Stable pairs and BPS invariants

Pandharipande, R.; Thomas, R.

doi:10.1090/S0894-0347-09-00646-8

Stable pairs and BPS invariants
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by R. Pandharipande and R. P. Thomas;

J. Amer. Math. Soc. 23 (2010), 267-297

DOI: https://doi.org/10.1090/S0894-0347-09-00646-8

Published electronically: October 1, 2009

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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

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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