Entanglement Entropy in Flat Holography

Transcription

1 Entanglement Entropy in Flat Holography Based on work with Qiang Wen, Jianfei Xu( THU), and Hongliang Jiang( The HKUST) East Asian Joint Workshop KEK Theory workshop 2017

2 Bekenstein Bardeen-Carter-Hawking Strominger-Vafa Maldacena Gubser-Klebanov-Polyakov, Witten Ryu-Takayanagi Hubeny-Rangamani-Takayanagai Casini-Huerta-Myers Lewkowycz-Maldacena Introduction S "# = A # 4G S (# AdS/CFT S *(/#*( = A,-. 4G Talk by Takayanagi S ))

3 Introduction S "# = A 4G Challenge: Holographic duality for nonasymptotically-ads spacetime AdS/CFT Holographic Dualities Kerr/CFT ds/cft Flat holography S (# unsual structures in the dual QFT: Not Lorentzian invariant Non-local

4 Question: what is the holographic entanglement entropy in holographic dualities for non-ads spacetimes? In this talk: Holographic EE in 3d asymptotically flat spacetime The strategy: use the proof to derive the bulk picture Assumptions: holography exists. Asymptotic symmetry in gravity =symmetry of the dual field theory A new geometric picture

5 ØThe generalized Rindler method Flat holography in 3d HEE in 3d flat spacetime

6 similarity between entanglement entropy( EE) and thermal entropy (TE) Modular Hamiltonian K 0 = log ρ 0 Hamiltonian Renyi entropy S (9) 0 = ; ;<9 0 e <9@ A thermal partition function Entanglement entropy S 0 = Tr 0 ρ 0 log ρ 0 thermal entropy In general, the modular Hamiltonian is non-local 6

7 The Rindler method in AdS/CFT : derivation of RT for spherical subregions Casini-Huerta-Myers Gravity Hyperbolic BH Horizon S "# AdS/CFT Field theory S (# conformal transformation AdS RT surface S *( RT( HRT) S )) for special regions 7

8 Main difficulties for generalizing the Rindler method (beyond AdS/CFT) : On the field theory side, how to map EE to TH in a general QFT( without Lorentzain invariance)? On the gravity side, find the analog of hyperbolic black hole ( also called the AdS-Rindler spacetime) for non-ads spacetime?

9 How to map EE to TE in a QFT( without Lorentzain invariance)? A prescription: to find the generalized Rindler transformation Jiang-WS-Wen Symmetry transformation x * xf = f(x) mapping is invariant under a `thermal x should identification be i x i x i + i i. The vacuum is invariant under translations hereafter. in xf - Modular flow generated by: k t x i keeps the domain of dependence invariant flow x i (s) We argue that if such transformation can be found, EE (vacuum) is mapped to TE. Comments: For special entangling regions, on the vacuum Independent of details of the field theory Algorithm to find Rindler transformations and modular Hamiltonians Works for both BMSFT and WCFT

10 Our prescription for generalizing the Rindler method (beyond AdS/CFT) : On the field theory side, how to map EE to TH the (generalized) Rindler transformation Jiang-WS-Wen Based on Castro-Hofman-Iqbal WS-Wen-XU On the field theory side, find the analog of hyperbolic black hole (also called the AdS-Rindler spacetime) construct the bulk `Rindler spacetime via quotient method WS-Wen-Xu

11 The generalized Rindler method (beyond AdS/CFT) Gravity Castro-Hofman-Iqbal WS-Wen-Xu Jiang-WS-Wen Field theory Black objects S "# Holographic dictionary S (# Quotient Non-AdS S #)) : = S "# New geometric picture for EE Rindler transforamation S )) (Special subregions) 11

12 üthe generalized Rindler method ØFlat holography in 3d HEE in 3d flat spacetime

13 BMS symmetry is the asymptotic symmetry for flat spacetime at null infinity. Bondi, van der Burg, Metzner, Sachs 62 Let us focus on the 3d version Barnich-Troessaert, Bagchi, 10 Bondi gauge at future null infinity ds K = P φ du K 2dudr + 2 J[φ] + u 2 UP φ dudφ + r K dφ K Zero mode solutions: ds K = Mdu K 2dudr + Jdudφ + r K dφ K M = 1, J = 0 global Minkowski spacetime M > 0, quotien of global Minkowski, Flat space cosmological solutions( FSC) S [\] = πj G M

14 The asymptotic symmetry for flat spacetime in 3d BMS` algebra L., L, L., M, = n m L.d, + c f n` n δ 12. d,,h = n m M.d, + c i n` n δ 12.d,,h M., M, = 0 For Einstein gravity c f = 0, c i = ` j For Topologically massive gravity c f 0

15 CFT: z = f z, wo = g w, Tr z = f <K (T z c f {f, z}) z 12 BMSFT: BMS invariant field theory z = f z, wo = f p z w + g z } ~ ;K {g, z}) Ju(z )= vw vx Pr(z )= f z <K (P z c i {f, z}) 12 <K y (J z z v{ {f, z})+ ;K vx <K (P z

16 `Cardy formula in BMSFT a torus with a `thermal circle and a `spatial circle (ũ, ) (ũ + iā, ia) (ũ +2 b, 2 b). Detournay-Hartman-Hofman Barnich Bagchi-Detournay-Fareghbal- Simón Jiang-WS-Wen S-transformation: 1. locally an allowed symmetry transformation 2. `thermal circle \<. wƒ, -ƒ. `spatial circle S b b (ā a) = 2 3 c L b a + c M (āb a b). `Cardy formula in BMSFT reproduces thermal entropy S [\] a 2 Note: keep track of all anomalies

17 üthe generalized Rindler method üflat holography in 3d ØHEE in 3d flat spacetime

18 Rindler method in BMSFT EE on A:, K K Rindler transformation: = ũ + u = arctanh tan( 2 ) tan( l 4 ), sin(l /2) 2 (cos cos(l /2)) u ( K, K ) p 1 2 l u csc( l 2 )sin. Jiang-WS-Wen BMS transformation Thermal circle (ũ, ) (ũ + i u, i ) flow is given by Entanglement entropy S )) A = S Br = 1 4G l u cot l 2 = S "<" [\ Br 2 u

19 Modular Hamiltonian in BMSFT Generators of modular flow Jiang-WS-Wen l k t = csc 2 +2 cos l 2 Modular Hamiltonian: K = Q(k ) +const. l l u csc + l u cos( ) cot 2 l cos( ) 2 l +2usin( u 2 k A = 0

20 The gravity story Extend the Rindler transformation to the bulk Global Minkovski 9Œƒ-. FSC symmetry: Poincare U 1 U(1) bulk transformation boundary transformation S #)) : = S [\] Bulk modular flow k Œ γ = 0

21 Holographic entanglement entropy in Flat` Jiang-WS-Wen Red A: a single interval at boundary Blue γ: spacelike geodesic all points are fixed points of the modular flow Green γ ± : null geodesics, the curve is invariant under the modular flow S #)) (A) = šj = šj = S )) (A) + d œ šj 21

22 Consistency Checks: 5 ways of calculating EE all agree Pure field theory calculation: Replica trick: twist operators Bagchi-Basu-Grumiller-Riegler Generalized Rindler method: S )) S (# S j. -x ] ž Jiang-WS-Wen Holographic calculation: Wilson lines in Chern-Simons formalism Basu-Riegler S )) S (# S "# S #)) Jiang-WS-Wen Flat limit from AdS` /Wigner contraction from CFT

23 Question: what is the holographic entanglement entropy in holographic dualities for non-ads spacetimes? In this talk: Summary A generalized Rindler method is developed Holographic EE in 3d asymptotically flat spacetime A new geometric picture( modifications of the homologous condition of RT) Modular Hamiltonian can be written down

24 What we have explored Dualities Isometry Asymptotic symmetry AdS` /CFT K Black Hole entropy EE in field theory HEE SL 2, R f Vir f Vir * S "# CC RT/HR U 1 * = S ] ž T Bootstrap Geodesic Witten Flat/BMSF T Kerr/CFT WAdS/CFT Poincare BMS S [\] = S "<" [\ BBGR JSW SL 2, R f Vir f S "# U 1 * = S ] ž SL 2, R f Vir f Vir * S "# CC SWX U 1 * = S ] ž (W)AdS/W CFT SL 2, R f Vir f U 1 * Kac Moody f S "# DHI SWX SX = S ## Relevant for astrophysical BHs

25 Welcome to Yau Mathematical Sciences Center Tsinghua University, Beijing Si Li Hai Lin Thank you!