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. The first paper is arXiv:2312.06818. 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 $P\to X$ over compact spin $n$-manifolds at once is equivalent to the vanishing of a certain morphism $\Omega_n^{\rm Spin}(\mathcal L BG)\to{\mathbb Z}_2$ 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. Moreover, we apply bordism techniques to prove that mod-$8$ Floer gradings exist for moduli spaces of $G_2$-instantons for all principal SU(2)-bundles. We also prove that there are canonical orientations for all principal U$(m)$-bundles $P\to X$ 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.