Deforming SW curve

TL;DR

Addresses the $\epsilon_2\to0$ limit of the Nekrasov partition function in ${\cal N}=2$ SYM in an $\\Omega$-background and derives a Bethe-Ansatz–type system selecting the dominant array of Young tableaux. Constructs an entire function $Y(z)$ whose zeros encode the column lengths and derives a Baxter-like functional equation that generalizes the Seiberg-Witten curve to finite $\epsilon_1$. Shows that the deformed curve and the prepotential $W(a,m,\epsilon_1)$ are connected via contour integrals of $Y$, and that chiral correlators $\langle\mathrm{tr}\,\phi^J\rangle$ can be expressed in terms of $Y$ and its zeros; provides an explicit $U(1)$ solution in terms of $_0F_1$ hypergeometric functions that reproduces the standard SW results as $\epsilon_1\to0$. The work links Nekrasov's $\{\epsilon_1,\epsilon_2\}$-deformed partition function to integrable-system structures and suggests possible TBA-like reformulations and connections to AGT-type semiclassical limits.

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 $ε_2\rightarrow 0$ limit is derived. It is shown that the prepotential with generic $ε_1$ is directly related to the (rescaled by $ε_1$) number of total boxes of these Young tableau. Moreover, all the expectation values of the chiral fields $\langle \tr φ^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.