Nekrasov Functions from Exact BS Periods: the Case of SU(N)

TL;DR

The paper investigates how Nekrasov functions with a single deformation parameter $\epsilon_1$ emerge from a quantized Seiberg-Witten (SW) framework for SU($N$). By applying a WKB expansion to the deformed differential $dS$ and introducing an operator $\hat{\cal O}$ that maps classical BS periods to quantum-corrected ones, the authors compute deformed A- and B-periods in terms of spectral roots $\lambda_i$ and verify that the deformed prepotential $F(\vec{a}|\epsilon_1)$ matches Nekrasov's $F(\vec{a}|\epsilon_1)$ (with $\epsilon_2=0$) to first nontrivial order in $\Lambda^{2N}$ and $\epsilon_1^2$. The key result is the consistency relation $\Pi_B(\hat{\cal O}[\Pi_A^{(0)}(\lambda)])=\hat{\cal O}[\Pi_B^{(0)}(\Pi_A^{(0)}(\lambda))]$, supported by explicit expressions for the periods and perturbative/instanton contributions. These findings bolster the conjecture that Nekrasov prepotentials arise from exact BS periods of a quantized integrable system and point to broader implications for AGT and related dualities in gauge/string theory.

Abstract

In arXiv:0910.5670 we suggested that the Nekrasov function with one non-vanishing deformation parameter εis obtained by the standard Seiberg-Witten contour-integral construction. The only difference is that the Seiberg-Witten differential pdx is substituted by its quantized version for the corresponding integrable system, and contour integrals become exact monodromies of the wave function. This provides an explicit formulation of the earlier guess in arXiv:0908.4052. In this paper we successfully check this suggestion in the first order in ε^2 and the first order in instanton expansion for the SU(N) model, where non-trivial is already consistency of the so deformed Seiberg-Witten equations.