The Poincaré invariant vacuum is not unique in quantum gravity. The BMS supertranslation symmetry originally defined at null infinity is spontaneously broken and results in inequivalent Poincaré vacua. In this paper we construct the unique vacua which interpolate between past and future null infinity in BMS gauge and which are entirely characterized by an arbitrary Goldstone boson defined on the sphere which breaks BMS invariance. We show that these vacua contain a defect which carries no Poincaré charges but which generically carries superrotation charges. We argue that there is a huge degeneracy of vacua with multiple defects. We also present the single defect vacua with its canonically conjugated source which can be constructed from a Liouville boson on the stereographic plane. We show that positivity of the energy forces the stress-tensor of the boson to vanish as a boundary condition. Finite superrotations, which turn on the sources, are therefore physically ruled out as canonical transformations around the vacua. Yet, infinitesimal superrotations are external symplectic symmetries which are associated with conserved charges which characterize the Goldstone boson.

The Kerr metric models the final classical black hole state after gravitational collapse of matter and radiation. Any stationary metric which is close to the Kerr metric has been proven to be diffeomorphic to it. Now, finite supertranslation diffeomorphisms are symmetries which map solutions to inequivalent solutions as such diffeomorphisms generate conserved superrotation charges. The final state of gravitational collapse is therefore parameterized by its mass, angular momentum and supertranslation field, signaled by its conserved superrotation charges.In this paper, we first derive the angle-dependent energy conservation law relating the asymptotic value of the supertranslation field of the final state to the details of the collapse and subsequent evolution of the system. We then generate the static solution with an asymptotic supertranslation field and we study some of its properties. Up to a caveat, the deviation from the Schwarzschild metric could therefore be predicted on a case-by-case basis from accurate modeling of the angular dependence of the ingoing and outgoing energy fluxes leading to the final state.

In this paper we investigate the holographic Rényi entropy of two disjoint intervals on complex plane with small cross ratio x for conformal field theory with W symmetry in the ground state, which could be dual to a higher spin AdS 3 gravity. We focus on the cases of W 3 and W 4 symmetries. In order to see the nontrivial contributions from the W fields, we calculate the Rényi entropy in the expansion of x to order x 8 in both the gravity and the CFT sides. In the gravity side the classical contributions to the entanglement entropy is still given by the Ryu-Takayanagi area formula under the reasonable assumption, while the 1-loop quantum corrections have to take into account of the contributions not only from massless gravitons, but also from massless higher spin fields. In the CFT side we still use the operator product expansion of twist operators in the small interval limit, but now we need to consider the quasiprimary fields constructed from W fields, besides the ones from Virasoro Verma module. In the large central charge limit, we obtain the classical, 1-loop, 2-loop, and 3-loop parts of the Rényi entropy. The classical and 1-loop results in the gravity and the CFT sides are in exact match. This confirms the higher spin gravity/CFT correspondence, and also supports the holographic computation of Rényi entanglement entropy, including the quantum correction, in both the AdS 3 gravity and the higher spin AdS 3 gravity.