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Published December 2013 | Published
Journal Article Open

Integrality of relative BPS state counts of toric del Pezzo surfaces

Abstract

Relative Bogomolny-Prasad-Sommerfield (BPS) state counts for log Calabi-Yau surface pairs were introduced by Gross-Pandharipande-Siebert in [4] and conjectured by the authors to be integers. For toric del Pezzo surfaces, we provide an arithmetic proof of this conjecture, by relating these invariants to the local BPS state counts of the surfaces. The latter were shown to be integers by Peng in [15]; and more generally for toric Calabi-Yau three-folds by Konishi in [8].

Additional Information

© 2013 International Press of Boston, Inc. Received January 10, 2014. M.v.G would like to thank T. Graber for introducing him to the subject of Gromov–Witten theory and for many enlightening discussions on curve counting that formed the basis for the present paper. M.v.G would like to extend special thanks to Y. Ruan, who has provided valuable guidance on the aspects of this paper relating to mirror symmetry. The authors would like to thank N. Yui, R. Abouaf and M. Florence for many helpful comments that improved the quality and readability of the paper. The authors would like to extend special thanks to the referees, whose reports greatly improved the quality of the present paper. Part of the research reported here was performed while the authors were students at the California Institute of Technology. This paper was completed while M.v.G was in residence at the Fields Institute for the thematic program: Calabi–Yau Varieties: Arithmetic, Geometry and Physics, July to December 2013. M.v.G would like to thank the Fields Institute for its hospitality and generous support. M.v.G. was supported by a Fields Postdoctoral Fellowship for the Fields major thematic program on Calabi–Yau Varieties: Arithmetic, Geometry and Physics from July to December 2013.

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Created:
August 19, 2023
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October 17, 2023