Infinitesimal moduli of G2 holonomy manifolds with instanton bundles

A bstractWe 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 Hd∨A1Y,EndV$$ {H}_{{\overset{\vee }{\mathrm{d}}}_A}^1\left(Y,\mathrm{End}(V)\right) $$ plus the moduli of the G2 structure preserving the instanton condition. The latter piece is contained in Hd∨θ1YTY$$ {H}_{\overset{\vee }{\mathrm{d}}\theta}^1\left(Y,TY\right) $$, and is given by the kernel of a map ℱ∨$$ \overset{\vee }{\mathrm{\mathcal{F}}} $$ which generalises the concept of the Atiyah map for holomorphic bundles on complex manifolds to the case at hand. In fact, the map ℱ∨$$ \overset{\vee }{\mathrm{\mathcal{F}}} $$ is given in terms of the curvature of the bundle and maps Hd∨θ1YTY$$ {H}_{\overset{\vee }{\mathrm{d}}\theta}^1\left(Y,TY\right) $$ into Hd∨A2Y,EndV$$ {H}_{{\overset{\vee }{\mathrm{d}}}_A}^2\left(Y,\mathrm{End}(V)\right) $$, 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.