| Summary: | We describe the infinitesimal moduli space of pairs (Y, V) where Y is a manifold with G2 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 G2 cohomology developed by Reyes-Carrión and Fernández and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli $H1d∨A(Y,End(V))Hd∨A1(Y,End(V))$ plus the moduli of the G2 structure preserving the instanton condition. The latter piece is contained in $H1d∨θ(Y,TY)Hd∨θ1(Y,TY),$ and is given by the kernel of a map $F∨F∨$ which generalises the concept of the Atiyah map for holomorphic bundles on complex manifolds to the case at hand. In fact, the map $F∨F∨$ is given in terms of the curvature of the bundle and maps $H1d∨θ(Y,TY)Hd∨θ1(Y,TY)$ into $H2d∨A(Y,End(V))Hd∨A2(Y,End(V))$, and moreover can be used to define a cohomology on an extension bundle of $TY$ by 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$α′ = 0$.
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