Gopakumar-Vafa BPS invariants, Hilbert schemes and quasimodular forms. I

TL;DR

This paper derives a closed formula for leading Gopakumar-Vafa BPS invariants of local Calabi-Yau geometries, linking Hilbert schemes and quasimodular forms, with initial focus on the projective plane.

Contribution

It provides a new explicit formula for BPS invariants of local Calabi-Yau geometries associated with toric Fano surfaces, connecting them to Hilbert schemes and quasimodular forms.

Findings

Closed formula for BPS invariants of local Calabi-Yau geometries

Connection established with Hilbert schemes and quasimodular forms

Initial case study on the projective plane

Abstract

We prove a closed formula for leading Gopakumar- Vafa BPS invariants of local Calabi-Yau geometries given by the canonical line bundles of toric Fano surfaces. It shares some similar features with Goettsche-Yau-Zaslow formula: Connection with Hilbert schemes, connection with quasimodular forms, and quadratic property after suitable transformation. In Part I of this paper we will present the case of projective plane, more general cases will be presented in Part II.