We discuss mirror symmetry in generalized Calabi-Yau compactifications of type II string theories with background NS fluxes. Starting from type IIB compactified on CalabiYau threefolds with NS three-form flux we show that the mirror type IIA theory arises from a purely geometrical compactification on a different class of six-manifolds. These mirror manifolds have SU(3) structure and are termed half-flat; they are neither complex nor Ricci-flat and their holonomy group is no longer SU (3). We show that type IIA appropriately compactified on such manifolds gives the correct mirror-symmetric lowenergy effective action.

In this paper we analyze the structure of supersymmetric vacua in compactifications of the heterotic string on certain manifolds with SU(3) structure. We first study the effective theories obtained from compactifications on half-flat manifolds and show that solutions which stabilise the moduli at acceptable values are hard to find. We then derive the effective theories associated with compactification on generalised half-flat manifolds. It is shown that these effective theories are consistent with four-dimensional N = 1 supergravity and that the superpotential can be obtained by a Gukov-Vafa-Witten type formula. Within these generalised models, we find consistent supersymmetric (AdS) vacua at weak gauge coupling, provided we allow for general internal gauge bundles. In simple cases we perform a counting of such vacua and find that a fraction of about 1/1000 leads to a gauge coupling consistent with gauge unification. * There is now a considerable body of work on moduli stabilization, facilitated by flux of the Neveu Schwarz-Neveu Schwarz (NSNS) and Ramond-Ramond (RR) anti-symmetric tensor fields, in the context of type II theories. Specifically, within type IIB it has been shown [1] that a combination of NSNS and RR three-form flux can stabilize the dilaton and all complex structure moduli, while the Kähler moduli have to be fixed by other effects such as non-perturbative contributions [2] or perhaps higher-order α ′ corrections [3]. The consistency of these procedures, including the interplay between α ′ and non-perturbative corrections, was analysed in Refs [4,5,6,7,8,9,10] Within type IIA theories, on the other hand, both odd and even degree form field strengths are available, so that flux potentials for complex structure moduli as well as Kähler moduli will typically be generated [11,12] (see also Ref. [13] for N = 1 models). One may therefore hope that all moduli can be stabilised by flux in some such models and specific examples have indeed been found [14,15,16,17,18], although it appears that in generic models of this kind some flat directions are still left over.Traditionally, the heterotic string has been considered the most attractive string theory, with the presence of (preferably E 8 × E 8 ) ten-dimensional gauge fields leading to a large number of supersymmetric compactifications with phenomenologically interesting properties [19]. It has also been known for a long time that heterotic NSNS three-form flux can stabilize all complex structure moduli of the theory [20,21]. More recently, this subject was addressed in Refs [22,23]. However, in the absence of any further (RR) antisymmetric tensor fields, the potential for stabilizing the remaining moduli seems rather limited compared to type II theories. This apparent problem can be overcome by departing from Calabi-Yau compactifications and by considering the heterotic string on general manifolds with SU(3) structure. Such models were analysed in Refs [24,25,26,27,28,29,30] where general aspects of compactifications on non-Kähler manifolds were studied....

In this note we derive the low-energy effective action of type IIB theory compactified on half-flat manifolds and we show that this precisely coincides with the low-energy effective action of type IIA theory compactified on a Calabi-Yau manifold in the presence of NS three-form fluxes. We provide in this way a further check of the recently formulated conjecture that half-flat manifolds appear as mirror partners of Calabi-Yau manifolds when NS fluxes are turned on.