Holographic entanglement entropy beyond AdS/CFT

Holographic entanglement entropy beyond AdS/CFT Edgar Shaghoulian Kavli Institute for the Physics and Mathematics of the Universe April 8, 014 Dionysios Anninos, Joshua Samani, and ES hep-th:1309.579

Contents Holographic entanglement entropy overview Warped AdS 3 and warped CFT overview Holographic entanglement entropy for WAdS 3 Holographic entanglement entropy for locally AdS 3 spacetime Perturbative holographic entanglement entropy for WAdS 3 Outlook

Entanglement entropy Geometric or entanglement entropy: S A = Tr A(ρ A log ρ A), ρ A = Tr Bρ. CFT in ground state on plane [Holzhey, Larsen, Wilczek; Cardy, Calabrese]: S A = c 3 CFT in ground state on cylinder: log Lx ɛ. S A = c 3 log sin(l θ/) ɛ CFT at finite left-moving and right-moving temperatures: S A = c ( ( ) ( )) βlβ 6 log R π ɛ sinh πlx πlx sinh β L β R

Holographic entanglement entropy For time-independent state in AdS/CFT, Ryu-Takayanagi (RT) proposed S A = Area(γA) 4G N for minimal surface γ A on a given time slice. Effectively proven by now [Casini, Huerta, Myers; Faulkner; Hartman; Lewkowycz, Maldacena]. Extended to quantum corrections [Barella, Dong, Hartnoll, Martin; Faulkner, Lewkowycz, Maldacena], higher spin theories [de Boer, Jottar; Ammon, Castro, Iqbal], higher curvature theories [Hung, Myers, Smolkin; Dong].

Time-dependent holographic entanglement entropy Hubeny-Rangamani-Takayanagi (HRT) proposed generalization S A = Area(γA) 4G N for extremal surface γ A not restricted to time slice. Formula unproven but satisfies nontrivial checks, e.g. strong subadditivity [Callan, He, Headrick; Wall], and reproduces CFT formulae at finite T L and T R. No extension of HRT proposal to non-ads UV asymptotics.

Warped AdS 3 Spacelike WAdS 3 written as fibration over Lorentzian AdS base space: ) ds = ( (1 l + r )dτ + dr 4 1 + r + a (du + r dτ) in global coordinates and ( ds = 1 4 in Poincaré-like coordinates. l dψ + dx x + a ( dφ + l dψ ) ) x All coordinates valued in R; a [0, ). R = (a 4) l. Solution of Einstein gravity plus matter; exists in string theory. Isometry group SL(, R) U(1), unless a = 1. No conformal boundary (but there exists anisotropic conformal infinity [Horava, Melby-Thompson]). Discrete identification gives warped BTZ black hole.

Warped AdS 3 Compactify fiber coordinate φ to get 3D part of NHEK geometry: ( ( ds = JΩ(θ) dψ + dx + dθ + a(θ) dφ + dψ ) ). x x Relevant for Kerr/CFT and understanding of astrophysical black holes.

Warped CFT Consider theories defined as having SL(, R) U(1) symmetry and proposed to be holographically dual to warped AdS 3. Symmetry automatically enhanced to infinite-dimensional V ir U(1) Kac-Moody [Hofman, Strominger]; this case referred to as warped CFT. Cardy-like formula can be derived for density of states [Detournay, Hartman, Hofman].

Trivial warping: AdS 3 spacetime Set a = 1 to get AdS 3 spacetime in fibered Poincaré-like coordinates: ( ds = 1 ( l dψ 4 x + dx l x + dφ + l dψ ) ). x In fibered global coordinates we have ) ds = ( (1 l + r )dτ + dr + (du + r dτ). 4 1 + r HRT proposal can be applied to these spacetimes! We stick to r = +. Coordinates on boundary are null. Figure : Adapted from 0905.61.

HRT proposal for fibered Poincaré-like AdS 3 spacetime Answer in terms of two null distances: ( S EE = c 3 log 1 L ψ l sinh ɛ = c 6 log L ψ ɛ ( ) ) Lφ l + c ( l 6 log ɛ sinh ( )) Lφ l ψ-movers in ground state and φ-movers at finite temperature l. Holographic renormalization shows T ψψ = T ψφ = 0; T φφ 0. Compactifying fiber coordinate gives near-horizon limit of extremal BTZ, which has T L = 0 and T R 0.

HRT proposal for fibered global AdS 3 spacetime ( S EE = c 3 log 1 sin ɛ ( Lτ = c 6 log ( 1 ɛ sin ( Lτ ) sinh ( ) ) Lu )) + c ( ( 1 6 log ɛ sinh Lu τ-movers in ground state on cylinder and u-movers at finite temperature. Coordinates all dimensionless. ))

WAdS 3 Geodesics Easy to solve for u(λ), τ(λ), and r(λ) in terms of conserved momenta c τ, c u and c v. For AdS 3, lim λ u(λ) = k for constant k, but for warped AdS 3 solution for fiber coordinate has piece linear in λ. We find Length λ = log [r f1(cu, cτ, a)] 1 + (1 1/a )c u and ( ) Lτ c τ = f (c u, a) cot, ( 1 + 1a ) c uλ + log cu + 1 + c u c u a = L u. c u 1 + c u c u a

Perturbation theory Troubling equation is ( 1 + 1a ) c uλ + log cu + 1 + c u c u a = L u, c u 1 + c u c u a transcendental in c u. Solve perturbatively instead for a = 1 + δ: with to assure convergence. Guaranteed if c u = c u,0 + δ c u,1 + δ c u, +, δ n c u,n δ n 1 c u,n 1 L u 1, λ δ 1. Latter requirement interpreted as remaining in AdS 3 part of geometry; this is just AdS/CFT in presence of infinitesimal, irrelevant source! HRT proposal should apply.

Answer Perturbative answer to all orders: l 4G N i= S EE = l 4G N [( 1 + δ coth L u δ i ( 1) i+1 coth L u L u csch(i 1) ) ( log r sin Lτ [ ( log r sin Lτ ( i )] Lu sinh + )] i sinh Lu ) c ij cosh(jl u). Taking L u 1 and δ > 1/ lets us sum the entire series to get ( S EE = l ( ) ) 1 (1 + δ) log sin Lτ G N ɛ exp Lu. j=0

Reading off the central charge Answer to all orders in δ in the large-l u limit: ( S EE = l 1 (1 + δ) log G N ɛ sin Lτ exp ( ) ) Lu. We have recovered universal CFT answer in large L u limit, with c L = c R = 3l G N (1 + δ). Peforming same perturbative expansion in Poincaré-like coordinates again gives universal CFT answer: ( S EE = l ( ) ) 1 Lφ (1 + δ) log L ψ l exp. G N ɛ l

Warped BTZ black hole Spacelike warped BTZ black holes are locally spacelike warped AdS 3 [Anninos, Li, Padi, Song, Strominger]: ds l = 3dt 4 a + dr 4(r r +)(r r + 6 3 ) (4 a ) (ar r +r ) dtdθ 3/ + 9r (4 a ) ( (a 1)r + r + + r a r +r ) dθ, θ θ + π Perturbative answer in large fiber-coordinate regime given by ( ( ) ) S EE = la r + r 3 π θ log exp t + sinh π θ, G N ɛ a (4 a ) β L β R with dimensionless temperatures ( β 1 3 L = T L = r π(4 a + + r ) r+r, ) a β 1 R = T R = 3(r+ r ) π(4 a ).

Nonperturbative (in δ) proposal Cardy formula is nontrivial check (even for finite δ): ( S = π 3πl (cltl + crtr) = 3 G N (4 a ) (ar+ ) r +r ) = A 4G N. Physically relevant range is a [0, ). Our expansion converges for a (1/, ), i.e. δ > 1/, so we propose it is valid in that range.

Open questions Full nonperturbative application of HEE proposal? Nonlocality in the UV with volume law? Independent way to see WCFT reproduce CFT behavior; correlation functions? φ i(x, x + )φ j(y, y + ) = fij(x y ) (x + y + ) λ i+λ j Universal entanglement entropy formulae in WCFT, without holography [ES, in progress (sort of)]. Extension to TMG [Castro, Detournay, Iqbal, Perlmutter, in progress]. Extend perturbative approach to spacetimes continuously connected to AdS d+. Produce c L = 1J in NHEK.

Take-away Sometimes useful to compute holographic entanglement entropy perturbatively. Warped CFT seems CFT -like at finite temperature, as long as we take an IR limit and large fiber coordinate separation. Central charge, T L, and T R predicted, with Cardy formula satisfied; first quantiatively successful application of (covariant) holographic entanglement entropy to non-asymptotically AdS spacetime! Many concrete directions for progress; outlook hopeful!

Additional Material Compactification of common bosonic sector of IIA/B and heterotic SUGRAs (Einstein frame): S = 1 d 10 x g (R κ 10 1 ( φ) 11 ) e Φ H MNP H MNP. 10 Compactify on S 3 T 3 S 1 and keep KK gauge field from S 1, with background ds 10 = e 3Y / ( e X ds 3 + e X (dφ + A) ) + e Y L Sds (S 3 ) + ds (T 3 ) H = h SL 3 SV ol(s 3 ) + Ĥ + ˆF (dφ + A) for Ĥ d ˆB ˆF A, L S the radius of S 3, h S a constant, and ds (S 3 ) and Vol(S 3 ) the metric and volume forms on S 3. We can thus reduce and consistently truncate the resulting 3D action to S 3D = 1 d 3 x ( g 3 R 3 1 ) 8 e3y Φ F + L kin (Φ, Y ) h 3e Φ 6Y κ 3 + 1 κ 3 d 3 x ( ) 1 g 3 e 4Y +Φ/ h Se 6Y Φ/ h3 L S 4κ 3 A F