A note on BPS structures and Gopakumar–Vafa invariants

We regard the work of Maulik and Toda, proposing a sheaf-theoretic approach to Gopakumar–Vafa invariants, as defining a BPS structure, that is, a collection of BPS invariants together with a central charge. Assuming their conjectures, we show that a canonical flat section of the flat connection corresponding to this BPS structure, at the level of formal power series, reproduces the Gromov–Witten partition function for all genera, up to some error terms in genus $0$ and $1$. This generalises a result of Bridgeland and Iwaki for the contribution from genus $0$ Gopakumar–Vafa invariants.

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Dedicated to the memory of Boris Dubrovin.

I would like to thank A. Barbieri, T. Bridgeland, M. Kool and J. Scalise for some discussions related to this note, and the anonymous Referees for several important comments on the manuscript. I am very grateful to the organisers and participants in the workshops Geometric correspondences of gauge theories, Vienna, August-September 2018, and D-modules, quantum geometry, and related topics, Kyoto, December 2018, especially to H. Iritani, K. Iwaki, T. Mochizuki, C. Sabbah and Y. Toda, for their interest in my talks and for their suggestions. My participation in these activities was supported by the Erwin Schrödinger Institute, Vienna and the Research Institute for the Mathematical Sciences, Kyoto.

Received 27 February 2019

Accepted 8 May 2019

Published 8 August 2022