Canonical orientations for moduli spaces of G2-instantons with gauge group SU(m) or U(m)

Dominic Joyce, Markus Upmeier

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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 BE for the infinite-dimensional moduli space of connections on E, modulo gauge. There is a natural principal Z2-bundle OED/g → BE parametrizing orientations of det D/gAd A for twisted elliptic operators D/gAd A at each [A] in BE. A theorem of Walpuski [33] shows OED/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 OED/g for all such bundles E → X, satisfying natural compatibilities. Now let (X, ϕ, g) be a compact G2-manifold, with d(∗ϕ) = 0. Then we can consider moduli spaces MGE2 of G2-instantons on E → X, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general, with MGE2 ⊂ BE. The restriction of OED/g to MGE2 is the Z2-bundle of orientations on MGE2 . Thus, our theorem induces canonical orientations on all such G2-instanton moduli spaces MGE2 . This contributes to the Donaldson–Segal programme [11], which proposes defining enumerative invariants of G2-manifolds (X, ϕ, g) by counting moduli spaces MGE2, with signs depending on a choice of orientation.

Original languageEnglish
Pages (from-to)199-229
Number of pages31
JournalJournal of Differential Geometry
Issue number2
Early online date16 Jun 2023
Publication statusPublished - 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.

Publisher Copyright:
© 2023 International Press of Boston, Inc.. All rights reserved.


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