de Sitter space and Holography

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Conformal AnomalyAdS/CFT instructs us to evaluate the bulk action on a givensolution in order to calculate the boundary partition functionfor a given boundary metric. This quantity has divergencesdue to the infinite volume of AdS. One needs to add localcounterterms.However in even boundary dimensions (odd bulk dimensiond) there are in addition log(z) terms and they represent theconformal anomaly.For the standard Einstein Hilbert action on the dS d−1 slicedAdS d background we get∫−2(d − 1) dzS on−shell =16πG N sinh d (z)Expanding in powers of z around z = 0 one finds∫dzsinh 3 = − 1(z) 2z 2 − log(z) + O(z 0 )∫2dzsinh 5 = − 1(z) 4z 4 + 512z 2 + 3 8 log(z) + O(z0 )∫dzsinh 7 = − 1(z) 6z 6 + 724z 2 − 259720z 2 − 5 16 log(z) + O(z0 )The log terms give the conformal anomaly evaluated on dS 2 ,dS 4 and dS 6 respectively.54
Now let us repeat the same exercise for the gravitationalaction with position dependent G N . The position dependenceof M d is M d (z) = tanh(z)M d which gives an extra factor oftanh d−2 (z). Up to terms that remain finite as z → 0S on−shell ==∫∫tanh d−2 dz(z)sinh d (z)1 dzsinh 2 (z) cosh d (z) = −1z + O(z)For all d the only divergent term is a universal − 1 zand thereare no logarithms. The conformal anomaly vanishes.One possible interpretation:• Lower dimensional gravity screens the central charge tobe zero, just like is well known from 2d gravity on stringtheory worldsheets. In this scenario one does not even need aconformal field theory beyond scales 1/R since the gravitationaldressing will also make any FT a CFT.• In the same spirit the universal UV dimension of the scalarfields can be understood as gravitational dressing.55
Page 1 and 2:
de Sitter space and HolographyThird
Page 3: 1. Historical review and motivation
Page 7 and 8: 5. Technologies:There are a number
Page 9 and 10: Let us know study different coordin
Page 11 and 12: Conformal coordinates: T, θ i , i
Page 13 and 14: Planer (Inflationary) coordinates:
Page 15 and 16: The corresponding metric isdS 2 =
Page 17 and 18: It is easy to see thatX 0 + X d =
Page 20 and 21: 1.2 Causal structure (Penrose diagr
Page 22 and 23: IINorth PoleOSouth PoleNorth PoleOS
Page 24 and 25: The entire region of the dS space i
Page 26 and 27: For example• In static coordinate
Page 28 and 29: An observer moving along a timelike
Page 30 and 31: 3. dS/CFT correspondenceFrom what w
Page 32 and 33: In d dimensions the Einstein equati
Page 34 and 35: One can generalize it for a general
Page 36 and 37: • One can always decompose the me
Page 38 and 39: Consider the Brown-York stress tens
Page 40 and 41: Using the definition of Brown-York
Page 42 and 43: Using this diffeomorphism one may s
Page 44 and 45: lim dtdtr→∞∫I ′ dφdφ ′
Page 46 and 47: dS static patch (spatial)dg =0 g =1
Page 48 and 49: • Lower dimensional Plank mass is
Page 50 and 51: One can use AdS/CFT correspondence
Page 52 and 53: Under this conformal map one has:
Page 56 and 57: • Naively one would think that gr
Page 58: This talk is based on the following
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de Sitter space and Holography

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