Deforming SW curve

TL;DR

This paper derives a Bethe-Ansatz type system for Young tableaux that captures the Nekrasov partition function in a specific limit, revealing a deformed Seiberg-Witten curve and connections to integrable models.

Contribution

It introduces a new Bethe-Ansatz type equation system for Young tableaux that generalizes the Seiberg-Witten curve in the presence of deformation parameters.

Findings

Derived a system of equations for Young tableaux in Nekrasov functions

Connected the prepotential to the total number of boxes in tableaux

Established a functional equation resembling Baxter's equation

Abstract

A system of Bethe-Ansatz type equations, which specify a unique array of Young tableau responsible for the leading contribution to the Nekrasov partition function in the $\epsilon_2\rightarrow 0$ limit is derived. It is shown that the prepotential with generic $\epsilon_1$ is directly related to the (rescaled by $\epsilon_1$) number of total boxes of these Young tableau. Moreover, all the expectation values of the chiral fields $\langle \tr \phi^J \rangle $ are simple symmetric functions of their column lengths. An entire function whose zeros are determined by the column lengths is introduced. It is shown that this function satisfies a functional equation, closely resembling Baxter's equation in 2d integrable models. This functional relation directly leads to a nice generalization of the equation defining Seiberg-Witten curve.