## Abstract

Suppose (X, g) is a compact, spin Riemannian 7-manifold, with Dirac operator D/^{g} : C^{∞}(X, S) / → C^{∞}(X, S). / Let G be SU(m) or U(m), and E → X be a rank m complex bundle with Gstructure. Write B_{E} for the infinite-dimensional moduli space of connections on E, modulo gauge. There is a natural principal Z_{2}-bundle O_{E}^{D/g} → B_{E} parametrizing orientations of det D/^{g}_{Ad A} for twisted elliptic operators D/^{g}_{Ad A} at each [A] in B_{E}. A theorem of Walpuski [33] shows O_{E}^{D/g} is trivializable. We prove that if we choose an orientation for det D/^{g}, and a flag structure on X in the sense of [17], then we can define canonical trivializations of O_{E}^{D/g} for all such bundles E → X, satisfying natural compatibilities. Now let (X, ϕ, g) be a compact G_{2}-manifold, with d(∗ϕ) = 0. Then we can consider moduli spaces M^{G}_{E}^{2} of G_{2}-instantons on E → X, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general, with M^{G}_{E}^{2} ⊂ B_{E}. The restriction of O_{E}^{D/g} to M^{G}_{E}^{2} is the Z_{2}-bundle of orientations on M^{G}_{E}^{2} . Thus, our theorem induces canonical orientations on all such G_{2}-instanton moduli spaces M^{G}_{E}^{2} . This contributes to the Donaldson–Segal programme [11], which proposes defining enumerative invariants of G_{2}-manifolds (X, ϕ, g) by counting moduli spaces M^{G}_{E}^{2}, with signs depending on a choice of orientation.

Original language | English |
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Pages (from-to) | 199-229 |

Number of pages | 31 |

Journal | Journal of Differential Geometry |

Volume | 124 |

Issue number | 2 |

Early online date | 16 Jun 2023 |

DOIs | |

Publication status | Published - 16 Jun 2023 |

### Bibliographical note

Funding Information:Acknowledgments. This research was partly funded by a Simons Collaboration Grant on ‘Special Holonomy in Geometry, Analysis and Physics’. The second author was funded by DFG grant UP 85/3-1 and by grant UP 85/2-1 of the DFG priority program SPP 2026 ‘Geometry at Infinity.’ The authors would like to thank Yalong Cao, Aleksander Doan, Sebastian Goette, Jacob Gross, Andriy Haydys, Johannes Nord-ström, Yuuji Tanaka, Richard Thomas and Thomas Walpuski for helpful conversations, and the referee.

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