On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit

TL;DR

This paper explores the Nekrasov-Shatashvili limit of N=2 supersymmetric gauge theories and topological strings, deriving differential equations that determine higher genus amplitudes and connecting them to holomorphic anomaly equations.

Contribution

It introduces differential equations from Seiberg-Witten and mirror geometries that determine topological amplitudes in the Nekrasov-Shatashvili limit, linking them to holomorphic anomaly equations.

Findings

Derived differential equations for topological amplitudes.

Showed consistency with previously known higher genus formulae.

Connected differential equations to holomorphic anomaly equations.

Abstract

We study the Nekrasov-Shatashvili limit of the N=2 supersymmetric gauge theory and topological string theory on certain local toric Calabi-Yau manifolds. In this limit one of the two deformation parameters \epsilon_{1,2} of the Omega background is set to zero and we study the perturbative expansion of the topological amplitudes around the remaining parameter. We derive differential equations from Seiberg-Witten curves and mirror geometries, which determine the higher genus topological amplitudes up to a constant. We show that the higher genus formulae previously obtained from holomorphic anomaly equations and boundary conditions satisfy these differential equations. We also provide a derivation of the holomorphic anomaly equations in the Nekrasov-Shatashvili limit from these differential equations.