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On Counting Flat Connections Over \(G_2\)-Orbifolds

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Abstract

We study the moduli space of \(G_2\)-instantons on (projectively) flat bundles over torsion-free \(G_2\)-orbifolds. We prove that the moduli space is compact and smooth at the irreducible locus after adding small and generic holonomy perturbations. Consequently, we define the \(G_2\)-Casson invariant that is invariant under \(C^0\)-deformation of torsion-free \(G_2\)-structures. We compute this invariant for some orbifolds that arise in Joyce’s construction of compact \(G_2\)-manifolds.

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Acknowledgements

The author is grateful for Simon Donaldson for suggesting this project with patient guidance and encouragement. He also wants to thank Donghao Wang, Daniel Platt, and Thomas Walpuski for enlightening discussions.

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  1. Shanghai Jiao Tong University, 800 Dongchuan Rd, Minghang, Shanghai, 200240, China

    Langte Ma

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Ma, L. On Counting Flat Connections Over \(G_2\)-Orbifolds. Commun. Math. Phys. 405, 124 (2024). https://doi.org/10.1007/s00220-024-05013-7

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