Bogomol’nyi-type mass formulas for a class of nonrotating black holes

In the presence of a Killing symmetry, various self-gravitating field theories with massless scalars (moduli) and vector fields reduce to $\sigma$ models, effectively coupled to three-dimensional gravity. We argue that this particular structure of the Einstein-matter equations gives rise to quadratic relations between the asymptotic flux integrals and the area and surface gravity (Hawking temperature) of the horizon. The method is first illustrated for the Einstein-Maxwell system. A derivation of the mass formula is then also presented for the Einstein-Maxwell-dilaton-axion model, which is relevant to the bosonic sector of heterotic string theory.