Stable pairs, flat connections and Gopakumar-Vafa invariants

TL;DR

This paper links Gopakumar-Vafa invariants to the asymptotic behavior of flat sections derived from Donaldson-Thomas invariants, revealing their role in Gromov-Witten theory through monodromy and flat connections.

Contribution

It demonstrates that Gopakumar-Vafa contributions emerge from the asymptotics of flat sections associated with Donaldson-Thomas invariants, connecting enumerative geometry and flat connection monodromy.

Findings

Gopakumar-Vafa invariants appear in flat section asymptotics.

The GW/DT change of variable is naturally derived from Fourier-Laplace integrals.

Monodromy of flat connections encodes Gromov-Witten invariants.

Abstract

Using the interpretation of certain generalised Donaldson-Thomas invariants, including stable pairs curve counts, as the monodromy of a flat connection on a formal principal bundle, we show that the conjectural Gopakumar-Vafa contributions of all genera to the Gromov-Witten partition function appear in the asymptotics of the corresponding flat sections. The Fourier-Laplace integrals used to produce flat sections lead naturally to the GW/DT change of variable -q = e^{i u}.