Black Hole Entropy in the presence of Chern-Simons Terms

TL;DR

This work extends Wald's Noether-charge approach to black hole entropy in theories with gravitational Chern-Simons terms, where strict diffeomorphism covariance is absent. It develops a generalized entropy construction using a modified Noether charge Q'_ξ and applies it to CS contributions, yielding explicit formulas for the entropy corrections. The results include ΔS = 8πβ ∫_Σ Γ_N in 3D, ΔS = 8π m ∫_Σ Γ_N R_N^{2m−2} P(F) in higher dimensions, and ΔS = 8π (λ/g^2) ∫_Σ Γ_N ∧ *H for Green-Schwarz couplings, with consistency checks against known BTZ and dyonic black hole cases. The framework broadens macroscopic entropy calculations in string/M-theory contexts and suggests avenues for integrating CS corrections with entropy function formalisms.

Abstract

We derive a formula for the black hole entropy in theories with gravitational Chern-Simons terms, by generalizing Wald's argument which uses the Noether charge. It correctly reproduces the entropy of three-dimensional black holes in the presence of Chern-Simons term, which was previously obtained via indirect methods.