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

Abstract

Suppose $(X,g)$ is a compact, spin Riemannian 7-manifold, with Dirac operator $D$. Let $G$ be SU$(m)$ or 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 $O_E^D\to {\mathcal{B}}_E$ parametrizing orientations of det$D_{\mathrm{Ad}A}$ for twisted elliptic operators $D_{\mathrm{Ad}A}$ at each $[A]$ in ${\mathcal{B}}_E$. A theorem of Walpuski shows $O_E^D$ 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 $O_E^D$ for all such bundles $E\to X$, satisfying natural compatibilities. Now let $(X,\phi ,g)$ be a compact $G_2$-manifold, with d$(\ast \phi )=0$. Then we can consider moduli spaces ${\mathcal{M}}_E^{G_2}$ of $G_2$-instantons on $E\to X$, 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 $O_E^D$ 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 $(X,\phi ,g)$ 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.

30 pages. Version 3: slightly revised presentation, fixed typos. Final version, to appear in the Journal of Differential Geometry

References in corpus (4)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Gauge Theory in higher dimensions, II
• Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds
• On orientations for gauge-theoretic moduli spaces

Cited by in corpus (3)

• Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds
• On orientations for gauge-theoretic moduli spaces
• Deformation theory of deformed Hermitian Yang-Mills connections and deformed Donaldson-Thomas connections