de Sitter space and Holography
de Sitter space and Holography de Sitter space and Holography
• Naively one would think that gravitational dressing shouldbring the operator dimension to d − 1 so that one can add itto the action. But the (d − 1)/2 ± 1/2 we find is consistent aslong as we only add products of the form Od/2 1 O2 d/2−1to theaction. We know that the coupling of the two CFTs has to beachieved via its boundary interactions.• Continuity across the UV brane in the original dS spacemeans that the value of the field at the boundary in one AdS(dual to CFT 1) appears as a boundary condition in the secondAdS (dual to CFT 2).• The discussion of multi-trace operators uses precisely theproduct operator O = Od/2 1 O2 d/2−1in order to achieve boundaryconditions of the type we want, at least in a folded version ofour duality: instead of one scalar field living in 2 copies of AdSthere one has 2 decoupled scalar fields in one copy of AdS.• Since we are dealing with gravity in addition to scalar fields,for us the 2 copies of AdS are more appropriate in order toavoid having 2 gravitons living in the same space.56
Some comments on dS/dS correspondence• The de Sitter space is a particularly symmetric example ofa Randall-Sundrum system. Bulk gravity calculations allow oneto probe some of the basic physics of induced gravity.• We computed several quantities determining how the lowerdimensional theory behaves at energies above the cutoff. Thisresulted in a holographic verification that the total centralcharge and heat capacity is zeroed out and that a simpleasymptotic dressing of operator dimensions arises. These areboth features familiar in 2d gravity plus matter systems. Directcouplings between the two CFTs are also required.• The duality naturally can also be extended to situationswith changing cosmological constant, such as inflation.• Repeated application of our duality allows one to go tosufficiently low dimensions, that is two or one, so that gravitybecomes non-dynamical and its effects reduce to constraints.• The motivation for this dual formulation is ultimately toprovide a framework for the physics of accelerated expansionin the real universe. Although at large radius and low energiesthe effective weakly coupled description remains the bulk ddimensional one, the description in terms of d − 1 dimensionalphysics may shed light on the physics of inflation and darkenergy.57
- 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 54 and 55: Conformal AnomalyAdS/CFT instructs
- Page 58: This talk is based on the following
• Naively one would think that gravitational dressing shouldbring the operator dimension to d − 1 so that one can add itto the action. But the (d − 1)/2 ± 1/2 we find is consistent aslong as we only add products of the form Od/2 1 O2 d/2−1to theaction. We know that the coupling of the two CFTs has to beachieved via its boundary interactions.• Continuity across the UV brane in the original dS <strong>space</strong>means that the value of the field at the boundary in one AdS(dual to CFT 1) appears as a boundary condition in the secondAdS (dual to CFT 2).• The discussion of multi-trace operators uses precisely theproduct operator O = Od/2 1 O2 d/2−1in or<strong>de</strong>r to achieve boundaryconditions of the type we want, at least in a fol<strong>de</strong>d version ofour duality: instead of one scalar field living in 2 copies of AdSthere one has 2 <strong>de</strong>coupled scalar fields in one copy of AdS.• Since we are <strong>de</strong>aling with gravity in addition to scalar fields,for us the 2 copies of AdS are more appropriate in or<strong>de</strong>r toavoid having 2 gravitons living in the same <strong>space</strong>.56
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