Heterotic string compactified on half-flat manifolds

We study the effective action of the heterotic string compactified on particular half-flat manifolds which arise in the context of mirror symmetry with Neveu-Schwarz\char21{}Neveu-Schwarz flux. We explicitly derive the superpotential and K\"ahler potential at lowest order in ${\ensuremath{\alpha}}^{\ensuremath{'}}$ by a reduction of the bosonic action. The superpotential contains new terms depending on the K\"ahler moduli which originate from the intrinsic geometrical flux of the half-flat manifolds. A generalized Gukov formula, valid for all manifolds with SU(3) structure, is derived from the gravitino mass term. For the half-flat manifolds it leads to a superpotential in agreement with our explicit bosonic calculation. We also discuss the inclusion of gauge fields.