Restrictions of Heterotic $G_2$ Structures and Instanton Connections

Abstract

This note revisits recent results regarding the geometry and moduli of solutions of the heterotic string on manifolds $Y$ with a $G_2$ structure. In particular, such heterotic $G_2$ systems can be rephrased in terms of a differential $\check{\mathcal{D}}$ acting on a complex ${\check{\Omega }}^{\ast }(Y,\mathcal{Q})$, where $\mathcal{Q}=T^{\ast }Y\oplus \mathrm{End}(TY)\oplus \mathrm{End}(V)$ and $\check{\mathcal{D}}$ is an appropriate projection of an exterior covariant derivative $\mathcal{D}$ which satisfies an instanton condition. The infinitesimal moduli are further parametrised by the first cohomology $H_{\check{\mathcal{D}}}^1(Y,\mathcal{Q})$. We proceed to restrict this system to manifolds $X$ with an $SU(3)$ structure corresponding to supersymmetric compactifications to four dimensional Minkowski space, often referred to as Strominger--Hull solutions. In doing so, we derive a new result: the Strominger-Hull system is equivalent to a particular holomorphic Yang-Mills covariant derivative on $\mathcal{Q}{|}_X=T^{\ast }X\oplus \mathrm{End}(TX)\oplus \mathrm{End}(V)$.

To appear in the proceedings for Nigel Hitchin's 70th birthday conference. 17 pages

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