N=1 geometries for M-theory and type IIA strings with fluxes

TL;DR

This work provides a systematic, constructive framework for N=1 flux backgrounds in M-theory and type IIA, recasting supersymmetry conditions as constraints on SU(3) and SU(2) structures and their intrinsic torsion. It develops generalized Hitchin-flow methods to build irreducible 7-manifolds with 4-form flux and analyzes reductions along isometries to yield Type IIA geometries with different flux combinations. The authors present explicit classes and examples, including Calabi–Yau bases, special-Hermitian bases with M5-branes, and Fayyazuddin–Smith-type solutions, illustrating how various fluxes, dilatons, and warps interplay with geometric torsion. The framework unifies previously known solutions and provides construction techniques for new vacua, including twisted-torus and K3-based geometries with all possible fluxes turned on.

Abstract

We derive a set of necessary and sufficient conditions for obtaining N=1 backgrounds of M-theory and type IIA strings in the presence of fluxes. Our metrics are warped products of four-dimensional Minkowski space-time with a curved internal manifold. We classify the different solutions for irreducible internal manifolds as well as for manifolds with $S^1$ isometries by employing the formalism of group structures and intrinsic torsion. We provide examples within these various classes along with general techniques for their construction. In particular, we generalize the Hitchin flow equations so that one can explicitly build irreducible 7-manifolds with 4-form flux. We also show how several of the examples found in the literature fit in our framework and suggest possible generalizations.