Canonical orientations for moduli spaces of G2-instantons with gauge group SU(m) or U(m)

Suppose (X, g) is a compact, spin Riemannian 7-manifold, with Dirac operator D/^g : C^∞(X, S) / → C^∞(X, S). / Let G be SU(m) or U(m), and E → X be a rank m complex bundle with Gstructure. Write B_E for the infinite-dimensional moduli space of connections on E, modulo gauge. There is a natural principal Z₂-bundle O_E^D/g → B_E parametrizing orientations of det D/^g_{Ad A} for twisted elliptic operators D/^g_{Ad A} at each [A] in B_E. A theorem of Walpuski [33] shows O_E^D/g is trivializable. We prove that if we choose an orientation for det D/^g, and a flag structure on X in the sense of [17], then we can define canonical trivializations of O_E^D/g for all such bundles E → X, satisfying natural compatibilities. Now let (X, ϕ, g) be a compact G₂-manifold, with d(∗ϕ) = 0. Then we can consider moduli spaces M^G_E² of G₂-instantons on E → X, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general, with M^G_E² ⊂ B_E. The restriction of O_E^D/g to M^G_E² is the Z₂-bundle of orientations on M^G_E² . Thus, our theorem induces canonical orientations on all such G₂-instanton moduli spaces M^G_E² . This contributes to the Donaldson–Segal programme [11], which proposes defining enumerative invariants of G₂-manifolds (X, ϕ, g) by counting moduli spaces M^G_E², with signs depending on a choice of orientation.