Anomalies in gravitational charge algebras of null boundaries and black hole entropy

Abstract

We revisit the covariant phase space formalism applied to gravitational theories with null boundaries, utilizing the most general boundary conditions consistent with a fixed null normal. To fix the ambiguity inherent in the Wald-Zoupas definition of quasilocal charges, we propose a new principle, based on holographic reasoning, that the flux be of Dirichlet form. This also produces an expression for the analog of the Brown-York stress tensor on the null surface. Defining the algebra of charges using the Barnich-Troessaert bracket for open subsystems, we give a general formula for the central — or more generally, abelian — extensions that appear in terms of the anomalous transformation of the boundary term in the gravitational action. This anomaly arises from having fixed a frame for the null normal, and we draw parallels between it and the holographic Weyl anomaly that occurs in AdS/CFT. As an application of this formalism, we analyze the near-horizon Virasoro symmetry considered by Haco, Hawking, Perry, and Strominger, and perform a systematic derivation of the fluxes and central charges. Applying the Cardy formula to the result yields an entropy that is twice the Bekenstein-Hawking entropy of the horizon. Motivated by the extended Hilbert space construction, we interpret this in terms of a pair of entangled CFTs associated with edge modes on either side of the bifurcation surface.

The integrability of Virasoro charges for axisymmetric Killing horizons

“…From the geometrical point of view, it selects a particular family of the AdS 3 folia by the relationships on α and β. The coefficient η = −1/2 agrees with the result in [31] with a different derivation. The pair of CFT tempratures for BTZ black hole [21] satisfy such condition, while the pair in Kerr/CFT does not [22,26].…”

“…Note that although the result has the same coefficient (η+1) as the piece of field space exact form (η + 1)δ(κ H ) in the symplectic potential, all the terms in (4.25) contribute to K m,−m and (4.27) relates them in a specific way to combine into (4.30). This is different with [31], in which the central extension purely localized on the noncovariance of the boundary term l b . Interestingly, there is also no linear term on m in K m,−m after (4.27) is imposed.…”

“…One can see immediately that for m = 0, as the vector field is a combination of Killing vectors, the surface gravity is invariant. Interestingly, due to the fixing of different structure on the horizon, we get a different result regarding δ ζ κ compared to [31] by the diffeomorphisms generated by the same vector field. Finally, we can evaluate the field variation of the red-shift factor (3.5) which appears in the corner term of the symplectic potential:…”

“…Such type of boundary term for null hypersurface has been studied in [31,[45][46][47]. With the boundary term, the symplectic potential will have a total shift by Θ H + λδ(κ H ).…”

“…It was initially expected that it is necessary to evoke the Wald-Zoupas counterterm prescription in order to make the Virasoro charges well-defined [22,24,29,30]. A comprehensive analysis was carried out in [31] by using the Barnich-Troessaert bracket for nonintegrable charges due to the gravitational flux [32]. The present article focuses on the situation in which the condition for integrable charges does exist without the need for a counterterm.…”