Bordism categories and orientations of moduli spaces
Dominic Joyce, Markus Upmeier
TL;DR
The paper develops a bordism-theoretic framework to study orientations of moduli spaces arising in gauge theory, calibrated geometry, and derived algebraic geometry, recasting orientability as a functorial problem on bordism categories and reducing computation to spin bordism groups of classifying spaces. It introduces a family of bordism categories (loop, submanifold, and cohomology variants) and transfer functors to relate complex orientation questions to simpler ones, and defines flag structures to produce canonical orientation data for moduli spaces, including DT4 settings on Calabi–Yau 4-folds. The main technical program computes low-dimensional spin bordism groups, analyzes maps from loop spaces, and provides explicit orientation results for a wide range of moduli spaces such as $G_2$-instantons, associative 3-folds, Cayley 4-folds, Spin(7)-instantons, and coherent sheaves, while correcting misconceptions in prior work. The framework yields practical criteria (in terms of bordism groups and Steenrod operations) to decide orientability and canonicalize orientations, enabling robust DT4 invariants and potential gradings in related Floer theories. Overall, this work lays foundational tools to define and compare orientation data across diverse geometric contexts, connecting differential geometry, topology, and derived algebraic geometry.
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'.