## Abstract

Let X be a compact manifold, a real elliptic operator on X, G a Lie group, a principal G-bundle, and the infinite-dimensional moduli space of all connections on P modulo gauge, as a topological stack. For each , we can consider the twisted elliptic operator on X. This is a continuous family of elliptic operators over the base , and so has an orientation bundle , a principal -bundle parametrizing orientations of at each . An orientation on is a trivialization .

In gauge theory one studies moduli spaces of connections on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds . Under good conditions is a smooth manifold, and orientations on pull back to orientations on in the usual sense of differential geometry under the inclusion . This is important in areas such as Donaldson theory, where one needs an orientation on to define enumerative invariants.

We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on , after fixing some algebro-topological information on X. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds, instantons, the Kapustin–Witten equations, and the Vafa–Witten equations on 4-manifolds, and the Haydys–Witten equations on 5-manifolds.

In gauge theory one studies moduli spaces of connections on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds . Under good conditions is a smooth manifold, and orientations on pull back to orientations on in the usual sense of differential geometry under the inclusion . This is important in areas such as Donaldson theory, where one needs an orientation on to define enumerative invariants.

We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on , after fixing some algebro-topological information on X. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds, instantons, the Kapustin–Witten equations, and the Vafa–Witten equations on 4-manifolds, and the Haydys–Witten equations on 5-manifolds.

Original language | English |
---|---|

Article number | 106957 |

Number of pages | 64 |

Journal | Advances in Mathematics |

Volume | 362 |

Early online date | 9 Jan 2020 |

DOIs | |

Publication status | Published - 4 Mar 2020 |

### Bibliographical note

AcknowledgementsThis research was partly funded by a Simons Collaboration Grant on ‘Special Holonomy in Geometry, Analysis and Physics’. The second author was partially supported by JSPS Grant-in-Aid for Scientific Research number JP16K05125. The third author was funded by DFG grants UP 85/2-1 of the DFG priority program SPP 2026 ‘Geometry at Infinity’ and UP 85/3-1.

The authors would like to thank Yalong Cao, Aleksander Doan, Simon Donaldson, Sebastian Goette, Andriy Haydys, Vicente Muñoz, Johannes Nordström, Cliff Taubes, Richard Thomas, and Thomas Walpuski for helpful conversations, and the referee for careful proofreading and useful comments.

## Keywords

- Moduli space
- Orientation
- Gauge theory
- Instanton
- Elliptic operator