An analytic invariant of G2 manifolds

Diarmuid Crowley, Sebastian Goette, Johannes Nordstrom

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Abstract

We prove that the moduli space of holonomy G2-metrics on a closed 7-manifold can be disconnected by presenting a number of explicit examples. We detect different connected components of the G2-moduli space by defining an analytic refinement ν¯(M,g)∈Z of the defect invariant ν(M,φ)∈Z/48 of G2-structures φ on a closed 7-manifold M introduced by the first and third authors. The ν¯-invariant is defined using η-invariants and Mathai-Quillen currents on M and we compute it for twisted connected sums à la Kovalev, Corti-Haskins-Nordström-Pacini and extra-twisted connected sums as constructed by the second and third authors. In particular, we find examples of G2-holonomy metrics in different components of the moduli space where the associated G2-structures are homotopic and other examples where they are not.

Original languageEnglish
Pages (from-to)865-907
Number of pages43
JournalInventiones Mathematicae
Volume239
Issue number3
Early online date23 Jan 2025
DOIs
Publication statusPublished - 31 Mar 2025

Acknowledgements

The authors thank Uli Bunke, Alessio Corti, Mark Haskins, Matthias Lesch, Arkadi Schelling and Thomas Walpuski for valuable discussions, and the referees for constructive comments. SG and JN would like to thank the Simons foundation for its support of their research under the Simons Collaboration on “Special Holonomy in Geometry, Analysis and Physics” (grants #488617, Sebastian Goette, and #488631, Johannes Nordström).

Funding

The authors thank Uli Bunke, Alessio Corti, Mark Haskins, Matthias Lesch, Arkadi Schelling and Thomas Walpuski for valuable discussions, and the referees for constructive comments. SG and JN would like to thank the Simons foundation for its support of their research under the Simons Collaboration on \u201CSpecial Holonomy in Geometry, Analysis and Physics\u201D (grants #488617, Sebastian Goette, and #488631, Johannes Nordstr\u00F6m).

FundersFunder number
Simons Foundation488617, 488631

    Keywords

    • math.GT
    • math.DG
    • 57R20 (Primary) 53C29, 58J28 (Secondary)

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