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Stable pairs and BPS invariants

Author(s): R. Pandharipande; R. P. Thomas
Journal: J. Amer. Math. Soc. 23 (2010), 267-297.
MSC (2000): Primary 14N35
Posted: October 1, 2009
Previous version: Original version posted September 24, 2009
Corrected version: Current version corrects publisher's error in rendering authors' coding \curly H\!om and removes stray coding in Contents
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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
Email: rahulp@math.princeton.edu

R. P. Thomas
Affiliation: Department of Mathematics, Imperial College, London, England
Email: rpwt@imperial.ac.uk

DOI: 10.1090/S0894-0347-09-00646-8
PII: S 0894-0347(09)00646-8
Received by editor(s): October 14, 2008
Posted: 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 of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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