Nu-invariants of extra-twisted connected sums

Sebastian Goette; Johannes Nordström; Don Zagier

doi:10.48550/arXiv.2010.16367

Nu-invariants of extra-twisted connected sums

Sebastian Goette
, Johannes Nordström
, Don Zagier

Department of Mathematical Sciences

University of Freiburg
Max-Planck-Institut für Mathematik

Research output: Contribution to journal › Article › peer-review

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Abstract

We analyse the possible ways of gluing twisted products of circles with asymptotically cylindrical Calabi-Yau manifolds to produce manifolds with holonomy G_2, thus generalising the twisted connected sum construction of Kovalev and Corti, Haskins, Nordstr\"om, Pacini. We then express the extended nu-invariant of Crowley, Goette, and Nordstr\"om arXiv:1505.02734 in terms of fixpoint and gluing contributions, which include different types of (generalised) Dedekind sums. Surprisingly, the calculations involve some non-trivial number-theoretical arguments connected with special values of the Dedekind eta-function and the theory of complex multiplication. One consequence of our computations is that there exist compact G_2-manifolds that are not G_2-nullbordant.

Original language	English
Journal	Communications on Pure and Applied Mathematics
DOIs	https://doi.org/10.48550/arXiv.2010.16367
Publication status	Submitted - 4 Apr 2025

Bibliographical note

73 pages; v2: Improved exposition and minor corrections, adjusted authorship

Funding

We 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). We gratefully acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University at which some of the research for this paper was carried out.

Funders
Simons Foundation

Keywords

math.GT
math.AG
math.DG
57R20 (Primary) 53C29, 58J28, 11F20 (Secondary)

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10.48550/arXiv.2010.16367

2010.16367v2Submitted manuscript, 0.99 MBLicence: Other

Cite this

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Nu-invariants of extra-twisted connected sums

Abstract

Bibliographical note

Funding

Keywords

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