Bordism categories and orientations of gauge theory moduli spaces
<p>This is the second paper of a series that develops a bordism-theoretic point of view on orientations in enumerative geometry. The first paper is arXiv:2312.06818. This paper focuses on those applications to gauge theory that can be established purely using formal arguments and calculations...
| Main Authors: | , |
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| Format: | Internet publication |
| Language: | English |
| Published: |
2023
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| Summary: | <p>This is the second paper of a series that develops a bordism-theoretic point of view on orientations in enumerative geometry. The first paper is arXiv:2312.06818. This paper focuses on those applications to gauge theory that can be established purely using formal arguments and calculations from algebraic topology.</p>
<p>We prove that the orientability of moduli spaces of connections in gauge theory for <i>all</i> principal <i>G</i>-bundles <i>P</i>→<i>X</i> over compact spin <i>n</i>-manifolds <i>at once</i> is equivalent to the vanishing of a certain morphism Ω<sup>Spin</sup><sub><i>n</i></sub>(<i>L</i><i>BG</i>)→<i>Z</i><sub>2</sub> on the <i>n</i>-dimensional spin bordism group of the free loop space of the classifying space of <i>G</i>, and we give a complete list of all compact, connected Lie groups <i>G</i> for which this holds. Moreover, we apply bordism techniques to prove that mod-8 <i>Floer gradings</i> exist for moduli spaces of <i>G</i><sub>2</sub>-instantons for all principal SU(2)-bundles.</p>
<p>We also prove that there are <i>canonical orientations</i> for all principal U(<i>m</i>)-bundles <i>P</i>→<i>X</i> over compact spin 8-manifolds satisfying <i>c</i><sub>2</sub>(<i>P</i>)−<i>c</i><sub>1</sub>(<i>P</i>)<sup>2</sup>=0. The proof is based on an interesting relationship to principal <i>E</i><sub>8</sub>-bundles. These canonical orientations play an important role in many conjectures about Donaldson-Thomas type invariants on Calabi-Yau 4-folds, and resolve an apparent paradox in these conjectures.</p> |
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