Asymptotic Form of Gopakumar-Vafa Invariants from Instanton Counting

TL;DR

This paper investigates the asymptotic behavior of Gopakumar-Vafa invariants for certain Calabi-Yau threefolds, linking it to instanton amplitudes in supersymmetric gauge theories via inverse Laplace transforms.

Contribution

It establishes a novel connection between the asymptotics of Gopakumar-Vafa invariants and instanton counting in N=2 gauge theories, providing a new computational approach.

Findings

Derived the asymptotic form as an inverse Laplace transform of instanton amplitudes.

Linked geometric invariants to gauge theory partition functions.

Provided explicit formulas for invariants in specific Calabi-Yau geometries.

Abstract

We study the asymptotic form of the Gopakumar-Vafa invariants at all genera for Calabi-Yau toric threefolds which have the structure of fibration of the A_n singularity over P^1. We claim that the asymptotic form is the inverse Laplace transform of the corresponding instanton amplitude in the prepotential of N=2 SU(n+1) gauge theory coupled to external graviphoton fields, which is given by the logarithm of the Nekrasov's partition function.