Topological terms and the Misner string entropy

Luca Ciambelli,Rodrigo Olea,José Figueroa,Cristóbal Corral,Gastón Giribet

doi:10.1103/physrevd.103.024052

Abstract

The method of topological renormalization in anti-de Sitter (AdS) gravity consists in adding to the action a topological term which renders it finite, defining at the same time a well-posed variational problem. Here, we use this prescription to work out the thermodynamics of asymptotically locally anti-de Sitter (AlAdS) spacetimes, focusing on the physical properties of the Misner strings of both the Taub-NUT-AdS and Taub-Bolt-AdS solutions. We compute the contribution of the Misner string to the entropy by treating on the same footing the AdS and AlAdS sectors. As topological renormalization is found to correctly account for the physical quantities in the parity preserving sector of the theory, we then investigate the holographic consequences of adding also the Chern-Pontryagin topological invariant to the bulk action; in particular, we discuss the emergence of the parity-odd contribution in the boundary stress tensor.

Highlights

Hawking and Hunter argued in [1] that the existence of gravitational entropy is associated to a topological obstruction to foliation of the Euclidean section of the space, which results in an obstruction for the existence of a unitary Hamiltonian evolution
Let us summarize our results: in this paper, we have considered the method of topological renormalization in asymptotically locally anti–de Sitter (AlAdS) spaces
The latter consists in adding to the gravitational action a topological term that, while suffices to render the Euclidean action finite and the variational principle well posed, provides a natural definition of the renormalized Noether charges

Summary

INTRODUCTION

Hawking and Hunter argued in [1] that the existence of gravitational entropy is associated to a topological obstruction to foliation of the Euclidean section of the space, which results in an obstruction for the existence of a unitary Hamiltonian evolution. We analyze the problem by using the method of topological renormalization; meaning, the method that consists of adding to the gravitational action a bulk piece of the Chern-Weil-Gauss-Bonnet (hereafter, Gauss-Bonnet) topological invariant, with the specific value of the coupling constant that renders the action equivalent to the MacDowell-Mansouri one [27,28] This procedure yields finite results for the Euclidean action, the Noether charges, and the thermodynamic quantities, while at the same time suffices to render the variational principle well-posed [29,30]; see [31,32,33,34].

NOETHER-WALD FORMALISM

TOPOLOGICAL RENORMALIZATION

Adding the Chern-Pontryagin invariant

Boundary stress tensors

CONCLUSIONS