifolds C2/G
In the dual gauge
theory, x4+ix5
makes the
complex scalar of
the vector
multiplet while
Susy is
halved
in the bulk
by
restricted
holonomy
x6, x 7 , x8 , x9
constitute the
scalar part of a
hypermultiplet
ALE manifolds as orbifold
resolutions
ALE manifolds are
related to the ADE
classification of Lie
algebras. They can be
obtained as suitable
Hyperkahler quotients
of flat HyperKahler
manifolds
Kronheimer construction of ALE spaces
is realized by String Theory
ALE Manifolds: main relevant
feature
The ansatz:
Note the
essential use
of the
harmonic
2 forms
dual to the
homology
cycles of ALE
Holomorphic field
Pinpointing the sources:
This is the real
problem of
interpretation:
what is the source
of B,C fields?
Even without D3 brane charge there is effective source for 5-form
Killing spinors I:
Killing spinors II:
Killing spinors III:
Two solutions because of SU(2) holonomy of ALE
Harmonic equation in the Eguchi
Hanson case
Explicit form of the
compact harmonic 2-form
and “intersection function”
Note factorization
of near cycle metric
The partial Fourier transform
Makes Laplace
Equation on
ALE x R2
inhomogeneous
Heun confluent equation
D3
brane
charge
Fixed by
boundary
conditions
Asymptotics and the charges
The D3 brane charge
determines the
coefficient of the
irregular solution
near the cycle
Asymptotic flatness fixes
the coefficient of the
irregular solution at infinity
Power series and questions
See picture
Interpolation and singularity
Far from the cycle we just see
the charge of a normal D3 in D=10
Near the cycle we see a
D3 brane in D=8. It is as
if we had compactified
sugra on an S2
The World Volume Action
Kappa supersymmetric:
the fermions are hidden
in the p—forms that are
superforms
Red=Auxiliary fields
Brown=Sugra bckg fields
Blue=World volume fields
The interactions of world—volume fields
(fluctuations from vacuum values) feel all
the bckg fields. Yet this action is source
only for C4.. Where is the source of A[2]L ?
This is the open question answering the which will shed light
on the real nature of fractional branes