On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit

TL;DR

The paper develops a unified NS-limit framework for both N=2 gauge theories and topological strings on local Calabi–Yau manifolds by deriving differential equations for deformed periods from Seiberg–Witten/mirror data. These equations fix higher-genus NS amplitudes up to constants and are shown to be compatible with holomorphic anomaly equations and gap conditions, with explicit derivations in pure and matter-coupled SU(2) theories and adjoint/massless cases. A core result is that holomorphic anomaly in the NS limit can be obtained from the deformed-dual-period equations and modular-derivative structures, and that the approach extends to local CY geometries such as local P^2 and local P^1×P^1, yielding genus-one and genus-two results consistent with BCOV-type formalisms. Overall, the work provides a detailed, cross-validated bridge between saddle-point/Nekrasov methods, Seiberg–Witten theory, and refined topological string theory in the NS limit, with explicit generating functions and modular structures.

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 ε_{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.