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



Xenia de la Ossa (Mathematical Institute, Oxford University, United Kingdom)

Magdalena Larfors (Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden)

Eirik E. Svanes (Institut Lagrange de Paris, Sorbonne Universités, Paris, France)


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.

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