Weyl symmetry for curve counting invariants via spherical twists

Buelles, Tim-Henrik; Moreira, Miguel

doi:10.1090/jag/829

Authors: Tim-Henrik Buelles and Miguel Moreira
Journal: J. Algebraic Geom. 33 (2024), 687-756
DOI: https://doi.org/10.1090/jag/829
Published electronically: May 10, 2024
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Abstract | References | Additional Information

Abstract:

We study the curve counting invariants of Calabi–Yau 3-folds via the Weyl reflection along a ruled divisor. We obtain a new rationality result and functional equation for the generating functions of Pandharipande–Thomas invariants. When the divisor arises as resolution of a curve of $A_1$-singularities, our results match the rationality of the associated Calabi–Yau orbifold.

The symmetry on generating functions descends from the action of an infinite dihedral group of derived auto-equivalences, which is generated by the derived dual and a spherical twist. Our techniques involve wall-crossing formulas and generalized Donaldson–Thomas invariants for surface-like objects.

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Additional Information

Tim-Henrik Buelles
Affiliation: The Division of Mathematics, Physics and Astronomy, California Institute of Technology, Pasadena, CA 91125
MR Author ID: 1422111
Email: tbuelles@caltech.edu

Miguel Moreira
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
MR Author ID: 1513017
Email: miguel@mit.edu

Received by editor(s): May 19, 2022
Received by editor(s) in revised form: October 3, 2023
Published electronically: May 10, 2024
Additional Notes: The authors were supported by ERC-2017-AdG-786580-MACI. The project received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation programme (grant agreement 786580).
Article copyright: © Copyright 2024 University Press, Inc.