Abstract
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} = 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.
Highlights
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 ChernSimons terms are included in the Kalb-Ramond field strength
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
In a recent paper [27] we studied the use of gauge-covariant Lie derivatives in the context of the Einstein-Maxwell theory using momentum maps to construct the derivatives
Summary
When the effective action of the Heterotic Superstring at leading order in α is compactified on a Tn, it describes the dynamics of the (10 − n)-dimensional (string-frame) metric gμν, Kalb-Ramond 2-form Bμν, dilaton field φ, Kaluza-Klein (KK) and winding 1-forms Amμ and Bm μ, respectively, and the scalars that parametrize the O(n, n)/O(n)×O(n) coset space, collected in the symmetric O(n, n) matrix M that we will write with upper O(n, n) indices I, J, . . . as M IJ. [27]), and calling the physical scalars in MIJ φx, the action of the d = (10 − n)-dimensional takes the form. In this action ea = eaμdxμ are the string-frame Vielbeins, stands for the Hodge dual and, (ea ∧ eb) =. The kinetic term of the scalars φx that parametrize the O(n, n)/(O(n)×O(n)) coset space can be written in the form. The equations of motion of the matter fields are given by EB = −d e−2φ H , Eφ = 8d e−2φ dφ −2L , EI.
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