  We re-derive the first law of black hole mechanics in the context of the
Heterotic Superstring effective action compactified on a torus to leading order
in alpha prime, 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.
  This work is a first step towards the derivation of the first law at first
order in alpha prime 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.
