## Stable pairs and BPS invariants

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- by R. Pandharipande and R. P. Thomas PDF
- J. Amer. Math. Soc.
**23**(2010), 267-297 Request permission

## Abstract:

We define the BPS invariants of Gopakumar-Vafa in the case of irreducible curve classes on Calabi-Yau 3-folds. The main tools are the theory of stable pairs in the derived category and Behrend’s constructible function approach to the virtual class. For irreducible curve classes, we prove that the stable pairs’ generating function satisfies the strong BPS rationality conjectures.

We define the contribution of each curve $C$ to the BPS invariants and show that the contributions lie between the geometric genus and arithmetic genus of $C$. Complete formulae are derived for nonsingular and nodal curves.

A discussion of primitive classes on $K3$ surfaces from the point of view of stable pairs is given in the Appendix via calculations of Kawai-Yoshioka. A proof of the Yau-Zaslow formula for rational curve counts is obtained. A connection is made to the Katz-Klemm-Vafa formula for BPS counts in all genera.

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## Additional Information

**R. Pandharipande**- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 357813
- Email: rahulp@math.princeton.edu
**R. P. Thomas**- Affiliation: Department of Mathematics, Imperial College, London, England
- MR Author ID: 636321
- Email: rpwt@imperial.ac.uk
- Received by editor(s): October 14, 2008
- Published electronically: October 1, 2009
- Additional Notes: The first author was partially supported by NSF grant DMS-0500187 and a Packard foundation fellowship

The second author was partially supported by a Royal Society University Research Fellowship. He also thanks the Leverhulme Trust and Columbia University for a visit to New York in the spring of 2007 when the project was started. Many of the results presented here were found during a visit to Lisbon in the summer of 2007. - © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**23**(2010), 267-297 - MSC (2000): Primary 14N35
- DOI: https://doi.org/10.1090/S0894-0347-09-00646-8
- MathSciNet review: 2552254