Moduli dependence of one-loop gauge couplings in (0,2) compactifications

We derive the moduli dependence of the one-loop gauge couplings for non-vanishing gauge background fields in a four-dimensional heterotic (0,2) string compactification. Remarkably, these functions turn out to have a representation as modular functions on an auxiliary Riemann surface on appropriate truncations of the full moduli space. In particular, a certain kind of one-loop functions is given by the free energy of two-dimensional solitons on this surface.