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We describe the infinitesimal moduli space of pairs $(Y, V)$ where $Y$ is amanifold with $G_2$ holonomy, and $V$ is a vector bundle on $Y$ with aninstanton connection. These structures arise in connection to the moduli spaceof heterotic string compactifications on compact and non-compact sevendimensional spaces, e.g. domain walls. Employing the canonical $G_2$ cohomologydeveloped by Reyes-Carri\'on and Fern\'andez and Ugarte, we show that themoduli space decomposes into the sum of the bundle moduli$H^1_{\check{d}_A}(Y,\mathrm{End}(V))$ plus the moduli of the $G_2$ structurepreserving 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 forholomorphic 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 $H^2_{\check{d}_A}(Y,\mathrm{End}(V))$, andmoreover 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 holomorphicAtiyah algebroid and connect the story to physics, in particular to heteroticcompactifications on $(Y,V)$ when $\alpha'=0$.

Infinitesimal moduli of G2 holonomy manifolds with instanton bundles