Nekrasov functions from exact Bohr-Sommerfeld periods: the case of SU(N)

We suggested in 2009 that the Nekrasov function with one non-vanishing deformation parameter epsilon is obtained by the standard Seiberg-Witten (SW) contour-integral construction. The only difference is that the SW differential pdx is substituted by its quantized version for the corresponding integrable system, and contour integrals become exact monodromies of the wavefunction. This provides an explicit formulation of the earlier guess by Nekrasov and Shatashvili in 2009. In this paper, we successfully check this suggestion in the first order in epsilon(2) and the first order in instanton expansion for the SU(N) model, where the consistency of the so-deformed SW equations is already non-trivial.