  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 3-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 quadratic formula is then
also presented for the Einstein-Maxwell-axion-dilaton model, which is relevant
to the bosonic sector of heterotic string theory.
