Bordism categories and orientations of moduli spaces

To define enumerative invariants in geometry, one often needs orientations on moduli spaces of geometric objects. This monograph develops a new bordism-theoretic point of view on orientations of moduli spaces. Let be a manifold with geometric structure, and a moduli space of geometric objects on . Our theory aims to answer the questions:
(i) Can we prove is orientable for all ?
(ii) If not, can we give computable sufficient conditions on that guarantee is orientable?
(iii) Can we specify extra data on which allow us to construct a canonical orientation on ?
We define 'bordism categories', such as with objects for a compact spin -manifold and a principal -bundle, for a Lie group. Bordism categories can be understood by computing bordism groups of classifying spaces using Algebraic Topology. Orientation problems are encoded in functors from a bordism category to -torsors.
We apply our theory to study orientability and canonical orientations for moduli spaces of -instantons and associative 3-folds in -manifolds, for moduli spaces of Spin(7)-instantons and Cayley 4-folds in Spin(7)-manifolds, and for moduli spaces of coherent sheaves on Calabi-Yau 4-folds. The latter are needed to define Donaldson-Thomas type invariants of Calabi-Yau 4-folds. In many cases we prove orientability of , and show canonical orientations can be defined using a 'flag structure'.