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

2. 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

3. 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

4. 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.

5. 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

6. 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)

7. 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

8. 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!

9. 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]

10. 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]

11. 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]

12. 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

13. 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

14. 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”

15. 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

16. 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

17. 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

18. 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; 0412043] 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; 0602124] 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.

19. Thank You