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 $D$. Let $G$ be SU$(m)$ or U$(m)$, and $Eto X$ be a rank $m$ complex bundle with $G$-structure. Write ${mathcal B}_E$ for the infinite-dimensional moduli space of connections on $E$, modulo gauge. There is a natural principal ${mathbb Z}_2$-bundle $O^D_Eto{mathcal B}_E$ parametrizing orientations of det$,D_{{rm Ad }A}$ for twisted elliptic operators $D_{{rm Ad }A}$ at each $[A]$ in ${mathcal B}_E$. A theorem of Walpuski shows $O^D_E$ is trivializable. We prove that if we choose an orientation for det$,D$, and a flag structure on X in the sense of Joyce arXiv:1610.09836, then we can define canonical trivializations of $O^D_E$ for all such bundles $Eto X$, satisfying natural compatibilities. Now let $(X,varphi,g)$ be a compact $G_2$-manifold, with d$(*varphi)=0$. Then we can consider moduli spaces ${mathcal M}_E^{G_2}$ of $G_2$-instantons on $Eto X$, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general, with ${mathcal M}_E^{G_2}subset{mathcal B}_E$. The restriction of $O^D_E$ to ${mathcal M}_E^{G_2}$ is the ${mathbb Z}_2$-bundle of orientations on ${mathcal M}_E^{G_2}$. Thus, our theorem induces canonical orientations on all such $G_2$-instanton moduli spaces ${mathcal M}_E^{G_2}$. This contributes to the Donaldson-Segal programme arXiv:0902.3239, which proposes defining enumerative invariants of $G_2$-manifolds $(X,varphi,g)$ by counting moduli spaces ${mathcal M}_E^{G_2}$, with signs depending on a choice of orientation. This paper is a sequel to Joyce-Tanaka-Upmeier arXiv:1811.01096, which develops the general theory of orientations on gauge-theoretic moduli spaces, and gives applications in dimensions 3,4,5 and 6. A third paper Cao-Gross-Joyce arXiv:1811.09658 studies orientations on moduli spaces in dimension 8.