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 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.

A single M5-brane probing G, an ADE-type singularity, leads to a system which has G × G global symmetry and can be viewed as "bifundamental" (G, G) matter. For the A N series, this leads to the usual notion of bifundamental matter. For the other cases it corresponds to a strongly interacting (1, 0) superconformal system in six dimensions. Similarly, an ADE singularity intersecting the Hořava-Witten wall leads to a superconformal matter system with E 8 × G global symmetry. Using the F-theory realization of these theories, we elucidate the Coulomb/tensor branch of (G, G ) conformal matter. This leads to the notion of fractionalization of an M5-brane on an ADE singularity as well as fractionalization of the intersection point of the ADE singularity with the Hořava-Witten wall. Partial Higgsing of these theories leads to new 6d SCFTs in the infrared, which we also characterize. This generalizes the class of (1, 0) theories which can be perturbatively realized by suspended branes in IIA string theory. By reducing on a circle, we arrive at novel duals for 5d affine quiver theories. Introducing many M5-branes leads to large N gravity duals.

In M-theory, the only AdS 7 supersymmetric solutions are AdS 7 × S 4 and its orbifolds. In this paper, we find and classify new supersymmetric solutions of the type AdS 7 × M 3 in type II supergravity. While in IIB none exist, in IIA with Romans mass (which does not lift to M-theory) there are many new ones. We use a pure spinor approach reminiscent of generalized complex geometry. Without the need for any Ansatz, the system determines uniquely the form of the metric and fluxes, up to solving a system of ODEs. Namely, the metric on M 3 is that of an S 2 fibered over an interval; this is consistent with the Sp(1) R-symmetry of the holographically dual (1,0) theory. By including D8 brane sources, one can numerically obtain regular solutions, where topologically M 3 ∼ = S 3 .

We show that the supersymmetry transformations for type II string theories on six-manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kähler form e iJ and the holomorphic form Ω. The equations are explicitly symmetric under exchange of the two pure spinors and a choice of even or odd-rank RR field. This is mirror symmetry for manifolds with torsion. Moreover, RR fluxes affect only one of the two equations: e iJ is closed under the action of the twisted exterior derivative in IIA theory, and similarly Ω is closed in IIB. Modulo a different action of the B-field, this means that supersymmetric SU(3)-structure manifolds are all generalized Calabi-Yau manifolds, as defined by Hitchin. An equivalent, and somewhat more conventional, description is given as a set of relations between the components of intrinsic torsions modified by the NS flux and the Clifford products of RR fluxes with pure spinors, allowing for a classification of type II supersymmetric vacua on six-manifolds. We find in particular that supersymmetric six-manifolds are always complex for IIB backgrounds while they are twisted symplectic for IIA.

We study superconformal and supersymmetric theories on Euclidean four-and threemanifolds with a view toward holographic applications. Preserved supersymmetry for asymptotically locally AdS solutions implies the existence of a (charged) "conformal Killing spinor" on the boundary. We study the geometry behind the existence of such spinors. We show in particular that, in dimension four, they exist on any complex manifold. This implies that a superconformal theory has at least one supercharge on any such space, if we allow for a background field (in general complex) for the R-symmetry. We also show that this is actually true for any supersymmetric theory with an R-symmetry. We also analyze the three-dimensional case and provide examples of supersymmetric theories on Sasaki spaces.

We show that the supersymmetry transformations for type II string theories on six-manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kähler form e iJ and the holomorphic form Ω. The equations are explicitly symmetric under exchange of the two pure spinors and a choice of even or odd-rank RR field. This is mirror symmetry for manifolds with torsion. Moreover, RR fluxes affect only one of the two equations: e iJ is closed under the action of the twisted exterior derivative in IIA theory, and similarly Ω is closed in IIB. Modulo a different action of the B-field, this means that supersymmetric SU(3)-structure manifolds are all generalized Calabi-Yau manifolds, as defined by Hitchin. An equivalent, and somewhat more conventional, description is given as a set of relations between the components of intrinsic torsions modified by the NS flux and the Clifford products of RR fluxes with pure spinors, allowing for a classification of type II supersymmetric vacua on six-manifolds. We find in particular that supersymmetric six-manifolds are always complex for IIB backgrounds while they are twisted symplectic for IIA.February 1, 2008

M-theory and string theory predict the existence of many six-dimensional SCFTs. In particular, type IIA brane constructions involving NS5-, D6-and D8-branes conjecturally give rise to a very large class of N = (1, 0) CFTs in six dimensions. We point out that these theories sit at the end of RG flows which start from six-dimensional theories which admit an M-theory construction as a M5 stack transverse to R 4 /Z k × R. The flows are triggered by Higgs branch expectation values and correspond to D6's opening up into transverse D8-branes via a Nahm pole. We find a precise correspondence between these CFT's and the AdS 7 vacua found in a recent classification in type II theories. Such vacua involve massive IIA regions, and the internal manifold is topologically S 3 . They are characterized by fluxes for the NS three-form and RR two-form, which can be thought of as the near-horizon version of the NS5's and D6's in the brane picture; the D8's, on the other hand, are still present in the AdS 7 solution, in the form of an arbitrary number of concentric shells wrapping round S 2 's.