The first law and Wald entropy formula of heterotic stringy black holes at first order in alpha prime

We derive the first law of black hole mechanics in the context of the Heterotic Superstring effective action to first order in alpha prime using Wald's formalism. We carefully take into account all the symmetries of the theory and, as a result, we obtain a manifestly gauge- and Lorentz-invariant entropy formula in which all the terms can be computed explicitly. An entropy formula with these properties allows unambiguous calculations of macroscopic black-hole entropies to first order in alpha prime that can be reliably used in a comparison with the microscopic ones. Such a formula was still lacking in the literature. In the proof we use momentum maps to define covariant variations and Lie derivatives and \textit{restricted generalized zeroth laws} which state the closedness of certain differential forms on the bifurcation sphere and imply the constancy of the associated potentials on it. We study the relation between our entropy formula and other formulae that have been used in the literature.