The infinitesimal moduli space of heterotic $G_2$ systems

Abstract

Heterotic string compactifications on integrable $G_2$ structure manifolds $Y$ with instanton bundles $(V,A),(TY,\tilde{\theta })$ yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative $\mathcal{D}$ and show that it is equivalent to a heterotic $G_2$ system encoding the geometry of the heterotic string compactifications. This operator $\mathcal{D}$ acts on a bundle $\mathcal{Q}=T^{\ast }Y\oplus \mathrm{End}(V)\oplus \mathrm{End}(TY)$ and satisfies a nilpotency condition ${\check{\mathcal{D}}}^2=0$, for an appropriate projection of $\mathcal{D}$. Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group ${\check{H}}_{\check{\mathcal{D}}}^1(\mathcal{Q})$. We comment on the similarities and differences of our result with Atiyah's well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the ${\alpha }^{\prime }$ expansion.

v1: 60 pages, including 3 appendices, v2: minor updates, references added, v3: matches published version

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