Gravity as a constrained BF theory: Noether charges and Immirzi parameter

TL;DR

This paper explores the Noether charges in a constrained BF theory of gravity, generalizing Wald's approach to include various topological terms and showing the black hole entropy is independent of the Immirzi parameter.

Contribution

It derives Noether charges in a constrained BF formulation of gravity, extending previous methods to include topological terms and analyzing their impact on black hole entropy.

Findings

Noether charges are derived for the constrained BF theory.

Differentiability of the action is naturally ensured by the model's structure.

Black hole entropy does not depend on the Immirzi parameter.

Abstract

We derive and analyze Noether charges associated with the diffeomorphism invariance for the constrained SO(2,3) BF theory. This result generalizes the Wald approach to the case of the first order gravity with a negative cosmological constant, the Holst modification and topological terms (Nieh-Yan, Euler, and Pontryagin). We show that differentiability of the action is automatically implemented by the the structure of the constrained BF model. Finally, we calculate the AdS--Schwarzschild black hole entropy from the Noether charge and we find that, unexpectedly, it does not depend on the Immirzi parameter.