We characterize N = 1 vacua of type II theories in terms of generalized complex structure on the internal manifold M . The structure group of T (M )⊕T * (M ) being SU(3)×SU(3) implies the existence of two pure spinors Φ1 and Φ2. The conditions for preserving N = 1 supersymmetry turn out to be simple generalizations of equations that have appeared in the context of N = 2 and topological strings. They are (d+H∧)Φ1 = 0 and (d+H∧)Φ2 = FRR. The equation for the first pure spinor implies that the internal space is a twisted generalized Calabi-Yau manifold of a hybrid complex-symplectic type, while the RR-fields serve as an integrability defect for the second. September 11, 2018 been noted already in [7] for cases in which T has SU(3) structure. The SU(3) structure vacua are however just special cases of the more general SU(3)×SU(3) on T ⊕ T * considered here. The SU(3) structure vacua correspond to either complex (and with vanishing c 1 ) or symplectic manifolds, which are the two particular cases which inspired the definition of generalized Calabi-Yau. In the generic SU(3)×SU(3) case, vacua can be a complex-symplectic hybrid, namely a manifold that is locally a product of a complex k-fold times an 6 − 2k symplectic manifold.Physically, the generalized Calabi-Yau condition has also been argued to imply the existence of a topological model [8,9], not necessarily coming from the twisting of a (2, 2) model, which generalizes the A and B models. In other words, we find that all N = 1 Minkowski vacua have an underlying topological model. When there is a (2, 2) model, both pure spinors are closed under (d + H∧) [2], which reflects the fact that two topological models can be defined. This condition unifies the Calabi-Yau case and the (2, 2) models of [10]. It corresponds to an unbroken N = 2 in the target space, and it has recently been found from supergravity in [3]. Although the N = 2 requirement of having two twisted closed pure spinors looks like our N = 1 equations (1.1) for F = A = 0, we stress again that (1.1) applies only when the RR fluxes are non zero. Therefore we cannot obtain from there the N = 1 equations for pure NS flux, which correspond to a (2, 1) model.Another feature of (1.1) is that they are essentially identical for IIA and IIB. This suggests there must exist some form of mirror symmetry for these compactifications exchanging the even and odd pure spinors [11][12][13]. As far as we know, mirror symmetry could even be present when supersymmetry is spontaneously broken [11,13,14]. For the case at hand of unbroken N = 1, however, this is made particularly concrete by the remark above that all vacua have an underlying topological model; mirror symmetry has long been viewed [15] as an exchange of topological models, without necessarily involving Calabi-Yau's.Presumably there are connections to more recent lines of thought relating Hitchin functionals [16] to topological theories [17][18][19]. Particularly promising seems the results in [20] about the quantization of the functional, which relate directly to ...

We find new, local, non-supersymmetric conformal field theories obtained by relevant deformations of the N=4 super Yang Mills theory in the large N limit. We contruct interpolating supergravity solutions that naturally represent the flow from the N=4 super Yang Mills UV theory to these non-supersymmetric IR fixed points. We also study the linearization around the N=4 superconformal point of N=1 supersymmetric, marginal deformations. We show that they give rise to N=1 superconformal fixed points, as expected from field-theoretical arguments.

We perform a systematic search for N = 1 Minkowski vacua of type II string theories on compact six-dimensional parallelizable nil-and solvmanifolds (quotients of six-dimensional nilpotent and solvable groups, respectively). Some of these manifolds have appeared in the construction of string backgrounds and are typically called twisted tori. We look for vacua directly in ten dimensions, using the a reformulation of the supersymmetry condition in the framework of generalized complex geometry. Certain algebraic criteria to establish compactness of the manifolds involved are also needed. Although the conditions for preserved N = 1 supersymmetry fit nicely in the framework of generalized complex geometry, they are notoriously hard to solve when coupled to the Bianchi identities. We find solutions in a large-volume, constant-dilaton limit. Among these, we identify those that are T-dual to backgrounds of IIB on a conformal T 6 with self-dual three-form flux, and hence conceptually not new. For all backgrounds of this type fully localized solutions can be obtained. The other new solutions need multiple intersecting sources (either orientifold planes or combinations of O-planes and D-branes) to satisfy the Bianchi identities; the full list of such new solution is given. These are so far only smeared solutions, and their localization is yet unknown. Although valid in a large-volume limit, they are the first examples of Minkowski vacua in supergravity which are not connected by any duality to a Calabi-Yau. Finally, we discuss a class of flat solvmanifolds that may lead to AdS 4 vacua of type IIA strings.