ℐ + NEARLY DE SITTER GRAVITY arXiv:1905.03780 (Cotler, KJ, Maloney) see also arXiv:1904.01911 (Maldacena, Turiaci, Yang) Non-Perturbative Methods in Quantum Field Theory ICTP Trieste, 3 September 2019 ℐ −
DE SITTER HOLOGRAPHY? By now we have a fairly good understanding of � ℐ + AdS holography: defined by a dual CFT. What about de Sitter? “dS/CFT”: a non-unitary CFT dual to an inflating patch. But basic observables of dS are transition amplitudes between infinite past and future (“metaobservables”). Is there a dual “CFT” which computes them? � 2
DE SITTER HOLOGRAPHY? But basic observables of dS are transition amplitudes between infinite past and future (“metaobservables”). Is there a dual “CFT” which computes them? ℐ + ? = ⟨ g + | 𝒱 | g − ⟩ = ∫ [ d ξ + ][ d ξ − ] e − S CFT [ ξ + , ξ − ] Immediate problem: There are nonzero correlations between � and � without local interactions ℐ − ℐ + which couple the boundaries. ℐ − � 3
JACKIW-TEITELBOIM GRAVITY Enter “JT” gravity, a toy model for 2d quantum gravity. AdS version: [KJ] [Maldacena, Stanford, Yang] [Engelsöy, Mertens, Verlinde] g φ ( R + 2 L 2 ) 16 π G ∫ d 2 x 1 S JT = − S 0 χ − “Dilaton” Usual Euler term familiar from worldsheet No bulk dof; however there is a string theory. boundary reparameterization mode. Loops can sometimes be summed to all orders in G. [Stanford, Witten] � 4
JACKIW-TEITELBOIM GRAVITY − g φ ( R − 2 L 2 ) 16 π G ∫ d 2 x 1 S JT = S 0 χ + There is also a version with positive cosmological constant. ds 2 = − dt 2 + cosh 2 ( L ) dx 2 t “Nearly dS 2 ” solutions: φ = sinh t ℓ Gives us a theoretical laboratory to study dS quantum gravity. � 5
GOAL: QUANTUM COSMOLOGY DONE RIGHT � 6
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + Boundary conditions: Smoothly caps off in the past Near � we have a cutoff slice � , ℐ + ε dS 2 ≈ dx 2 x ∼ x + β , ε 2 , φ ≈ 1 ? J ε , � 7
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + Classical solution is complex: ds 2 = − dt 2 + cosh 2 t d θ 2 , φ = 2 π t ≥ 0 : β J sinh t , t = 0 ds 2 = d τ 2 + cos 2 τ d θ 2 , τ ∈ [0, π /2] : φ = − 2 π i β J sin τ , � 8
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + Classical solution is complex: ds 2 = − dt 2 + cosh 2 t d θ 2 , φ = 2 π β J sinh t , t = 0 t � t = − i τ − i π 2 � 9
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + 16 π G ∫ d 2 x 1 − g φ ( R − 2) S JT = S 0 χ + + 1 8 π G ∫ d θ h φ ( K − 1) t = 0 � 10
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + 16 π G ∫ d 2 x 1 − g φ ( R − 2) S JT = S 0 χ + + 1 8 π G ∫ d θ h φ ( K − 1) − iS 0 t = 0 � 11
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + 16 π G ∫ d 2 x 1 S JT = − iS 0 + − g φ ( R − 2) + 1 8 π G ∫ d θ h φ ( K − 1) t = 0 � 12
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ � 0 ℐ + 16 π G ∫ d 2 x 1 S JT = − iS 0 + − g φ ( R − 2) + 1 8 π G ∫ d θ h φ ( K − 1) t = 0 � 13
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + 8 π G ∫ d θ 1 S JT = − iS 0 + h φ ( K − 1) t = 0 � 14
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + 8 π G ∫ d θ 1 S JT = − iS 0 + h φ ( K − 1) gives Schwarzian action t = 0 � 15
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + ∫ [ Df ] e iS [ f ] Z HH = e S 0 d θ ( { f ( θ ), θ } + 1 2 π ) 4 G β J ∫ 1 2 f ′ � ( θ ) 2 S [ f ] = 0 t = 0 2 2 ( f ′ � ( θ ) ) { f ( θ ), θ } = f ′ � ′ � ′ � ( θ ) f ′ � ( θ ) − 3 f ′ � ′ � ( θ ) = Schwarzian derivative f ( θ + 2 π ) = f ( θ ) + 2 π a tan ( n.b. 2 ) + b f tan ( 2 ) ∼ f f ∈ Diff ( 𝕋 1 )/ PSL (2; ℝ ) ⇒ , ad − bc = 1 c tan ( 2 ) + d f � 16
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + ∫ [ Df ] e iS [ f ] Z HH = e S 0 1 π i 2 π ( − 2 i β J ) 3/2 e S 0 + = 4 G β J t = 0 exact to all orders in G! Not clear how to normalize � or � , | β ⟩ | ∅⟩ but we do see relative suppression to nucleate at large � , i.e. large universes. β � 17
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + 1 π i 2 π ( − 2 i β J ) 3/2 e S 0 + Z HH = 4 G β J = Z disc ( − i β J ) continuation of Euclidean AdS 2 result! t = 0 � 18
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + Classical solution is complex: ds 2 = − dt 2 + cosh 2 t d θ 2 , φ = 2 π β J sinh t , t = 0 t � t = − i τ − i π t = ρ − i π 2 2 � 19
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + Classical solution is complex: ds 2 = − ( d ρ 2 + sinh 2 ρ d θ 2 ) , φ = − 2 π i β J cosh ρ , t = 0 t � t = − i τ − i π t = ρ − i π 2 2 � 20
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + ds 2 = − ( d ρ 2 + sinh 2 ρ d θ 2 ) , φ = − 2 π i β J cosh ρ , Hyperbolic disc in (-,-) signature. Continuation from dS to EAdS [Maldacena, ’10] � 21
NO-BOUNDARY WAVEFUNCTION Consider the “disk” partition function: Z HH ≈ ⟨ β | 𝒱 | ∅⟩ ℐ + Z HH = ∫ ∞ dE ρ ( E ) e i β E ∼ tr ( e i β H ) 0 2 π 3/2 J sinh ( GJ ) ρ ( E ) = e S 0 G π E � 22
GLOBAL NEARLY DS 2 Now the annulus partition function: Z global ≈ ⟨ β + | 𝒱 | β − ⟩ ℐ + Classical solutions (for � ): β + β + = β − = β ds 2 = − dt 2 + α 2 cosh 2 t d Ψ 2 , φ = 2 πα 2 πα β J sinh t Ψ = θ + γ Θ ( t ) β − ℐ − � 23
GLOBAL NEARLY DS 2 Now the annulus partition function: Z global ≈ ⟨ β + | 𝒱 | β − ⟩ Z global = − ∫ α 2 π ℐ + d α α 2 G ∫ d γ ∫ [ Df + ][ Df − ] e iS [ f + , f − ] β + 0 0 d θ ( { f + ( θ ), θ } + α 2 ) − ( + → − ) 2 π 4 G β + J ∫ 1 + ( θ ) 2 2 f ′ � S [ f + , f − ] = 2 πα 0 ∞ Z global = − 2 π ∫ d α α 2 G Z T ( β + J , α ) Z * T ( β − J , α ) β − 0 1 π i α 2 ℐ − Z T ( β J , α ) = 2 π ( − 2 i β J ) 1/2 e 4 G β J � 24
GLOBAL NEARLY DS 2 Now the annulus partition function: Z global ≈ ⟨ β + | 𝒱 | β − ⟩ Z global = − ∫ α 2 π ℐ + d α α 2 G ∫ d γ ∫ [ Df + ][ Df − ] e iS [ f + , f − ] β + 0 0 β + β − = i 2 πα β + − β − 2 π Again exact to all orders in G. β − Can interpret as propagator for the universe. ℐ − � 25
GLOBAL NEARLY DS 2 Now the annulus partition function: Z global ≈ ⟨ β + | 𝒱 | β − ⟩ ℐ + β + β − β + Z global = i β + − β − 2 π = Z 0,2 ( β 1 J → − i β + J , β 2 J → i β − J ) 2 πα Continuation of annulus Z of EAdS 2 [Saad, Shenker, Stanford] β − ℐ − � 26
GLOBAL NEARLY DS 2 ds 2 = − dt 2 + α 2 cosh 2 t d θ 2 t − i π 2 t = ρ − i π 2 ds 2 = − ( d ρ 2 + α 2 sinh 2 ρ d θ 2 ) � 27
TOPOLOGICAL GAUGE THEORY Another way of thinking about it: JT gravity in dS is equivalent to a � � theory. PSL (2; ℝ ) BF So is JT in Euclidean AdS! For the annulus partition function of � one integrates over BF Wilson loops around the circle. Integral over � = integral over elliptic monodromies of � . PSL (2; ℝ ) α � 28
HIGHER TOPOLOGIES This viewpoint is ideally situated to tackle more complicated topologies, and so the genus expansion of JT dS gravity. There are no non-singular Lorentzian R=2 geometries beyond the annulus. However we can define the gravity on more complicated topologies by integrating over smooth, flat gauge configurations. After some work (assuming a conjecture [Do ’11] ), the genus expansion coefficients are the continuation from those recently obtained for Euclidean AdS. � 29
MATRIX INTEGRAL INTERPRETATION Let us return to the question of de Sitter holography. What dual structure can compute the various amplitudes? [Saad, Shenker, Stanford] recently showed that the genus expansion of EAdS JT gravity coincides with the genus expansion of an appropriate double scaled one Hermitian-matrix integral Z MM = ∫ dH exp ( − L tr ( V ( H )) ) (whose leading density of states coincides with that of the Schwarzian theory). � 30
MATRIX INTEGRAL INTERPRETATION Our result implies that the genus β + β + 1 2 expansion of JT dS gravity is encoded in the same ensemble. An example of the dictionary: ⋮ g ⟨ tr ( e i β + 1 H ) tr ( e i β + 2 H ) tr ( e − i β − H ) ⟩ conn,MM,g = β − � 31
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