Entanglement entropy in flat holography

Published for SISSA by Springer Received: July 2, 2017 Accepted: July 15, 2017 Published: July 28, 2017 Hongliang Jiang,a Wei Songb and Qiang Wenb a Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, P.R. China b Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China E-mail: , , Abstract: BMS symmetry, which is the asymptotic symmetry at null infinity of flat spacetime, is an important input for flat holography. In this paper, we give a holographic calculation of entanglement entropy and Rényi entropy in three dimensional Einstein gravity and Topologically Massive Gravity. The geometric picture for the entanglement entropy is the length of a spacelike geodesic which is connected to the interval at null infinity by two null geodesics. The spacelike geodesic is the fixed points of replica symmetry, and the null geodesics are along the modular flow. Our strategy is to first reformulate the Rindler method for calculating entanglement entropy in a general setup, and apply it for BMS invariant field theories, and finally extend the calculation to the bulk. Keywords: Black Holes, Field Theories in Lower Dimensions, Gauge-gravity correspondence, Models of Quantum Gravity ArXiv ePrint: 1706.07552 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP07(2017)142 JHEP07(2017)142 Entanglement entropy in flat holography Contents 1 2 Generalized Rindler method 2.1 Field theory calculation of entanglement entropy 2.1.1 Generalized Rindler method 2.1.2 “Cardy” formula 2.2 Holographic entanglement entropy 3 4 4 6 6 3 Review of BMS group and asymptotically flat spacetime 3.1 BMS invariant field theory 3.2 BMS as asymptotic symmetry group 3.3 Global Minkowski, null-orbifold and FSC 3.4 Poincaré coordinates 3.5 Solutions with general spatial circle 7 7 8 8 10 10 4 Entanglement entropy in field theory side 4.1 Rindler transformations and the modular flow in BMSFT 4.2 Entanglement entropy for BMSFT 4.2.1 Zero temperature BMSFT on the plane 4.2.2 Finite temperature BMSFT 4.2.3 Zero temperature BMSFT on the cylinder 11 11 14 14 15 16 5 Holographic entanglement entropy in gravity side 5.1 The strategy 5.2 Poincaré coordinate 5.3 FSC 5.4 Global Minkowski 16 16 17 19 21 6 The geometric description for holographic entanglement entropy 6.1 Three special curves 6.2 The modular flow and its bulk extension 6.3 The geometric picture of holographic entanglement entropy 21 21 24 25 7 Holographic entanglement entropy from flat limit of AdS3 7.1 Holographic entanglement entropy for AdS3 with Brown-Henneaux boundary conditions 7.2 Flat limit of AdS3 7.2.1 Transformation to Bondi gauge and the flat limit 7.3 Holographic entanglement entropy in flat limit 25 –i– 26 28 28 29 JHEP07(2017)142 1 Introduction 30 30 32 32 33 9 Rényi entropy 33 A “Cardy” formula in BMSFT revisited 35 B Killing vectors B.1 FSC B.2 Poincaré patch B.3 Global Minkowski 37 38 38 39 C Rindler transformations for BMSFT C.1 Thermal BMSFT C.2 BMSFT on the cylinder 39 39 40 1 Introduction Holography [1, 2], which relates a theory containing gravity in higher spacetime dimensions to a quantum field theory in lower dimensions, is believed to be a promising way to understand quantum gravity. In particular, holography for asymptotically locally AdS (AlAdS) spacetimes, the so-called AdS/CFT correspondence [3–5] is one of the most active research fields. A priori, it is not clear whether the rich conceptual achievements from AdS/CFT are contingent to AlAdS. To understand the generality, it is important to extend the great success of holography beyond the context of AdS/CFT. Progresses for non-AlAdS holography includes dS/CFT correspondence [6, 7], the Shrödinger or Lifshitz spacetime/non-relativistic field theory duality [8–11], the Kerr/CFT correspondence [12–16], the WAdS/CFT [17] or WAdS/WCFT [18, 19] correspondence, and illuminating results toward flat holography in four dimensions [20–23] and three dimensions [24–27]. A general item in the dictionary of holography is that the asymptotic symmetry for the gravitational theory in the bulk agrees with the symmetry of the dual field theory, if the later exists. For four dimensional flat spacetime, the asymptotic symmetry group at null future (past) is the BMS± group, first studied by Bondi, van der Burg, Metzner and Sachs [20, 21]. In a recent resurgence, Strominger [22] pointed out that the diagonal elements of BMS+ × BMS− → BMS is the symmetry of the S−matrix. BMS group is connected to infra properties of scattering amplitude [28, 29], and memory effects [30], in a triangle [31]. See the lecture notes [32] for a review. As a simple toy model, the three ± dimensional version of BMS group has also generated lots of interested. BMS ± 3 on I –1– JHEP07(2017)142 8 Topologically massive gravity 8.1 Topologically massive gravity in flat space 8.2 Thermal entropy formula from flat limit 8.3 Thermal entropy formula from direct calculation 8.4 CS contribution to holographic entanglement entropy –2– JHEP07(2017)142 was discussed in [24, 33–36]. Interesting developments include connections with Virasoro algebra [34], isomorphism between BMS algebra and Galileo conformal algebra [25], representations and bootstrap [37–45]. Flat holography based on BMS3 symmetry was proposed in [25, 26] and supporting evidence can be found in [46–48]. The antipodal identification in three dimensions was discussed in [49, 50]. See [51–56] for more discussions for this flat holography and related topics. One useful probe of holography is the entanglement entropy, which describes the correlation structure of a quantum system. In the context of AdS/CFT correspondence, Ryu and Takayanagi [57, 58] (RT) proposed that the entanglement entropy is given by the area of a codimension-two minimal surface in the bulk, which anchored on the entangling surface of the subsystem on the boundary. A covariant version was proposed by Hubeny, Rangamani and Takayanagi (HRT) [59]. Using AdS/CFT for Einstein gravity, RT and HRT proposal have been proved by [60–64]. It is interesting to ask if the connection between spacetime structure in the bulk and entanglement in the boundary still exist beyond the context of AdS/CFT, and if so, how it works for non-AlAdS spacetimes. So far, in the literature, there are three approaches. The first approach is to start with the RT or HRT proposal, and study the implications in the holographic dual, see [65–69]. The second approach is to directly propose a prescription in the bulk, and check its consistency [70, 71]. The third approach, which we will advocated in the current paper, is to derive an analog of RT proposal using the dictionary of holography, along the lines of [60–64]. In [72, 73], holographic entanglement entropy in Warped AdS3 spacetime was derived by generalizing the gravitational entropy [63] and Rindler method [60, 74], respectively. (...truncated)