Nu-invariants of extra-twisted connected sums

Sebastian Goette; Johannes Nordström; Don Zagier

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arXiv · 2010.16367

Nu-invariants of extra-twisted connected sums

Sebastian Goette·Johannes Nordström·Don Zagier

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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.

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Sebastian Goette, Johannes Nordström, Don Zagier. 2020-10-30. Nu-invariants of extra-twisted connected sums. https://arxiv.org/abs/2010.16367

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