  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.

High Energy Physics - Theory