New Variables for Classical and Quantum Gravity in all Dimensions V. Isolated Horizon Boundary Degrees of Freedom

TL;DR

This paper extends the isolated horizon framework in loop quantum gravity to higher dimensions, formulating a Chern-Simons boundary theory that may lead to finite entropy calculations.

Contribution

It generalizes the boundary conditions and symplectic structure of isolated horizons to all dimensions, proposing a new higher-dimensional Chern-Simons theory approach.

Findings

The symplectic structure matches higher-dimensional SO(2(n+1))-Chern-Simons theory.

The approach closely resembles the 3+1 dimensional treatment.

Potential for finite entropy through a stronger boundary condition.

Abstract

In this paper, we generalise the treatment of isolated horizons in loop quantum gravity, resulting in a Chern-Simons theory on the boundary in the four-dimensional case, to non-distorted isolated horizons in 2(n+1)-dimensional spacetimes. The key idea is to generalise the four-dimensional isolated horizon boundary condition by using the Euler topological density of a spatial slice of the black hole horizon as a measure of distortion. The resulting symplectic structure on the horizon coincides with the one of higher-dimensional SO(2(n+1))-Chern-Simons theory in terms of a Peldan-type hybrid connection and resembles closely the usual treatment in 3+1 dimensions. We comment briefly on a possible quantisation of the horizon theory. Here, some subtleties arise since higher-dimensional non-Abelian Chern-Simons theory has local degrees of freedom. However, when replacing the natural generalisation to higher dimensions of the usual boundary condition by an equally natural stronger one, it is conceivable that the problems originating from the local degrees of freedom are avoided, thus possibly resulting in a finite entropy.