Superpotentials from flux compactifications of M-theory

In flux compactifications of M-theory a superpotential is generated whose explicit form depends on the structure group of the 7-dimensional internal manifold. In this note, we discuss superpotentials for the structure groups: G_2, SU(3) or SU(2). For the G_2 case all internal fluxes have to vanish. For SU(3) structures, the non-zero flux components entering the superpotential describe an effective 1-dimensional model and a Chern-Simons model if there are SU(2) structures.Comment: 10 page

Supersymmetric D-branes and calibrations on general N=1 backgrounds

We study the conditions to have supersymmetric D-branes on general {\cal N}=1 backgrounds with Ramond-Ramond fluxes. These conditions can be written in terms of the two pure spinors associated to the SU(3)\times SU(3) structure on T_M\oplus T^\star_M, and can be split into two parts each involving a different pure spinor. The first involves the integrable pure spinor and requires the D-brane to wrap a generalised complex submanifold with respect to the generalised complex structure associated to it. The second contains the non-integrable pure spinor and is related to the stability of the brane. The two conditions can be rephrased as a generalised calibration condition for the brane. The results preserve the generalised mirror symmetry relating the type IIA and IIB backgrounds considered, giving further evidence for this duality.Comment: 23 pages. Some improvements and clarifications, typos corrected and references added. v3: Version published in JHE

Supersymmetric sources, integrability and generalized-structure compactifications

In the context of supersymmetric compactifications of type II supergravity to four dimensions, we show that orientifold sources can be compatible with a generalized SU(3) x SU(3)-structure that is neither strictly SU(3) nor static SU(2). We illustrate this with explicit examples, obtained by suitably T-dualizing known solutions on the six-torus. In addition we prove the following integrability statements, valid under certain mild assumptions: (a) for general type II supergravity backgrounds with orientifold and/or D-brane generalized-calibrated sources, the source-corrected Einstein and dilaton equations of motion follow automatically from the supersymmetry equations once the likewise source-corrected form equations of motion and Bianchi identities are imposed; (b) in the special case of supersymmetric compactifications to four-dimensional Minkowski space, the equations of motion of all fields, including the NSNS three-form, follow automatically once the supersymmetry and the Bianchi identities of the forms are imposed. Both (a) and (b) are equally valid whether the sources are smeared or localized. As a byproduct we obtain the calibration form for a space-filling NS5-brane.Comment: 32 pages, 1 table, v2: added references, v3: corrected mistake in (4.1) leading to factor 2 mistake in (B.6), corrected (B.5), smaller typo

Killing spinors in supergravity with 4-fluxes

We study the spinorial Killing equation of supergravity involving a torsion 3-form \T as well as a flux 4-form \F. In dimension seven, we construct explicit families of compact solutions out of 3-Sasakian geometries, nearly parallel \G_2-geometries and on the homogeneous Aloff-Wallach space. The constraint \F \cdot \Psi = 0 defines a non empty subfamily of solutions. We investigate the constraint \T \cdot \Psi = 0, too, and show that it singles out a very special choice of numerical parameters in the Killing equation, which can also be justified geometrically

Perturbing gauge/gravity duals by a Romans mass

We show how to produce algorithmically gravity solutions in massive IIA (as infinitesimal first order perturbations in the Romans mass parameter) dual to assigned conformal field theories. We illustrate the procedure on a family of Chern--Simons--matter conformal field theories that we recently obtained from the N=6 theory by waiving the condition that the levels sum up to zero.Comment: 30 page

G2 Hitchin functionals at one loop

We consider the quantization of the effective target space description of topological M-theory in terms of the Hitchin functional whose critical points describe seven-manifolds with G2 structure. The one-loop partition function for this theory is calculated and an extended version of it, that is related to generalized G2 geometry, is compared with the topological G2 string. We relate the reduction of the effective action for the extended G2 theory to the Hitchin functional description of the topological string in six dimensions. The dependence of the partition functions on the choice of background G2 metric is also determined.Comment: 58 pages, LaTeX; v2: Acknowledgments adde

The general (2,2) gauged sigma model with three--form flux

We find the conditions under which a Riemannian manifold equipped with a closed three-form and a vector field define an on--shell N=(2,2) supersymmetric gauged sigma model. The conditions are that the manifold admits a twisted generalized Kaehler structure, that the vector field preserves this structure, and that a so--called generalized moment map exists for it. By a theorem in generalized complex geometry, these conditions imply that the quotient is again a twisted generalized Kaehler manifold; this is in perfect agreement with expectations from the renormalization group flow. This method can produce new N=(2,2) models with NS flux, extending the usual Kaehler quotient construction based on Kaehler gauged sigma models.Comment: 24 pages. v2: typos fixed, other minor correction

D-branes on general N=1 backgrounds: superpotentials and D-terms

We study the dynamics governing space-time filling D-branes on Type II flux backgrounds preserving four-dimensional N=1 supersymmetry. The four-dimensional superpotentials and D-terms are derived. The analysis is kept on completely general grounds thanks to the use of recently proposed generalized calibrations, which also allow one to show the direct link of the superpotentials and D-terms with BPS domain walls and cosmic strings respectively. In particular, our D-brane setting reproduces the tension of D-term strings found from purely four-dimensional analysis. The holomorphicity of the superpotentials is also studied and a moment map associated to the D-terms is proposed. Among different examples, we discuss an application to the study of D7-branes on SU(3)-structure backgrounds, which reproduces and generalizes some previous results.Comment: 50 pages; v2: table of contents, some clarifications and references added; v3: typos corrected and references adde

Effective actions and N=1 vacuum conditions from SU(3) x SU(3) compactifications

We consider compactifications of type II string theory on general SU(3) x SU(3) structure backgrounds allowing for a very large set of fluxes, possibly nongeometric ones. We study the effective 4d low energy theory which is a gauged N=2 supergravity, and discuss how its data are obtained from the formalism of the generalized geometry on T+T*. In particular we relate Hitchin's special Kaehler metrics on the spaces of even and odd pure spinors to the metric on the supergravity moduli space of internal metric and B-field fluctuations. We derive the N=1 vacuum conditions from this N=2 effective action, as well as from its N=1 truncation. We prove a direct correspondence between these conditions and an integrated version of the pure spinor equations characterizing the N=1 backgrounds at the ten dimensional level.Comment: 54 pages. v2, v3: minor change

Generalized structures of N=1 vacua

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) x SU(3) implies the existence of two pure spinors Phi_1 and Phi_2. 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 wedge) Phi_1=0 and (d + H wedge) Phi_2 = F_RR. 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.Comment: 21 pages. v2, v3: minor changes and correction