higher genus partition functions from three dimensional gravity Henry Maxfield 1601.00980 with Simon Ross (Durham) and Benson Way (DAMTP) 21 March 2016 McGill University 1
motivation
None of these address the qualitative structure of entanglement shared between many parties, e.g. W 100 010 001 vs GHZ 000 111 entanglement and geometry Hints from holography: emergence of geometry is closely related to entanglement structure of CFT. A • Entropy and area: S = 4 G N [Bekenstein-Hawking’80s][Ryu-Takayanagi ’06] • Entanglement wedge hypothesis: CFT subregion encodes gravitational EFT in region up to minimal surface • Consistency of entanglement restricts geometry and gravitational dynamics 3
entanglement and geometry Hints from holography: emergence of geometry is closely related to entanglement structure of CFT. A • Entropy and area: S = 4 G N [Bekenstein-Hawking’80s][Ryu-Takayanagi ’06] • Entanglement wedge hypothesis: CFT subregion encodes gravitational EFT in region up to minimal surface • Consistency of entanglement restricts geometry and gravitational dynamics None of these address the qualitative structure of entanglement shared between many parties, e.g. | W ⟩ ∝ | 100 ⟩ + | 010 ⟩ + | 001 ⟩ vs | GHZ ⟩ ∝ | 000 ⟩ + | 111 ⟩ 3
a simple set of geometries and states Topologically nontrivial solutions to pure 3D gravity: multiboundary black holes [Brill ’95] Dual to entangled state on several copies of the CFT | Σ ⟩ ∈ H 1 ⊗ H 2 ⊗ H 3 naturally defined in any theory by the path integral on a bordered Riemann surface Σ . [Skenderis-van Rees ’11] AdS dual is connected geometry only for some moduli. E.g. thermofield double, Hawking-Page phase transition. 4
edges of moduli space ‘Cold’ limit [Balasubramanian,Hayden,Maloney,Marolf,Ross ’14] C ijk e − β 1 H 1 / 2 e − β 2 H 2 / 2 e − β 3 H 3 / 2 | i ⟩ 1 | j ⟩ 2 | k ⟩ 3 ∑ | ψ ⟩ = ijk Dual: disconnected copies of AdS, entanglement is O ( c 0 ) . ‘Hot’ limit [Marolf,HM,Peach,Ross ’15] Each region in local TFD, purified by some other region 3 3 1 2 1 2 Entanglement is local and bipartite. Dual: ℓ horizons ≫ ℓ AdS 5
phases and partition functions Wavefunction evaluated on field configuration ϕ computed by ∫ D Φ e − I Σ [Φ] ⟨ ϕ | Σ ⟩ = Φ( ∂ Σ)= φ Norm ⟨ Σ | Σ ⟩ computed by inserting complete set of field configurations: path integral on Σ and a reflected copy, sewn along boundaries. Calculates the partition function on ‘Schottky double’ X of Σ , so ⟨ Σ | Σ ⟩ = Z ( X ) (generalise by inserting operators). Phases come from dominance of different saddle point geometries in dual gravitational path integral for Z ( X ) . 6
motivations • Phase structure of geometric states • Symmetry breaking and non-handlebodies [Yin ’07] • Computation of Rényi entropies [Faulkner ’13] • Universal (vacuum module) part of any CFT • Mathematical: Kähler potential for Weil-Petersson metric on Teichmüller space [Takhtajan-Zograf ’88] 7
problem and solution
For CFT, interesting dependence is on the conformal structure of X . In 2 dimensions, equivalent to complex structure, so X is naturally a Riemann surface. Each CFT gives a function on moduli space of Riemann surfaces. Holography: on-shell action of bulk with boundary X . background Partition function: may do the path integral on any geometry ∫ D Φ e − I X [Φ] Z ( X ) = Example: for X = space × S 1 E e − β E β , get Z = ∑ 9
background Partition function: may do the path integral on any geometry ∫ D Φ e − I X [Φ] Z ( X ) = Example: for X = space × S 1 E e − β E β , get Z = ∑ For CFT, interesting dependence is on the conformal structure of X . In 2 dimensions, equivalent to complex structure, so X is naturally a Riemann surface. Each CFT gives a function on moduli space of Riemann surfaces. Holography: on-shell action of bulk M with boundary ∂ M = X . 9
pure 3d gravity Possible to find solutions M with ∂ M = X in 3D pure gravity because it’s locally trivial: M = H 3 / Γ for Γ ⊆ ISO ( H 3 ) • ISO ( H 3 ) = SO ( 3 , 1 ) ≡ PSL ( 2 , C ) • Acts on boundary ∂ H 3 = P 1 by Möbius maps w �→ aw + b cw + d • Need X ≈ P 1 / Γ as quotient of Riemann sphere The appropriate construction is Schottky uniformisation 10
schottky uniformisation Cut 2 g holes in the sphere and glue them in pairs with some Möbius maps L 1 , . . . , L g . This makes a genus g surface: 11
schottky uniformisation Cut 2 g holes in the sphere and glue them in pairs with some Möbius maps L 1 , . . . , L g . This makes a genus g surface: The action of L i extends into H 3 . Fundamental region of bulk bounded by hemispheres, identified in pairs. ( Handlebodies ) 11
schottky uniformisation Cut 2 g holes in the sphere and glue them in pairs with some Möbius maps L 1 , . . . , L g . This makes a genus g surface: The action of L i extends into H 3 . Fundamental region of bulk bounded by hemispheres, identified in pairs. ( Handlebodies ) Multiple solutions for any given Riemann surface boundary X : choice of g independent cycles to fill 11
action! Now evaluate action: − 1 [∫ d 3 x √ g ( R + 2 ) + 2 d 2 x √ γ ( κ − 1 ) + constant ] ∫ I = 16 π G N M ∂ M Divergent! Cutoff depends on choice of boundary metric ds 2 = e 2 φ ( w , ¯ w ) dwd ¯ ⇒ cutoff at z = ϵ e − φ + · · · w = Dependence on choice of metric gives the conformal anomaly: c d 2 x √ γ ∫ ( ∇ ω ) 2 + R ω log Z [ e 2 ω γ ] = log Z [ γ ] + ( ) 24 π 12
Metric invariant under quotient group: for L , 1 Lw d Lw d Lw w dwdw 2 e 2 e 2 Lw w 2 log L w Multiple solutions for given X : helps to match moduli action! Canonical choice of metric: constant curvature R = − 2. R = − 2 e − 2 φ ∇ 2 ϕ = ∇ 2 ϕ = e 2 φ ⇒ 13
Multiple solutions for given X : helps to match moduli action! Canonical choice of metric: constant curvature R = − 2. R = − 2 e − 2 φ ∇ 2 ϕ = ∇ 2 ϕ = e 2 φ ⇒ Metric invariant under quotient group: for L ∈ Γ , ⇒ ϕ ( Lw ) = ϕ ( w ) − 1 � 2 e 2 φ ( Lw ) d ( Lw ) d ( Lw ) = e 2 φ ( w ) dwd ¯ w = 2 log � � L ′ ( w ) � 13
action! Canonical choice of metric: constant curvature R = − 2. R = − 2 e − 2 φ ∇ 2 ϕ = ∇ 2 ϕ = e 2 φ ⇒ Metric invariant under quotient group: for L ∈ Γ , ⇒ ϕ ( Lw ) = ϕ ( w ) − 1 � 2 e 2 φ ( Lw ) d ( Lw ) d ( Lw ) = e 2 φ ( w ) dwd ¯ w = 2 log � � L ′ ( w ) � Multiple solutions for given X : ϕ helps to match moduli 13
the recipe 1. Solve ∇ 2 ϕ = e 2 φ on a fundamental region D for Γ 2 log | L ′ ( w ) | 2 2. With boundary conditions ϕ ( Lw ) = ϕ ( w ) − 1 3. Match moduli by geodesic lengths in canonical metric 4. Evaluate on-shell action I = − c ∫ d 2 w ( ∇ ϕ ) 2 + (boundary and constant terms) 24 π D Action of [Takhtajan,Zograf ’88], holography by [Krasnov ’00] 14
analytic example: the torus
genus 1 schottky groups A genus 1 Schottky group is generated by a single Möbius map, which we may choose to be w �→ qw , for 0 < | q | < 1. Canonical metric flat: ϕ harmonic, with ϕ ( qw ) = ϕ ( w ) − log | q | Solution: ϕ = − log ( 2 π | w | ) dwd ¯ w ds 2 = e 2 φ dwd ¯ w = w = dzd ¯ z ( 2 π ) 2 w ¯ where w = exp ( 2 π iz ) . Now z is identified as z ∼ z + 1 ∼ z + τ , with q = exp ( 2 π i τ ) . Evaluating action is straightforward: get I = c 12 log | q | 16
phases for the torus Different τ related by PSL ( 2 , Z ) give the same complex structure, but different solutions. As the moduli change smoothly, the dominant solution may change. First-order phase transitions at large c . log Z ( τ ) = 2 π c ( a τ + b ) 12 max ℑ c τ + d When τ = i β 2 π is pure imaginary: β ≥ 2 π vacuum log Z = c β 12 ( 2 π ) 2 β ≤ 2 π Cardy β This is the familiar Hawking-Page phase transition. 17
numerical solution
numerical solution We need to solve Liouville’s equation on this domain: 19
numerical solution We need to solve Liouville’s equation on this domain: Nasty shaped region! Use finite element methods Approximate domain by triangles. Discretise the equation on these elements, and solve by Newton’s method. 19
numerical solution Solution for ϕ : 19
genus 2
surfaces considered Solve explicitly for a two-dimensional subspace of genus 2 moduli. Corresponds to three-boundary wormhole with two equal horizon sizes λ 1 = λ 2 . Use moduli ℓ 12 , ℓ 3 . Conformal automorphisms Z 2 × Z 2 . Three phases: connected, disconnected (3 × AdS), partially connected (AdS + BTZ) [Same family of surfaces: single-exterior black hole with rectangular torus behind horizon; three different Rényi entropies] 21
phase diagram 3.0 2.5 2.0 Enhanced symmetries: D 6 along line ℓ 1 = ℓ 2 = ℓ 3 , and D 4 at connected/ ℓ 3 1.5 ℓ Sym Connected disconnected phase boundary. 1.0 Modular transformation swaps Disconnected connected and disconnected phases. 0.5 Partially Connected 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ℓ 12 ℓ Sym 22
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