Domain-wall/QFT correspondence Wen-Yu Wen Academia Sinica Feb 24, 2006 A Bridge Connecting Gravity and Gauge Theory

Outline Holographic Principle Holographic Principle AdS/CFT correspondence AdS/CFT correspondence D-brane: open/closed string duality D-brane: open/closed string duality DW/QFT correspondence DW/QFT correspondence Applications Applications

Holographic Principle GR: inevitable existence of black holes GR: inevitable existence of black holes After formation of BH, the entropy is proportional to the area of event horizon [Bekenstein-Hawking] 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] 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. A gravitational theory should be related to a non-gravitational theory in one fewer dimension.

AdS/CFT correspondence Start with 4d gauge theories with some additional properties: Large Nc [‘tHooft 02] Large Nc [‘tHooft 02] Strong coupling λ = g 2 Nc Strong coupling λ = g 2 Nc Supersymmetry N = 4 Supersymmetry N = 4 This may still serve a good approximation of QCD 3 colors (charges) Electrodynamics U(1) SU(3) Chromodynamics electron photon g g gluon g g

AdS/CFT correspondence Now we have N=4 SU(Nc) SCFT Scale invariant: β(μ)=0 Scale invariant: β(μ)=0 Symmetry group: Poincar`e + scaling + others SO(4,2)~SU(2,2) Symmetry group: Poincar`e + scaling + others SO(4,2)~SU(2,2) 4 supercharge rotated by R-symmetry SU(4)~SO(6) 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) These make superconformal group SU(2,2|4) S-duality [Montonen-Olive] SL(2,Z) 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 5 th coordinate associates with energy scale in CFT… If the 5 th coordinate associates with energy scale in CFT… This is exactly a 5d anti-de Sitter (AdS) space (Poincar`e patch). Boundary ∞ 0 ultraviolet, high energies infrared, low energies R4R4 AdS 5 r

AdS/CFT correspondence AdS 5 x S 5 Isometry of AdS SO(4,2) Isometry of AdS SO(4,2) Isometry of S SO(6) Isometry of S SO(6) IIB N=8 susy IIB N=8 susy Gravitons Gravitons Mass of particles Mass of particles CFT 4 Conformal group SO(4,2) Conformal group SO(4,2) R-symmetry SU(4) R-symmetry SU(4) N=4 linear +4 conformal N=4 linear +4 conformal Energy stress tensor Energy stress tensor Scaling dimension Scaling dimension In fact, we need 5 more spatial dimensions… 10d type IIB supergravity in AdS5 x S5 More than that, we need IIB superstring!

AdS/CFT correspondence N=4 SYM on M4 gs=g YM 2 gs=g YM 2 Large λ, large Nc Large λ, large Nc Finite λ, large Nc Finite λ, large Nc Finite λ, finite Nc Finite λ, finite Nc IIB on AdS5 x S5 l 4 =4πgsNcα’ 2 l 4 =4πgsNcα’ 2 Classical SUGRA Classical SUGRA Classical string Classical string Full quantum string Full quantum string Three forms of correspondence with increasing strength: [Maldacena, Witten, Gubser, Klebanov, Polyakov]

Applications Regulate mass of dilatonic black hole [ Michael Duff, James Liu, et al ‘99 ] by Hamilton-Jacobi counterterms [Wen, Liu, et al. ‘04] Regulate mass of dilatonic black hole [ Michael Duff, James Liu, et al ‘99 ] by Hamilton-Jacobi counterterms [Wen, Liu, et al. ‘04] Construct AdS 5 [Lin, Lunin, Maldacena] and AdS 3 bubbles solution [Wen, Liu, Vaman ‘05] Construct AdS 5 [Lin, Lunin, Maldacena] and AdS 3 bubbles solution [Wen, Liu, Vaman ‘05] Generalize to DW/QFT correspondence in null-like background [Wen, Lin; Ho, Chu ‘06] 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 AdS/CFT is one realization of gravity/gauge field In term of string theory, this is open/closed string duality In term of string theory, this is open/closed string duality D-brane plays an essential role [Polchinski ‘95] D-brane plays an essential role [Polchinski ‘95]

D-brane physics As a SUGRA solution, Dp-brane interpolates between flat Minkowski and AdSxS geometry. As a SUGRA solution, Dp-brane interpolates between flat Minkowski and AdSxS geometry. Dilaton decouples only for p=3, where AdS/CFT is applicable. Dilaton decouples only for p=3, where AdS/CFT is applicable. R 1,9 AdS p+2 x S 8-p

Domain-Wall solutions Low energy limit keep fixed Low energy limit keep fixed Fixed `tHooft coupling let Fixed `tHooft coupling let Weyl transforms metric to “dual frame” [Boonstra,Skenderis,Townsend ;Itzhaki,Maldacena ‘98] Weyl transforms metric to “dual frame” [Boonstra,Skenderis,Townsend ;Itzhaki,Maldacena ‘98] P<3 P>3 Wall as a probe brane

Null-like DW/QFT Functions a,b,h need to satisfy equations of motion Functions a,b,h need to satisfy equations of motion Solution preserves ¼ BPS Solution preserves ¼ BPS Geodesic is not complete Geodesic is not complete X + is a Killing direction, can be seen as (light-cone) “time” X + is a Killing direction, can be seen as (light-cone) “time”

Probing DW solution This DW background can be probed by a test brane This DW background can be probed by a test brane There are two ways, which give slightly different energy-distance relation: There are two ways, which give slightly different energy-distance relation: 1. Open string stretching in between 2. Closed string propagates Nc-1

QFT living on the wall DBI action of “wall” = action of SU(Nc) QFT DBI action of “wall” = action of SU(Nc) QFT Effective coupling is running with U and x + Effective 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. Classical gravity is good at large g eff and small dilaton 2. Perturbative gauge theory is good at small g eff

Boundary Cosmology rewrite equation of motion for a(x + ), where P, Q are related to h. rewrite equation of motion for a(x + ), where P, Q are related to h. Two simple solutions: Two simple solutions: 1. P=Q=0 2. P=0, Q=const Boundary theory valids at E > H Boundary theory valids at E > H

References related papers by authors mentioned in this talk, especially related papers by authors mentioned in this talk, especially 1. Bubbling 1/4 BPS solutions in type IIB and supergravity reduction on Sn x Sn [Wen, Liu, Vaman; ] 2. Black hole mass and Hamilton-Jacobi counterterms [Wen,Batrachenko,Liu,McNees,Sabra; JHEP 0505 (2005) 034] 3. Supersymmetric null-like holographic cosmologies [Wen,Lin; ] some pictures are taken from online resources, such as Scientific American, Wikipedia, NOVA, etc. some pictures are taken from online resources, such as Scientific American, Wikipedia, NOVA, etc.

Thank You