On orientations for gauge-theoretic moduli spaces

Let X be a compact manifold, a real elliptic operator on X, G a Lie group, a principal G-bundle, and the infinite-dimensional moduli space of all connections on P modulo gauge, as a topological stack. For each , we can consider the twisted elliptic operator on X. This is a continuous family of elliptic operators over the base , and so has an orientation bundle , a principal -bundle parametrizing orientations of at each . An orientation on is a trivialization .

In gauge theory one studies moduli spaces of connections on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds . Under good conditions is a smooth manifold, and orientations on pull back to orientations on in the usual sense of differential geometry under the inclusion . 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 , 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.