Domain-wall/QFT Domain-wall/QFT correspondencecorrespondence

Wen-Yu WenWen-Yu Wen

Academia SinicaAcademia Sinica

Feb 24, 2006Feb 24, 2006

A Bridge Connecting Gravity and Gauge Theory

OutlineOutline

Holographic PrincipleHolographic Principle AdS/CFT correspondenceAdS/CFT correspondence D-brane: open/closed string D-brane: open/closed string

dualityduality DW/QFT correspondenceDW/QFT correspondence ApplicationsApplications

Holographic PrincipleHolographic Principle

GR: GR: inevitable existence inevitable existence of black holesof black holes

After formation of After formation of BH, the entropy is BH, the entropy is proportional to the proportional to the area of event area of event horizon [Bekenstein-horizon [Bekenstein-Hawking]Hawking]

S=S=1/4 1/4 AA

Holographic PrincipleHolographic Principle

Holographic PrincipleHolographic Principle: : the universe is really the universe is really a hologram which is a hologram which is isomorphic to the isomorphic to the information information "inscribed" on its "inscribed" on its boundaries… boundaries… [‘tHooft [‘tHooft 93; Susskind 94]93; Susskind 94]

A gravitational theory A gravitational theory should be related to should be related to a non-gravitational a non-gravitational theory in one fewer theory in one fewer dimension.dimension.

AdS/CFT AdS/CFT correspondencecorrespondence

Start with Start with 4d4d gauge gauge theories with some theories with some additional properties:additional properties:

Large Large Nc Nc [‘tHooft 02][‘tHooft 02] Strong coupling Strong coupling λλ = g = g22

NcNc Supersymmetry N = 4Supersymmetry N = 4

This may still serve a good This may still serve a good approximation of QCDapproximation of QCD

3 colors (charges)

Electrodynamics U(1)

SU(3)Chromodynamics

electron

photon

g g

gluongg

AdS/CFT AdS/CFT correspondencecorrespondenceNow we have N=4 SU(Now we have N=4 SU(NcNc) SCFT) SCFT Scale invariant: β(μ)=0 Scale invariant: β(μ)=0 Symmetry group: Poincar`e + scaling + others Symmetry group: Poincar`e + scaling + others

SO(4,2)~SU(2,2)SO(4,2)~SU(2,2) 4 supercharge rotated by R-symmetry 4 supercharge rotated by R-symmetry

SU(4)~SO(6)SU(4)~SO(6)1 gauge field 1 gauge field AμAμ 4 Weyl fermions 4 Weyl fermions ψaψa6 real scalars 6 real scalars φiφi These make superconformal group SU(2,2|4)These make superconformal group SU(2,2|4) S-duality [Montonen-Olive] SL(2,Z)S-duality [Montonen-Olive] SL(2,Z)

AdS/CFT AdS/CFT correspondencecorrespondenceCan it also be described by an effective Can it also be described by an effective

(gravitational) theory in (gravitational) theory in 5d5d? (Holographic ? (Holographic principle)principle)

If the 5If the 5thth coordinate associates with energy coordinate associates with energy scale in CFT…scale in CFT…

This is exactly a 5d anti-de Sitter (AdS) space This is exactly a 5d anti-de Sitter (AdS) space (Poincar`e patch).(Poincar`e patch).

22

2

2

22 dr

rdxdx

rds

Boundary

∞ 0 ultraviolet, high energies

infrared,

low energies

R4

AdS5

r

AdS/CFT AdS/CFT correspondencecorrespondence

AdSAdS55 x S x S55

Isometry of AdS Isometry of AdS SO(4,2)SO(4,2)

Isometry of S Isometry of S SO(6)SO(6)

IIB N=8 susyIIB N=8 susy

GravitonsGravitons Mass of particlesMass of particles

CFTCFT44

Conformal groupConformal groupSO(4,2)SO(4,2)

R-symmetryR-symmetrySU(4)SU(4)

N=4 linear +4 N=4 linear +4 conformalconformal

Energy stress tensorEnergy stress tensor Scaling dimensionScaling dimension

In fact, we need 5 more spatial dimensions…10d type IIB supergravity in AdS5 x S5More than that, we need IIB superstring!

AdS/CFT AdS/CFT correspondencecorrespondence

N=4 SYM on M4N=4 SYM on M4

gs=ggs=gYMYM22

Large λ, large NcLarge λ, large Nc Finite λ, large NcFinite λ, large Nc Finite λ, finite NcFinite λ, finite Nc

IIB on AdS5 x S5IIB on AdS5 x S5

ll4 4 =4πgsNcα’=4πgsNcα’22

Classical SUGRAClassical SUGRA Classical stringClassical string Full quantum stringFull quantum string

Three forms of correspondence with increasing strength:[Maldacena, Witten, Gubser, Klebanov, Polyakov]

ApplicationsApplications

Regulate mass of dilatonic black hole Regulate mass of dilatonic black hole [[Michael Michael

Duff, James Liu, et al ‘99Duff, James Liu, et al ‘99]] by Hamilton-Jacobi by Hamilton-Jacobi counterterms counterterms [Wen, Liu, et al. ‘04][Wen, Liu, et al. ‘04]

Construct AdSConstruct AdS55 [Lin, Lunin, Maldacena][Lin, Lunin, Maldacena] and AdS and AdS3 3

bubbles solution bubbles solution [Wen, Liu, Vaman ‘05][Wen, Liu, Vaman ‘05] Generalize to DW/QFT correspondence in Generalize to DW/QFT correspondence in

null-like background null-like background [Wen, Lin; Ho, Chu ‘06][Wen, Lin; Ho, Chu ‘06]

D-brane physicsD-brane physics

AdS/CFT is one AdS/CFT is one realization of realization of gravity/gauge fieldgravity/gauge field

In term of string In term of string theory, this is theory, this is open/closed string open/closed string dualityduality

D-brane plays an D-brane plays an essential roleessential role [Polchinski ‘95][Polchinski ‘95]

D-brane physicsD-brane physics

psrp

p

pp

rNcgHHF

He

drdrHdxHds

71)2(

4/)3(

2)8(

222/121

2/12

)(1

)(

As a SUGRA As a SUGRA

solution, Dp-brane solution, Dp-brane interpolates interpolates between flat between flat Minkowski and Minkowski and AdSxS geometry.AdSxS geometry.

Dilaton decouples Dilaton decouples only for p=3 , where only for p=3 , where AdS/CFT is AdS/CFT is applicable. applicable.

R1,9

AdSp+2 x S8-p

Domain-Wall solutionsDomain-Wall solutions

0' Low energy limitLow energy limitkeep keep fixed fixed

Fixed `tHooft Fixed `tHooft couplingcoupling

let let Weyl transforms Weyl transforms

metric to “dual metric to “dual frame” frame” [Boonstra,Skenderis,Towns[Boonstra,Skenderis,Townsend;Itzhaki,Maldacena ‘98]end;Itzhaki,Maldacena ‘98]

'r

U

NcNcgYM2

4/)7(31

2)8(

2221

5121

][ pp

ppp

UNe

ddUUdsUds

stringp

dual gNeg )7/(2)(

e

U

P<3 P>3

Wall as a probe brane

Null-like DW/QFTNull-like DW/QFT

)(][

])())(,,(2)[(4/)7(31

2)8(

22225121

xbUNe

ddUUdxdxUxxhdxdxxaUdspp

piip

Functions a,b,h need to satisfy equations of motionFunctions a,b,h need to satisfy equations of motion Solution preserves ¼ BPSSolution preserves ¼ BPS Geodesic is not completeGeodesic is not complete XX++ is a Killing direction, can be seen as (light-cone) is a Killing direction, can be seen as (light-cone)

“time”“time”

Probing DW solutionProbing DW solution

This DW background can be probed by a This DW background can be probed by a test branetest brane

There are two ways, which give slightly There are two ways, which give slightly different energy-distance relation:different energy-distance relation:

1.1. Open string stretching in betweenOpen string stretching in between

2.2. Closed string propagatesClosed string propagates

UE ~

2/)5( pUE

Nc-1)7/(222

2

)( pYMYM

YM

xbgg

Ng

QFT living on the wallQFT living on the wall

DBI action of “wall” = action of SU(Nc) QFTDBI action of “wall” = action of SU(Nc) QFT Effective coupling is running with U and xEffective coupling is running with U and x++ Gravity/gauge theory are only valid in certain regimes:Gravity/gauge theory are only valid in certain regimes:

1.1. Classical gravity is good at large gClassical gravity is good at large geffeff and small dilaton and small dilaton

2.2. Perturbative gauge theory is good at small gPerturbative gauge theory is good at small geffeff

323

211

4

1det]det[

p

YM

peff

pp

pp

NcEgEg

FgxdFgxd

TrS

Boundary CosmologyBoundary Cosmology

rewrite equation of motion rewrite equation of motion for a(xfor a(x++), where P, Q are ), where P, Q are related to h.related to h.

Two simple solutions:Two simple solutions:1.1. P=Q=0P=Q=02.2. P=0, Q=constP=0, Q=const

Boundary theory valids at E Boundary theory valids at E > H> H

a

aH

xPa

xQH

a

a

)()(

22

0),||exp()||exp(

0),sin()cos(

)(

102

102

/110

QxQcxQca

QxQcxQca

xcca

ReferencesReferences

related papers by authors mentioned in this talk, related papers by authors mentioned in this talk, especiallyespecially

1.1. Bubbling 1/4 BPS solutions in type IIB and Bubbling 1/4 BPS solutions in type IIB and supergravity reduction on Sn x Sn [Wen, Liu, supergravity reduction on Sn x Sn [Wen, Liu, Vaman; 0412043] Vaman; 0412043]

2.2. Black hole mass and Hamilton-Jacobi counterterms Black hole mass and Hamilton-Jacobi counterterms [Wen,Batrachenko,Liu,McNees,Sabra; JHEP 0505 [Wen,Batrachenko,Liu,McNees,Sabra; JHEP 0505 (2005) 034] (2005) 034]

3.3. Supersymmetric null-like holographic cosmologies Supersymmetric null-like holographic cosmologies [Wen,Lin; 0602124][Wen,Lin; 0602124]

some pictures are taken from online resources, such some pictures are taken from online resources, such as Scientific American, Wikipedia, NOVA, etc. as Scientific American, Wikipedia, NOVA, etc.

Thank YouThank You