2025/03/26 by Dominic Joyce, Joyce, Dominic, Markus Upmeier +1 · 2 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2503.20456
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 BordnSpin(BG) with objects (X,P) for 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 \mathbb Z2-torsors. We apply our theory to study orientability and canonical orientations for moduli spaces of G2-instantons and associative 3-folds in G2-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'.