Canonical orientations for moduli spaces of $G_2$-instantons with gauge group $\mathrm{SU}(m)$ or $\mathrm{U}(m)$

Suppose $(X, g)$ is a compact, spin Riemannian $7$-manifold, with Dirac operator ${\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g : C^\infty (X, {\mathrm{S}\mspace{-10mu}/\mspace{2mu}}) \to {\mathrm{S}\mspace{-10mu}/\mspace{2mu}})$. Let $G$ be $\mathrm{SU}(m)$ or $\mathrm{U}(m)$, and $E X$ rank $m$ complex bundle $G$-structure. Write $\mathcal{B}_E$ for the infinite-dimensional moduli space of connections on $E$, modulo gauge. There natural principal $\mathbb{Z}_2$-bundle $O^{{\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g}_E \mathcal{B}_E$ parametrizing orientations $\operatorname{det} {\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g_{\mathrm{Ad} \, A}$ twisted elliptic operators ${\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g_{\mathrm{Ad} at each $[A]$ in $\mathcal{B}_E$. A theorem Walpuski [33] shows $O^{{\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g}_E$ trivializable. We prove that if we choose an orientation {\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g$ , flag structure $X$ sense [17], then can define canonical trivializations all such bundles X$, satisfying compatibilities. Now let \varphi, compact $G_2$-manifold, $d (\ast \varphi) = 0$. Then consider spaces $\mathcal{M}^{G_2}_E$ $G_2$-instantons which are smooth manifolds under suitable transversality conditions, derived general, $\mathcal{M}^{G_2}_E \subset \mathcal{B}_E$. The restriction to $\mathcal{M}^{G_2}_E$. Thus, our induces $G_2$-instanton This contributes Donaldson–Segal programme [11], proposes defining enumerative invariants $G_2$-manifolds by counting $\mathcal{M}^{G_2}_E$, signs depending choice orientation.