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.

In [1] it was observed that asymptotic boundary conditions play an important role in the study of holographic entanglement beyond AdS/CFT. In particular, the RyuTakayanagi proposal must be modified for warped AdS 3 (WAdS 3 ) with Dirichlet boundary conditions. In this paper, we consider AdS 3 and WAdS 3 with Dirichlet-Neumann boundary conditions. The conjectured holographic duals are warped conformal field theories (WCFTs), featuring a Virasoro-Kac-Moody algebra. We provide a holographic calculation of the entanglement entropy and Rényi entropy using AdS 3 /WCFT and WAdS 3 /WCFT dualities. Our bulk results are consistent with the WCFT results derived by CastroHofman-Iqbal using the Rindler method. Comparing with [1], we explicitly show that the holographic entanglement entropy is indeed affected by boundary conditions. Both results differ from the Ryu-Takayanagi proposal, indicating new relations between spacetime geometry and quantum entanglement for holographic dualities beyond AdS/CFT.

We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS 3 /CFT 2 . With the guidance from the boundary and bulk modular flows we find a natural slicing of the entanglement wedge with the modular planes, which are codimension one bulk surfaces tangent to the modular flow everywhere. This gives an one-to-one correspondence between the points on the boundary interval A and the points on the Ryu-Takayanagi (RT) surface E A . In the same sense an arbitrary subinterval A 2 of A will correspond to a subinterval E 2 of E A . This fine correspondence indicates that the length of E 2 captures the contribution s A (A 2 ) from A 2 to the entanglement entropy S A , hence gives the contour function for entanglement entropy. Furthermore we propose that s A (A 2 ) in general can be written as a simple linear combination of entanglement entropies of single intervals inside A. This proposal passes several nontrivial tests.

Based on the Lewkowycz-Maldacena prescription and the fine structure analysis of holographic entanglement proposed in [1], we explicitly calculate the holographic entanglement entropy for warped CFT that duals to AdS 3 with a Dirichlet-Neumann type of boundary conditions. We find that certain type of null geodesics emanating from the entangling surface ∂A relate the field theory UV cutoff and the gravity IR cutoff. Inspired by the construction, we furthermore propose an intrinsic prescription to calculate the generalized gravitational entropy for general spacetimes with non-Lorentz invariant duals. Compared with the RT formula, there are two main differences. Firstly, instead of requiring that the bulk extremal surface E should be anchored on ∂A, we require the consistency between the boundary and bulk causal structures to determine the corresponding E. Secondly we use the null geodesics (or hypersurfaces) emanating from ∂A and normal to E to regulate E in the bulk. We apply this prescription to flat space in three dimensions and get the entanglement entropies straightforwardly. arXiv:1810.11756v3 [hep-th] 29 Jan 20191 The extremal condition is the result of imposing the equations of motion and replica symmetry on all the fields in the action. In [13], as the gauge fields are nondynamical and do not appear in the symplectic structure, thus should not be imposed with the replica symmetry (or periodic) condition. As a result, in that case the geometric quantity E that measures the entanglement entropy is not an extremal surface. See [14] for a simpler discussion on the extremal condition.2 A proof for the homology constraint at topological level in AdS/CFT is given in [19] 3 Although the AdS/CFT has attracted most of the attentions, the holographic principle is assumed to be hold for general spacetimes. So far the holography beyond AdS/CFT that has been proposed include the dS/CFT correspondence [20], the Lifshitz spacetime/Lifshitz-type field theory duality [21][22][23][24], the Kerr/CFT correspondence [25], the WAdS/CFT [26,27] or WAdS/WCFT [28,29] correspondence, and flat holography in four dimensions [30][31][32] and three dimensions [33][34][35].