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# Advances in Theoretical and Mathematical Physics

## Volume 19 (2015)

### Number 4

### Exploring $SU(3)$ structure moduli spaces with integrable $G_2$ structures

Pages: 837 – 903

DOI: http://dx.doi.org/10.4310/ATMP.2015.v19.n4.a5

#### Authors

#### Abstract

We study the moduli space of $SU(3)$ structure manifolds $X$ that form the internal compact spaces in four-dimensional $N = \frac{1}{2}$ domain wall solutions of heterotic supergravity with flux. Together with the direction perpendicular to the four-dimensional domain wall, $X$ forms a non-compact $7$-manifold $Y$ with torsionful $G_2$ structure. We use this $G_2$ embedding to explore how $X(t)$ varies along paths $C(t)$ in the $SU(3)$ structure moduli space. Our analysis includes the Bianchi identities which strongly constrain the flow. We show that requiring that the $SU(3)$ structure torsion is preserved along the path leads to constraints on the $G_2$ torsion and the embedding of $X$ in $Y$. Furthermore, we study flows along which the torsion classes of $X$ go from zero to non-zero values. In particular, we present evidence that the flow of half-flat $SU(3)$ structures may contain Calabi–Yau loci, in the presence of non-vanishing $H$-flux.