Bordism categories and orientations of moduli spaces

Abstract

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 $X$ be a manifold with geometric structure, and $\cal M$ a moduli space of geometric objects on $X$. Our theory aims to answer the questions: (i) Can we prove $\cal M$ is orientable for all $X,\cal M$? (ii) If not, can we give computable sufficient conditions on $X$ that guarantee $\cal M$ is orientable? (iii) Can we specify extra data on $X$ which allow us to construct a canonical orientation on $\cal M$? We define 'bordism categories', such as $Bord_n^{Spin}(BG)$ with objects $(X,P)$ for $X$ a compact spin $n$-manifold and $P\to X$ a principal $G$-bundle, for $G$ 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 ${\mathbb Z}_2$-torsors. We apply our theory to study orientability and canonical orientations for moduli spaces of $G_2$-instantons and associative 3-folds in $G_2$-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 $\cal M$, and show canonical orientations can be defined using a 'flag structure'.