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
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Holographic Principle
AdS/CFT correspondence
D-brane: open/closed string duality
DW/QFT correspondence
Applications
Holographic Principle
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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
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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
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electron
SU(3)
Chromodynamics
g
g
gluon
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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

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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
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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
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Regulate mass of dilatonic black hole [Michael Duff,
James Liu, et al ‘99] by Hamilton-Jacobi counterterms
[Wen, Liu, et al. ‘04]
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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
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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
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( 3 p ) / 4
F( p  2 )   r H 1
H  1  g s Nc(
R1,9
AdSp+2 x S8-p

r
)7  p
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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
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P>3
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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 /( p7) 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  )
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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
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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
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1.
2.
S  pd
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
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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
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Boundary theory valids at E > H
References
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1.
2.
3.
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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