Bordism categories and orientations of moduli spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910894624079872 |
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| author | Joyce, Dominic Upmeier, Markus |
| author_facet | Joyce, Dominic Upmeier, Markus |
| contents | 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'. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bordism categories and orientations of moduli spaces Joyce, Dominic Upmeier, Markus Algebraic Topology Algebraic Geometry Differential Geometry 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'. |
| title | Bordism categories and orientations of moduli spaces |
| topic | Algebraic Topology Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2503.20456 |