Bordism categories and orientations of gauge theory moduli spaces

This is the second paper of a series that develops a bordism-theoretic point of view on orientations in enumerative geometry. This paper focuses on those applications to gauge theory that can be established purely using formal arguments and calculations from algebraic topology.

We prove that the orientability of moduli spaces of connections in gauge theory for all principal G-bundles over compact spin n-manifolds at once is equivalent to the vanishing of a certain morphism on the n-dimensional spin bordism group of the free loop space of the classifying space of G, and we give a complete list of all compact, connected Lie groups G for which this holds.

We also prove that there are canonical orientations for all principal U(m)-bundles P over compact spin 8-manifolds satisfying c_2(P)-c_1(P)^2=0. The proof is based on an interesting relationship to principal E_8-bundles. These canonical orientations play an important role in many conjectures about Donaldson-Thomas type invariants on Calabi-Yau 4-folds, and resolve an apparent paradox in these conjectures.