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

Abstract

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