M-theory Moduli from Exceptional Complex Structures

TL;DR

This paper analyzes the moduli spaces of M-theory flux compactifications using exceptional generalised geometry, classifying backgrounds into two types and calculating their infinitesimal moduli, revealing flux effects and dualities.

Contribution

It introduces a cohomological framework for computing moduli in M-theory compactifications, extending previous methods to include flux effects and duality checks.

Findings

Type-0 backgrounds have moduli from de Rham cohomology.

Flux reduces the number of moduli compared to fluxless cases.

Moduli of heterotic M-theory match those of the Hull-Strominger system.

Abstract

We continue the analysis of the geometry of generic Minkowski $\mathcal{N} = 1$, $D = 4$ flux compactifications in M-theory using exceptional generalised geometry, including the calculation of the infinitesimal moduli spaces. The backgrounds can be classified into two classes: type-0 and type-3. For type-0, we review how the moduli arise from standard de Rham cohomology classes. We also argue that, under reasonable assumptions, there are no appropriate sources to support compact flux backgrounds for this class and so the only solutions are in fact $G_2$ geometries. For type-3 backgrounds, given a suitable $\partial ' \bar{\partial} ' $-lemma, we show that the moduli can be calculated from a cohomology based on an involutive sub-bundle of the complexified tangent space. Using a simple spectral sequence we prove quite generally that the presence of flux can only reduce the number of moduli compared with the fluxless case. We then use the formalism to calculate the moduli of heterotic M-theory and show they match those of the dual Hull-Strominger system as expected.