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c-Theorem and Universal bound in AdS/ non -CFT correspondence

c-Theorem and Universal bound in AdS/ non -CFT correspondence. Based on arXiv:0804.0779. 溫文鈺 Wen-Yu Wen (NTU) 4/29/08 @ CYCU. Outline. Universal ratio in AdS/CFT Universal lower bound in AdS/ non -CFT Examples Proof by c-Theorem. AdS 5 / CFT 4 correspondence.

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c-Theorem and Universal bound in AdS/ non -CFT correspondence

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  1. c-Theorem and Universal bound in AdS/non-CFT correspondence Based on arXiv:0804.0779 溫文鈺Wen-Yu Wen (NTU) 4/29/08 @ CYCU

  2. Outline • Universal ratio in AdS/CFT • Universal lower bound in AdS/non-CFT • Examples • Proof by c-Theorem

  3. AdS5/CFT4 correspondence • Consider N coincident D3 branes in IIB string • Open string degrees of freedom give rise to N=4 SU(N) super Yang-Mills theory in 3+1 dimensions • Field contents: gauge field Aμ, 6 scalars Xi, 4 fermions λa • R-symmetry SU(4) ~SO(6)

  4. AdS5/CFT4 correspondence • It was conjectured that same degrees of freedom are described by closed string (supergravity) on its (D3 branes’) near horizon geometry AdS5×S5

  5. AdS5/CFT4 correspondence • Dictionary of correspondence between operators (CFT) and states (Gravity) can be built.

  6. d-dimensional Conformal Field TheoryCentral charge • Cardy’s two-point function [correlator]: • This can also be calculated via boundary-boundary propagator of scalar field in AdS

  7. AdSd+1-BH/thermal CFTdEntropy density • Schwarzschild black hole in AdS • Bekenstein-Hawking

  8. Conjecture 1: Universal ratio • CFTs which admit a dual gravitational description via the AdS/CFT correspondence, the central charge is equal to the normalized entropy density (0801.2785:P.Kovtun,A.Ritz) • d=2 CFTs are always true due to symmetry regardless existence of gravity dual • This is a nontrivial claim for d > 2

  9. AdS/non-CFT • What happens to those non-CFT with gravitational dual description? • There appears additional scale(s) in field theory, set by non-trivial geometry or dilaton/scalar field in gravity side.

  10. Conjecture 2: Universal bound • Define c via correlator at short distance, therefore c is the same as before. • However, entropy density may be different. • We claim a lower bound by observing examples and prove by c-Theorem.[0804.0779:Wen]

  11. Example 1Hard-wall AdS/QCD Model • Hard IR cut-off in AdS, • Deconfinement phase transition set by • Entropy has been calculated [0705.1529:L.Pando Zayas]

  12. Example 2N=2* Pilch-Warner solution • Breaking N=4 to N=2 by introducing mass m to hypermultiplet, corresponding to two additional scalar fields with potential P • Entropy was calculated perturbatively at high T

  13. c-Theorem • Consider a general background with asymptotic AdS at infinity r (UV). • c-Theorem states that there exists a non-increasing function along the flow toward the IR. • This was proved by weak energy condition, saying [9904017:Freedman,Gubser,Warner]

  14. Proof for universal bound • Consider a multi-kink (Ki) geometry with AdS UV(IR) • Entropy is given in each kinkand • C-Theorem implies

  15. Thank You

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