Canonical orientations for moduli spaces of G_2-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 BE for the infinite-dimensional moduli space of connections on E, modulo gauge. There is a natural principal Z2-bundle O D/ g E → BE parametrizing orientations of det D/ g Ad A for twisted elliptic operators D/ g Ad A at each [A] in BE. A theorem of Walpuski [33] shows O D/ g E 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 D/ g E 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 MG2 E of G2-instantons on E → X, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general, with MG2 E ⊂ BE. The restriction of O D/ g E to MG2 E is the Z2-bundle of orientations on MG2 E . Thus, our theorem induces canonical orientations on all such G2-instanton moduli spaces MG2 E . This contributes to the Donaldson–Segal programme [11], which proposes defining enumerative invariants of G2-manifolds (X, ϕ, g) by counting moduli spaces MG2 E , with signs depending on a choice of orientation.