Degree two Gopakumar-Vafa invariants of local curves

TL;DR

This paper studies degree two Gopakumar-Vafa invariants of local curves, providing a computation mechanism under generic conditions, confirming the GV/GW correspondence for genus two, and offering evidence for higher genus cases.

Contribution

It introduces a method to compute degree two GV invariants of local curves and verifies the GV/GW correspondence for genus two and some higher genera.

Findings

Computed all degree two GV invariants for genus two curves.

Established GV/GW correspondence in the genus two case.

Provided evidence supporting the GV/GW conjecture for higher genus curves.

Abstract

We investigate the Gopakumar-Vafa (GV) theory of local curves, namely, the total spaces of rank two vector bundles with canonical determinant on smooth projective curves. Under a certain genericity condition on the rank two bundles, we propose a general mechanism to compute the degree two GV invariants of local curves. In particular, we determine all the degree two GV invariants when the base curve has genus two. Combined with previous work by Bryan and Pandharipande, we obtain the GV/GW correspondence in this case. When the base curve has genus greater than two, we calculate GV invariants for some extremal genera, providing evidence for the GV/GW conjecture for curves of higher genus.