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

TL;DR

This paper establishes criteria for N=1 supersymmetric backgrounds in M-theory and type IIA string theory with fluxes, classifies solutions based on internal manifold structures, and generalizes Hitchin flow equations for constructing such manifolds.

Contribution

It provides a comprehensive classification of flux backgrounds using group structures, generalizes Hitchin flow equations, and offers explicit construction techniques for internal manifolds.

Findings

Derived necessary and sufficient conditions for N=1 flux backgrounds.

Classified solutions based on internal manifold structures and isometries.

Generalized Hitchin flow equations for building 7-manifolds with flux.

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.