Nekrasov Functions and Exact Bohr-Sommerfeld Integrals

TL;DR

This work demonstrates that the one-parameter Nekrasov prepotential ${\cal F}(a|\epsilon_1)$ with ${\epsilon_2=0}$ can be described within the Seiberg-Witten framework by replacing the classical SW differential with a quantum deformed differential $Pdx$, where the exact Bohr-Sommerfeld periods (monodromies) of the 1d sine-Gordon system encode ${\cal F}(a|\epsilon_1)$. By deriving the exact BS periods through WKB-type expansions and the Picard-Fuchs equation, the authors show that the leading SW prepotential and its first quantum corrections reproduce the known ${\epsilon_1}$-expansion of the Nekrasov prepotential, with ${\hbar=\epsilon_1}$ and ${\gamma=\Lambda^2}$. The results establish a concrete bridge between Liouville theory (via AGT), sine-Gordon integrable dynamics, and Nekrasov functions, and lay the groundwork for a systematic double-parameter quantization involving both ${\epsilon_1}$ and ${\epsilon_2}$. The work also points to extensions to higher rank groups and deeper algebraic structures (double-loop/p-adic frameworks) as promising avenues for future research.

Abstract

In the case of SU(2), associated by the AGT relation to the 2d Liouville theory, the Seiberg-Witten prepotential is constructed from the Bohr-Sommerfeld periods of 1d sine-Gordon model. If the same construction is literally applied to monodromies of exact wave functions, the prepotential turns into the one-parametric Nekrasov prepotential F(a,ε_1) with the other epsilon parameter vanishing, ε_2=0, and ε_1 playing the role of the Planck constant in the sine-Gordon Shroedinger equation, \hbar=ε_1. This seems to be in accordance with the recent claim in arXiv:0908.4052 and poses a problem of describing the full Nekrasov function as a seemingly straightforward double-parametric quantization of sine-Gordon model. This also provides a new link between the Liouville and sine-Gordon theories.