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 study the P -V criticality and phase transition in the extended phase space of charged anti-de Sitter black holes in canonical ensemble of ghost-free massive gravity, where the cosmological constant is viewed as a dynamical pressure of the black hole system. We give the generalized thermodynamic first law and the Smarr relation with massive gravity correction. We find that not only when the horizon topology is spherical but also in the Ricci flat or hyperbolic case, there appear the P -V criticality and phase transition up to the combination k+c 2 0 c2m 2 in the four-dimensional case, where k characterizes the horizon curvature and c2m 2 is the coefficient of the second term of massive potential associated with the graviton mass. The positivity of such combination indicate the van der Waals-like phase transition. When the spacetime dimension is larger then four, the Maxwell charge there seems unnecessary for the appearance of critical behavior, but a infinite repulsion effect needed, which can also be realized through negative valued c3m 2 or c4m 2 , which is third or fourth term of massive potential. When c3m 2 is positive, a Hawking-Page like black hole to vacuum phase transition is shown in the five-dimensional chargeless case. For the van der Waals like phase transition in four and five spacetime dimensions, we calculate the critical exponents near critical point and find they are the same as those in the van der Waals liquid-gas system.

Warped conformal field theory (WCFT) is a two dimensional quantum field theory whose local symmetry algebra consists of a Virasoro algebra and a U(1) Kac-Moody algebra. In this paper, we study correlation functions for primary operators in WCFT. Similar to conformal symmetry, warped conformal symmetry is very constraining. The form of the two and three point functions are determined by the global warped conformal symmetry while the four point functions can be determined up to an arbitrary function of the cross ratio. The warped conformal bootstrap equation are constructed by formulating the notion of crossing symmetry. In the large central charge limit, four point functions can be decomposed into global warped conformal blocks, which can be solved exactly. Furthermore, we revisit the scattering problem in warped AdS spacetime (WAdS), and give a prescription on how to match the bulk result to a WCFT retarded Green's function. Our result is consistent with the conjectured holographic dualities between WCFT and WAdS.

For spacetimes that are not asymptotic to anti-de Sitter (non AAdS) space, we adapt the Lewkowycz-Maldacena procedure to find the holographic entanglement entropy. The key observation, which to our knowledge is not very well appreciated, is that asymptotic boundary conditions play an essential role on extending the replica trick to the bulk. For non AAdS, we expect the following three main modifications: (1) the expansion near the special surface has to be compatible with the asymptotic expansion; (2) periodic conditions are imposed to coordinates on the phase space with diagonalized symplectic structure, not to all fields appearing in the action; (3) evaluating the entanglement functional using the boundary term method amounts to evaluating the presymplectic structure at the special surface, where some additional exact form may contribute. An explicit calculation is carried out for three-dimensional warped anti-de Sitter spacetime (WAdS_{3}) in a consistent truncation of string theory, the so-called S-dual dipole theory. It turns out that the generalized gravitational entropy in WAdS_{3} is captured by the least action of a charged particle in WAdS_{3} space, or equivalently, by the geodesic length in an auxiliary AdS_{3}. Consequently, the bulk calculation agrees with the CFT results, providing another piece of evidence for the WAdS_{3}/CFT_{2} correspondence.

In this paper, we explore means which can be used to change qualitatively the phase structure of charged black systems. For this, we consider a system of black D1/D5 (or its S-dual F/NS5). We find that the delocalized charged black D-strings (F-strings) alone share the same phase structure as the charged black D5 branes (NS5-branes), having no van der Waals-Maxwell liquid-gas type. However, when the two are combined to form D1/D5 (F/NS5), the resulting phase diagram has been changed dramatically to a richer one, containing now the above liquid-gas type. The effect of adding the charged D-strings (F-strings) on the phase structure can also be effectively described as a slight increase of the transverse dimensions to the original D5 (NS5). This may be viewed as a connection between a brane charge and a fraction of spatial dimension at least in a thermodynamical sense.