In many situations in Differential or Algebraic Geometry, one forms moduli spaces $\cal M$ of geometric objects, such that $\cal M$ is a manifold, or something close to a manifold (a derived manifold, Kuranishi space, …). Then we can ask whether $\cal M$ is orientable, and if so, whether there is a natural choice of orientation.
This is important in the definition of enumerative invariants: we arrange that the moduli space $\cal M$ is a compact oriented manifold (or derived manifold), so it has a fundamental class in homology, and the invariants are the integrals of natural cohomology classes over this fundamental class.
For example, if $X$ is a compact oriented Riemannian 4-manifold, we can form moduli spaces $\cal M$ of instanton connections on some principal $G$-bundle $P$ over $X$, and the Donaldson invariants of $X$ are integrals over $\cal M$.
In the paper arXiv:2503.20456, Markus Upmeier and I develop a theory of "bordism categories”, which are a new tool for studying orientability and canonical orientations of moduli spaces. It uses a lot of Algebraic Topology, and computation of bordism groups of classifying spaces. We apply it to study orientability and canonical orientations of moduli spaces of $G_2$ instantons and associative 3-folds on $G_2$ manifolds, and of Spin(7) instantons and Cayley 4-folds on Spin(7) manifolds, and of coherent sheaves on Calabi-Yau 4-folds. These have applications to enumerative invariants, in particular, to Donaldson-Thomas type invariants of Calabi-Yau 4-folds.
All this is joint work with Markus Upmeier.
