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

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c-Theorem and Universal bound in AdS/non-CFT
correspondence
Based on arXiv0804.0779

• ???Wen-Yu Wen (NTU)
• 4/29/08 _at_ CYCU

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Outline

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

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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 31 dimensions
• Field contents gauge field Aµ, 6 scalars Xi, 4
  fermions ?a
• R-symmetry SU(4) SO(6)

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AdS5/CFT4 correspondence

• It was conjectured that same degrees of freedom
  are described by closed string (supergravity) on
  its (D3 branes) near horizon geometry AdS5S5

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AdS5/CFT4 correspondence

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

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d-dimensional Conformal Field TheoryCentral
charge

• Cardys two-point function correlator
• This can also be calculated via boundary-boundary
  propagator of scalar field in AdS

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AdSd1-BH/thermal CFTdEntropy density

• Schwarzschild black hole in AdS
• Bekenstein-Hawking

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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
  (0801.2785P.Kovtun,A.Ritz)
• d2 CFTs are always true due to symmetry
  regardless existence of gravity dual
• This is a nontrivial claim for d gt 2

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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.

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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.0804.0779Wen

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Example 1Hard-wall AdS/QCD Model

• Hard IR cut-off in AdS,
• Deconfinement phase transition set by
• Entropy has been calculated 0705.1529L.Pando
  Zayas

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Example 2N2 Pilch-Warner solution

• Breaking N4 to N2 by introducing mass m to
  hypermultiplet, corresponding to two additional
  scalar fields with potential P
• Entropy was calculated perturbatively at high T

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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
  9904017Freedman,Gubser,Warner

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Proof for universal bound

• Consider a multi-kink (Ki) geometry with AdS
  UV(IR)
• Entropy is given in each kinkand
• C-Theorem implies

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Thank You