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.

Compactifications of type II theories on Calabi-Yau threefolds including electric and magnetic background fluxes are discussed. We derive the bosonic part of the four-dimensional low energy effective action and show that it is a non-canonical N = 2 supergravity which includes a massive two-form. The symplectic invariance of the theory is maintained as long as the flux parameters transform as a symplectic vector and a massive two-form which couples to both electric and magnetic field strengths is present. The mirror symmetry between type IIA and type IIB compactified on mirror manifolds is shown to hold for R-R fluxes at the level of the effective action. We also compactify type IIA in the presence of NS three-form flux but the mirror symmetry in this case remains unclear.

We study the vacuum structure of compactifications of type II string theories on orientifolds with SU (3) × SU (3) structure. We argue that generalised geometry enables us to treat these non-geometric compactifications using a supergravity analysis in a way very similar to geometric compactifications. We find supersymmetric Minkowski vacua with all the moduli stabilised at weak string coupling and all the tadpole conditions satisfied. Generically the value of the moduli fields in the vacuum is parametrically controlled and can be taken to arbitrarily large values. † An example of a generalised almost complex structure is one induced by an almost complex structure IAnother example is one that is induced by an almost symplectic two-form JIn general, a generalised almost complex structure will be some combination of the two. Given a generalised almost complex structure it is possible to generate a new one by a B-transformation defined as 4) where B is a real two-form, B ∈ Λ 2 T * . Similarly we can also generate a new one through a β-transform 5)

We study perturbative aspects of noncommutative field theories. This work is arranged in two parts. First, we review noncommutative field theories in general and discuss both canonical and path integral quantization methods. In the second part, we consider the particular example of noncommutative Φ 4 theory in four dimensions and work out the corresponding effective action and discuss renormalizability of the theory, up to two loops.

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.

In this paper, we continue the analysis of heterotic string compactifications on half-flat mirror manifolds by including the 10-dimensional gauge fields. It is argued, that the heterotic Bianchi identity is solved by a variant of the standard embedding. Then, the resulting gauge group in four dimensions is still E 6 despite the fact that the Levi-Civita connection has SO(6) holonomy. We derive the associated four-dimensional effective theories including matter field terms for such compactifications. The results are also extended to more general manifolds with SU(3) structure. *

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 -Neveu-Schwarz flux. We explicitly derive the superpotential and Kä hler potential at lowest order in 0 by a reduction of the bosonic action. The superpotential contains new terms depending on the Kä hler 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.