On orientations for gauge-theoretic moduli spaces

Let X be a compact manifold, D:Γ∞(E0)→Γ∞(E1) a real elliptic operator on X, G a Lie group, P→X a principal G-bundle, and BP the infinite-dimensional moduli space of all connections ∇P on P modulo gauge, as a topological stack. For each [∇P]∈BP, we can consider the twisted elliptic operator D∇Ad(P):Γ∞(Ad(P)⊗E0)→Γ∞(Ad(P)⊗E1) on X. This is a continuous family of elliptic operators over the base BP, and so has an orientation bundle OPE•→BP, a principal Z2-bundle parametrizing orientations of KerD∇Ad(P)⊕CokerD∇Ad(P) at each [∇P]. An orientation on (BP,E•) is a trivialization OPE•≅BP×Z2. In gauge theory one studies moduli spaces MPga of connections ∇P on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds (X,g). Under good conditions MPga is a smooth manifold, and orientations on (BP,E•) pull back to orientations on MPga in the usual sense of differential geometry under the inclusion MPga↪BP. This is important in areas such as Donaldson theory, where one needs an orientation on MPga to define enumerative invariants. We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on (BP,E•), 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.