Bordism categories and orientations of moduli spaces

Internet publication

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 $\mathcal{X}$ be a manifold with geometric structure, and $\mathcal{M}$ a moduli space of geometric objects on $\mathcal{X}$. Our theory aims to answer the questions:

(i) Can we prove $\mathcal{M}$ is orientable for all $\mathcal{X,M}$?

(ii) If not, can we give computable sufficient conditions on $\mathcal{X}$ that guarantee $\mathcal{M}$ is orientable?

(iii) Can we specify extra data on $\mathcal{X}$ which allow us to construct a canonical orientation on $\mathcal{M}$?

We define 'bordism categories', such as BordSpin n BG with objects $\mathcal{(X,P)}$ for $\mathcal{X}$ a compact spin n-manifold and P→ 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 $\mathcal{Z}$2-torsors.

We apply our theory to study orientability and canonical orientations for moduli spaces of G2-instantons and of associative 3-folds in G2-manifolds, and for moduli spaces of Spin(7)-instantons and of Cayley 4-folds in Spin(7)-manifolds, and for moduli spaces of coherent sheaves and perfect complexes on Calabi–Yau 4-folds. The latter are needed to define Donaldson–Thomas type invariants of 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 $\mathcal{M}$, and show canonical orientations can be defined using a 'flag structure'.

Authors: Joyce, D; Upmeier, M

• Publication status: Published
• Peer review status: Not peer reviewed
• Host title: arXiv
• Publication date: 2025-03-26
• DOI: 10.48550/arXiv.2503.20456
• EISSN: 2331-8422
• Language: English