The first law of heterotic stringy black hole mechanics at zeroth order in α′
Zachary Elgood, Dimitrios Mitsios, Tomás Ortı́n, David Pereñíguez
Abstract
A bstract We prove the first law of black hole mechanics in the context of the Heterotic Superstring effective action compactified on a torus to leading order in α ′, using Wald’s formalism, covariant Lie derivatives and momentum maps. The Kalb-Ramond field strength of this theory has Abelian Chern-Simons terms which induce Nicolai-Townsend transformations of the Kalb-Ramond field. We show how to deal with all these gauge symmetries deriving the first law in terms of manifestly gauge-invariant quantities. In presence of Chern-Simons terms, several definitions of the conserved charges exist, but the formalism picks up only one of them to play a role in the first law. We study explicitly a non-extremal, charged, black ring solution of pure $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 1 , d = 5 supergravity embedded in the Heterotic Superstring effective field theory. This work is a first step towards the derivation of the first law at first order in α ′ where, more complicated, non-Abelian, Lorentz (“gravitational”) and Yang-Mills Chern-Simons terms are included in the Kalb-Ramond field strength. The derivation of a first law is a necessary step towards the derivation of a manifestly gauge-invariant entropy formula which is still lacking in the literature. In its turn, this entropy formula is needed to compare unambiguously macroscopic and microscopic black hole entropies.