Supersymmetry breaking,heterotic strings and fluxes

In this paper we consider compactifications of heterotic strings in the presence of background flux. The background metric is a T² fibration over a K3 base times four-dimensional Minkowski space. Depending on the choice of three-form flux different amounts of supersymmetry are preserved (N=2,1,0$N=2,1,0$). For supersymmetric solutions unbroken space–time supersymmetry determines all background fields except one scalar function which is related to the dilaton. The heterotic Bianchi identity gives rise to a differential equation for the dilaton which we discuss in detail for solutions preserving an N=2$N=2$ supersymmetry. In this case the differential equation is of Laplace type and as a result the solvability is guaranteed.