Weyl symmetry for curve counting invariants via spherical twists
Creators
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1.
California Institute of Technology
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2.
Massachusetts Institute of Technology
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 A₁-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.
Copyright and License
© 2024 University Press, Inc.
Acknowledgement
The authors are grateful to Y. Bae, T. Beckmann, G. Oberdieck, R. Pandharipande, D. Nesterov, J. Rennemo, E. Scheidegger, and R. Thomas for discussions on stable pairs in the derived category and curve counting on Calabi-Yau 3-folds. Conversations with E. Scheidegger on the STU model were very helpful. The authors thank G. Oberdieck for pointing out the connection to the DT crepant resolution conjecture and R. Thomas for discussions on spherical twists and wall-crossing. The authors thank the referee for a very detailed reading and for many corrections and suggestions of improvements in the exposition. The first author thanks the IHES for hospitality during the final stage of this work.
Funding
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).
Additional details
Related works
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- Discussion Paper: arXiv:2108.06751 (arXiv)
Funding
- European Research Council
- ERC-2017-AdG-786580-MACI
- European Research Council
- 786580