Infinitesimal moduli of G2 holonomy manifolds with instanton bundles

TL;DR

This paper characterizes the infinitesimal moduli space of G2 holonomy manifolds with instanton bundles, revealing its decomposition into bundle and geometric moduli, and generalizing the Atiyah map in this context.

Contribution

It introduces a cohomological framework for the moduli space of G2 manifolds with instanton bundles, extending the Atiyah map concept to this setting.

Findings

The moduli space decomposes into bundle and G2 structure moduli.

The G2 structure moduli are characterized by the kernel of a generalized Atiyah map.

A cohomology on an extension bundle is defined, linking bundle curvature to moduli.

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-Carri\'on and Fern\'andez and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli $H^1_{\check{d}_A}(Y,\mathrm{End}(V))$ plus the moduli of the $G_2$ structure preserving the instanton condition. The latter piece is contained in $H^1_{\check{d}_\theta}(Y,TY)$, and is given by the kernel of a map ${\cal\check F}$ which generalises the concept of the Atiyah map for holomorphic bundles on complex manifolds to the case at hand. In fact, the map ${\cal\check F}$ is given in terms of the curvature of the bundle and maps $H^1_{\check{d}_\theta}(Y,TY)$ into $H^2_{\check{d}_A}(Y,\mathrm{End}(V))$, 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 $\alpha'=0$.