On orientations for gauge-theoretic moduli spaces

Abstract

Let $X$ be a compact manifold, $D$ a real elliptic operator on $X$, $G$ a Lie group, $P\to X$ a principal $G$-bundle, and ${\mathcal B}_P$ the infinite-dimensional moduli space of all connections $\nabla_P$ on $P$ modulo gauge, as a topological stack. For each $[\nabla_P]\in{\mathcal B}_P$, we can consider the twisted elliptic operator $D^{\nabla_{Ad(P)}}$ on X. This is a continuous family of elliptic operators over the base ${\mathcal B}_P$, and so has an orientation bundle $O^D_P\to{\mathcal B}_P$, a principal ${\mathbb Z}_2$-bundle parametrizing orientations of Ker$D^{\nabla_{Ad(P)}}\oplus$Coker$D^{\nabla_{Ad(P)}}$ at each $[\nabla_P]$. An orientation on $({\mathcal B}_P,D)$ is a trivialization $O^D_P\cong{\mathcal B}_P\times{\mathbb Z}_2$. In gauge theory one studies moduli spaces $\mathcal M$ of connections $\nabla_P$ on $P$ satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds $(X, g)$. Under good conditions $\mathcal M$ is a smooth manifold, and orientations on $({\mathcal B}_P,D)$ pull back to orientations on $\mathcal M$ in the usual sense under the inclusion ${\mathcal M}\hookrightarrow{\mathcal B}_P$. This is important in areas such as Donaldson theory, where one needs an orientation on $\mathcal M$ to define enumerative invariants. We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on $({\mathcal B}_P,D)$, after fixing some algebro-topological information on $X$. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds, instantons, Kapustin-Witten and Vafa-Witten equations on 4-manifolds, and the Haydys-Witten equations on 5-manifolds. Two sequels arXiv:1811.02405, arXiv:1811.09658 discuss orientations in 7 and 8 dimensions.