Domain-wall/QFT correspondence
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Transcript Domain-wall/QFT correspondence
Domain-wall/QFT
correspondence
A Bridge Connecting Gravity and Gauge Theory
Wen-Yu Wen
Academia Sinica
Feb 24, 2006
Outline
Holographic Principle
AdS/CFT correspondence
D-brane: open/closed string duality
DW/QFT correspondence
Applications
Holographic Principle
GR:
inevitable existence of
black holes
After formation of BH,
the entropy is
proportional to the
area of event horizon
[Bekenstein-Hawking]
S=1/4 A
Holographic Principle
Holographic Principle:
the universe is really a
hologram which is
isomorphic to the
information "inscribed"
on its boundaries…
[‘tHooft 93; Susskind 94]
A gravitational theory
should be related to a
non-gravitational
theory in one fewer
dimension.
AdS/CFT correspondence
U(1)
Electrodynamics
photon
g
Start with 4d gauge theories
with some additional
properties:
g
electron
SU(3)
Chromodynamics
g
g
gluon
Large Nc [‘tHooft 02]
Strong coupling λ = g2 Nc
Supersymmetry N = 4
This may still serve a good
approximation of QCD
3 colors (charges)
AdS/CFT correspondence
Now we have N=4 SU(Nc) SCFT
Scale invariant: β(μ)=0
Symmetry group: Poincar`e + scaling + others
SO(4,2)~SU(2,2)
4 supercharge rotated by R-symmetry SU(4)~SO(6)
1 gauge field Aμ
4 Weyl fermions ψa
6 real scalars φi
These make superconformal group SU(2,2|4)
S-duality [Montonen-Olive] SL(2,Z)
AdS/CFT correspondence
Can it also be described by an effective (gravitational)
theory in 5d? (Holographic principle)
If the 5th coordinate associates with energy scale in
CFT…
2
2
r
2
ds 2 dx dx 2 dr 2
r
R4
AdS5
∞
ultraviolet,
high energies
This is exactly a 5d anti-de Sitter (AdS) space
(Poincar`e patch).
Boundary
r
0
infrared,
low energies
AdS/CFT correspondence
In fact, we need 5 more spatial dimensions…
10d type IIB supergravity in AdS5 x S5
More than that, we need IIB superstring!
AdS5 x S5
Isometry of AdS
SO(4,2)
Isometry of S
SO(6)
IIB N=8 susy
Gravitons
Mass of particles
CFT4
Conformal group
SO(4,2)
R-symmetry
SU(4)
N=4 linear +4
conformal
Energy stress tensor
Scaling dimension
AdS/CFT correspondence
Three forms of correspondence with increasing strength:
[Maldacena, Witten, Gubser, Klebanov, Polyakov]
N=4 SYM on M4
gs=gYM2
Large λ, large Nc
Finite λ, large Nc
Finite λ, finite Nc
IIB on AdS5 x S5
l4 =4πgsNcα’2
Classical SUGRA
Classical string
Full quantum string
Applications
Regulate mass of dilatonic black hole [Michael Duff,
James Liu, et al ‘99] by Hamilton-Jacobi counterterms
[Wen, Liu, et al. ‘04]
Construct AdS5 [Lin, Lunin, Maldacena] and AdS3
bubbles solution [Wen, Liu, Vaman ‘05]
Generalize to DW/QFT correspondence in null-like
background [Wen, Lin; Ho, Chu ‘06]
D-brane physics
AdS/CFT is one
realization of
gravity/gauge field
In term of string
theory, this is
open/closed string
duality
D-brane plays an
essential role [Polchinski
‘95]
D-brane physics
ds2 H 1 / 2dx2p 1 H 1 / 2 (dr 2 r 2 d (28 p ) )
e H
( 3 p ) / 4
F( p 2 ) r H 1
H 1 g s Nc(
R1,9
AdSp+2 x S8-p
r
)7 p
As a SUGRA solution,
Dp-brane interpolates
between flat Minkowski
and AdSxS geometry.
Dilaton decouples only
for p=3 , where
AdS/CFT is applicable.
Domain-Wall solutions
1ds2 1U 5 p ds2p 1 U 2 dU 2 d (28 p ) Low energy limit
keep
e N 1[U p 3 ]( 7 p ) / 4
Wall as a probe brane
e
P<3
P>3
U
r
'
' 0
fixed
Fixed `tHooft coupling
2
gYM
Nc let Nc
Weyl transforms metric
to “dual frame”
[Boonstra,Skenderis,Townsend
;Itzhaki,Maldacena ‘98]
U
gdual (e N )2 /( p7) gstring
Null-like DW/QFT
1ds2 1U 5 p a( x )[2dx dx h( x , x i , U )(dx )2 (dxi )2 ] U 2dU 2 d (28 p )
e N 1[U p 3 ]( 7 p ) / 4 b( x )
Functions a,b,h need to satisfy equations of motion
Solution preserves ¼ BPS
Geodesic is not complete
X+ is a Killing direction, can be seen as (light-cone)
“time”
Probing DW solution
1.
2.
This DW background can be probed by a test
brane
There are two ways, which give slightly different
energy-distance relation:
Open string stretching in between
E ~U
Closed string propagates
2
E U ( 5 p) / 2 gYM
N
2
gYM gYM
b( x )2 /(7 p ) Nc-1
2
QFT living on the wall
1.
2.
S pd
p 1
x de t[g F ] p d
geff E
p 3
p 1
1
x de t g TrF 2
4
gYM NcE p 3
2
DBI action of “wall” = action of SU(Nc) QFT
Effective coupling is running with U and x+
Gravity/gauge theory are only valid in certain regimes:
Classical gravity is good at large geff and small dilaton
Perturbative gauge theory is good at small geff
Boundary Cosmology
1.
2.
rewrite equation of motion for
a(x+), where P, Q are related to
h.
Two simple solutions:
P=Q=0
P=0, Q=const
a
Q( x )
2
H
P
(
x
)
2
a
a
a
H
a
a (c0 c1 x )1 /
a 2 c0 cos( Q x ) c1 sin( Q x ), Q 0
a 2 c0 e xp( | Q | x ) c1 e xp( | Q | x ), Q 0
Boundary theory valids at E > H
References
1.
2.
3.
related papers by authors mentioned in this talk, especially
Bubbling 1/4 BPS solutions in type IIB and supergravity
reduction on Sn x Sn [Wen, Liu, Vaman; 0412043]
Black hole mass and Hamilton-Jacobi counterterms
[Wen,Batrachenko,Liu,McNees,Sabra; JHEP 0505 (2005)
034]
Supersymmetric null-like holographic cosmologies [Wen,Lin;
0602124]
some pictures are taken from online resources, such as
Scientific American, Wikipedia, NOVA, etc.
Thank You