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, ϵ2 = 0, and ϵ1 playing the role of the Planck constant in the sine-Gordon Shrodinger equation, ℏ = ϵ1. 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.

Nekrasov functions and exact Bohr-Sommerfeld integrals

On AGT relation in the case of U(3)

Matrix models of two-dimensional gravity and Toda theory

We investigate the structure of the Fock modules overA1(1) introduced by Wakimoto. We show that irreducible highest weight modules arise as degree zero cohomology groups in a BRST-like complex of Fock modules. Chiral primary fields are constructed as BRST invariant operators acting on Fock modules. As a result, we obtain a free field representation of correlation functions of theSU(2) WZW model on the plane and on the torus. We also consider representations of fractional level arising in Polyakov's 2D quantum gravity. Finally, we give a geometrical, Borel-Weil-like interpretation of the Wakimoto construction.

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Fock representations and BRST cohomology in SL (2) current algebra

Quantum field theory for the multi-variable Alexander-Conway polynomial

The free field representation or "bosonization" rule1 for Wess-Zumino-Witten model (WZWM) with arbitrary Kac-Moody algebra and arbitrary central charge is discussed. Energy-momentum tensor, arising from Sugawara construction, is quadratic in the fields. In this way, all known formulae for conformal blocks and correlators may be easily reproduced as certain linear combinations of correlators of these free fields. Generalization to conformal blocks on arbitrary Riemann surfaces is straightforward. However, projection rules in the spirit of Ref. 2 are not specified. The special role of βγ systems is emphasized. From the mathematical point of view, the construction involved represents generators of Kac-Moody (KM) algebra in terms of generators of a Heisenberg one. If WZW Lagrangian is considered as d−1 of Kirillov form on an orbit of KM algebra,3 then the free fields of interest (i.e. generators of the Heisenberg algebra) diagonalize Kirillov form and the action. Reduction of KM algebra within the same construction should naturally lead to arbitrary coset models.

S. Shatashvili

Papers

Wess-Zumino-Witten model as a theory of free fields