On orientations for gauge-theoretic moduli spaces - Joyce, D; Tanaka, Y; Upmeier, M (Elsevier, 2020)
Let X be a compact manifold, D a real elliptic operator on X, G a Lie group, P→X a principal G-bundle, and P the infinite-dimensional moduli space of all connections ∇P on P modulo gauge, as a topological stack. For each [∇P]∈P, we can consider the twisted elliptic operator D∇Ad(P) on X. This is a continuous family of elliptic operators over the base P, and so has an orientation bundle ODP→P, a principal ℤ2-bundle parametrizing orientations of KerD∇Ad(P)⊕CokerD∇Ad(P) at each [∇P]. An orientation on (P,D) is a trivialization ODP≅P×ℤ2. In gauge theory one studies moduli spaces  of connections ∇P on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds (X,g). Under good conditions  is a smooth manifold, and orientations on (P,D) pull back to orientations on  in the usual sense of differential geometry under the inclusion ↪P. This is important in areas such as Donaldson theory, where one needs an orientation on  to define enumerative invariants. We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on (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, the Kapustin-Witten equations, and the Vafa-Witten equations on 4-manifolds, and the Haydys-Witten equations on 5-manifolds.