New Variables for Classical and Quantum Gravity in all Dimensions V. Isolated Horizon Boundary Degrees of Freedom
Norbert Bodendorfer, Thomas Thiemann, Andreas Thurn
TL;DR
The paper extends loop quantum gravity’s isolated horizon framework to even spacetime dimensions by exploiting the Euler topological density on horizon slices, which induces a higher-dimensional Chern-Simons boundary theory with internal gauge groups SO$(D+1)$ or SO$(1,D)$. By performing a canonical transformation to higher-dimensional connection variables and enforcing a non-distortion condition, the horizon symplectic structure acquires a CS form with a level tied to the horizon area via $nA_S/ig angle E^{(2n)}ig angle$. Distortion is explored via Beetle–Engle and Perez–Pranzetti methods, with the latter offering a potential multi-CS defense but leaving quantisation in higher dimensions unresolved. The authors discuss finite-entropy prospects, noting that full higher-dimensional CS quantisation is unknown and may require restricting to a topological sub-sector or dimensional reduction to 3+1D, potentially yielding an area law similar to the 4D case. The work thus provides a classical backbone for higher-dimensional IH physics and highlights key challenges and possible routes toward a finite, horizon-area–proportional entropy.
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.