Nekrasov functions and exact Bohr-Sommerfeld integrals

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 \(\mathcal{F} (a, \epsilon_{1}) \) with the other epsilon parameter vanishing, ?₂ = 0, and ?₁ playing the role of the Planck constant in the sine-Gordon Shrödinger equation, ? = ?₁. This seems to be in accordance with the recent claim in 1] 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.