Asymptotic form of Gopakumar–Vafa invariants from instanton counting

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 An 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.  2004 Elsevier B.V. All rights reserved.