Stable pairs and BPS invariants

TL;DR

This work defines Gopakumar–Vafa–style BPS invariants for irreducible curve classes on Calabi–Yau 3-folds using stable pairs and Behrend’s weighted Euler characteristic, establishing that the stable-pairs generating series $Z_{\beta}(q)$ is rational and admits a deformation-invariant BPS expansion $Z_{\beta}(q)=\sum_{r=0}^g n_{r,\beta}\,q^{1-r}(1+q)^{2r-2}$. It develops a local theory of BPS invariants via fixed curves $C$, Chow decompositions, and germ-level singularity data, proving that contributions from a curve $C$ can be isolated into local integers $n_{r,C}$ with support only for $r$ between the geometric genus $g(\bar C)$ and the arithmetic genus $g(C)$. The paper provides complete results for nonsingular and nodal curves, extends the analysis to singular and reducible cases through Serre duality and duality on Ext groups, and connects to Yau–Zaslow and Katz–Klemm–Vafa while offering a framework potentially extensible to Fano cases. Overall, the approach yields deformation-invariant BPS counts tied to stable-pair geometry and provides new proofs of classical formulas in the K3 context via a sheaf-theoretic lens.

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.

Theorems & Definitions (38)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1.1
• Lemma 1.3
• proof
• Theorem 4
• proof
• Proposition 2.1
• Proposition 2.2