Infinitesimal moduli of G2 holonomy manifolds with instanton bundles

Abstract

We describe the infinitesimal moduli space of pairs $(Y,V)$ where $Y$ is a manifold with $G_2$ holonomy, and $V$ is a vector bundle on $Y$ with an instanton connection. These structures arise in connection to the moduli space of heterotic string compactifications on compact and non-compact seven dimensional spaces, e.g. domain walls. Employing the canonical $G_2$ cohomology developed by Reyes-Carrión and Fernández and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli $H_{{\check{d}}_A}^1(Y,\mathrm{End}(V))$ plus the moduli of the $G_2$ structure preserving the instanton condition. The latter piece is contained in $H_{{\check{d}}_{\theta }}^1(Y,TY)$, and is given by the kernel of a map $\check{\mathcal{F}}$ which generalises the concept of the Atiyah map for holomorphic bundles on complex manifolds to the case at hand. In fact, the map $\check{\mathcal{F}}$ is given in terms of the curvature of the bundle and maps $H_{{\check{d}}_{\theta }}^1(Y,TY)$ into $H_{{\check{d}}_A}^2(Y,\mathrm{End}(V))$, and moreover can be used to define a cohomology on an extension bundle of $TY$ by $\mathrm{End}(V)$. We comment further on the resemblance with the holomorphic Atiyah algebroid and connect the story to physics, in particular to heterotic compactifications on $(Y,V)$ when ${\alpha }^{\prime }=0$.

arXiv comment51 pages, minor corrections, references added

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• Marginal deformations of heterotic $G_2$ sigma models
• Superpotential of Three Dimensional $\mathcal{N}=1$ Heterotic Supergravity
• The heterotic ${\mathrm{G}}_2$ system on contact Calabi--Yau $7$-manifolds
• $\mathcal{G}$-structure symmetries and anomalies in $(1,0)$ non-linear $\sigma$-models
• Heterotic Line Bundle Models on Generalized Complete Intersection Calabi Yau Manifolds
• Supersymmetric gauge theory on curved 7-branes
• G-structures and Superstrings from the Worldsheet
• Transversality for the moduli space of Spin(7)-instantons