Stable pairs and BPS invariants

TL;DR

This paper defines Gopakumar-Vafa BPS invariants for irreducible curve classes on Calabi-Yau 3-folds using stable pairs and proves their rationality, providing formulas for various curve types and connecting to known enumerative formulas.

Contribution

It introduces a new approach to defining BPS invariants via stable pairs and proves their rationality for irreducible classes, linking to established enumerative formulas.

Findings

BPS invariants satisfy strong rationality conjectures.

Complete formulas derived for nonsingular and nodal curves.

Connections established to Yau-Zaslow and Katz-Klemm-Vafa formulas.

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. We prove that for irreducible classes the stable pairs generating function satisfies the strong BPS rationality conjectures. We define the contribution of each curve to the BPS invariants. A curve $C$ only contributes to the BPS invariants in genera lying 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.