On orientations for gauge-theoretic moduli spaces

Let $X$ be a compact manifold, $D$ a real elliptic operator on $X$, $G$ a Lie group, $Pto X$ a principal $G$-bundle, and ${mathcal B}_P$ the infinite-dimensional moduli space of all connections $ abla_P$ on $P$ modulo gauge, as a topological stack. For each $[ abla_P]in{mathcal B}_P$, we can consider the twisted elliptic operator $D^{ abla_{Ad(P)}}$ on X. This is a continuous family of elliptic operators over the base ${mathcal B}_P$, and so has an orientation bundle $O^D_Pto{mathcal B}_P$, a principal ${mathbb Z}_2$-bundle parametrizing orientations of Ker$D^{ abla_{Ad(P)}}oplus$Coker$D^{ abla_{Ad(P)}}$ at each $[ abla_P]$. An orientation on $({mathcal B}_P,D)$ is a trivialization $O^D_Pcong{mathcal B}_Ptimes{mathbb Z}_2$. In gauge theory one studies moduli spaces $mathcal M$ of connections $ abla_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.