On relation between Nekrasov functions and BS periods in pure SU(N) case

TL;DR

This work provides thorough, computable checks of the duality between the Nekrasov function with $\epsilon_2=0$ and the quantized Seiberg-Witten prepotential with $\epsilon_2=\hbar$ in pure SU($N$) gauge theories. It develops a perturbative Baxter-equation approach to construct a quantum SW differential $Pdx$, representing it as a differential operator $\hat{\mathcal{O}}$ acting on the classical differential, and verifies the matching of A- and B-periods against Nekrasov data up to $o(\hbar^6)$, including $\ln(\Lambda)$-dependent terms. The Nekrasov side is organized into perturbative and instanton components, with explicit two-instanton contributions, enabling automated checks for arbitrary $N$ and higher-precision tests for $N=2,3,4$. The results substantiate the proposed duality in the pure gauge case and set the stage for extensions to matter and deeper structural understanding of the quantum-geometry operator $\hat{\mathcal{O}}$.

Abstract

We investigate the duality between the Nekrasov function and the quantized Seiberg-Witten prepotential, first guessed in [1] and further elaborated in [2] and [3]. We concentrate on providing more thorough checks than the ones presented in [3] and do not discuss the motivation and historical context of this duality. The check of the conjecture up to $o (\hbar^6, \ln (Λ))$ is done by hands for arbitrary $N$ (explicit formulas are presented). Moreover, details of the calculation that are essential for the computerization of the check are worked out. This allows us to test the conjecture up to $\hbar^6$ and up to higher powers of $Λ$ for $N = 2,3,4$. Only the case of pure SU(N) gauge theory is considered.