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.

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

This paper begins investigation of the concept of ``generalized $\tau$-function'', defined as a generating function of all the matrix elements of a group element $g \in G$ in a given highest-weight representation of a universal enveloping algebra ${\cal G}$. In the generic situation, the time-variables correspond to the elements of maximal nilpotent subalgebras rather than Cartanian elements. Moreover, in the case of quantum groups such $\tau$-``functions'' are not $c$-numbers but take their values in non-commutative algebras (of functions on the quantum group $G$). Despite all these differences from the particular case of conventional $\tau$-functions of integrable (KP and Toda lattice) hierarchies (which arise when $G$ is a Kac-Moody (1-loop) algebra of level $k=1$), these generic $\tau$-functions also satisfy bilinear Hirota-like equations, which can be deduced from manipulations with intertwining operators. The most important applications of the formalism should be to $k>1$ Kac-Moody and multi-loop algebras, but this paper contains only illustrative calculations for the simplest case of ordinary (0-loop) algebra $SL(2)$ and its quantum counterpart $SL_q(2)$, as well as for the system of fundamental representations of $SL(n)$.

Generalized Hirota Equations and Representation Theory. I. The case of $SL(2)$ and $SL_q(2)$"

This paper begins investigation of the concept of a “generalized τ function,” defined as a generating function of all the matrix elements of a group element g∈G in a given highest weight representation of a universal enveloping algebra . In the generic situation, the time variables correspond to the elements of maximal nilpotent subalgebras rather than Cartanian elements. Moreover, in the case of quantum groups such τ “functions” are not c numbers but take their values in noncommutative algebras (of functions on the quantum group G). Despite all these differences from the particular case of conventional τ functions of integrable (KP and Toda lattice) hierarchies [which arise when G is a Kac-Moody (one-loop) algebra of level k=1], these generic τ functions also satisfy bilinear Hirota-like equations, which can be deduced from manipulations with intertwining operators. The most important applications of the formalism should be to k>1 Kac-Moody and multiloop algebras, but this paper contains only illustrative calculations for the simplest case of ordinary (zero-loop) algebra SL(2) and its quantum counterpart SLq(2), as well as for the system of fundamental representations of SL(n).

GENERALIZED HIROTA EQUATIONS AND REPRESENTATION THEORY I: THE CASE OF SL(2) AND SLq(2)

In this series of papers we represent the "Whittaker" wave functional of the (d + 1)-dimensional Liouville model as a correlator in (d + 0)-dimensional theory of the sine–Gordon type (for d = 0 and 1). The asymptotics of this wave function is characterized by the Harish-Chandra function, which is shown to be a product of simple Γ function factors over all positive roots of the corresponding algebras (finite-dimensional for d = 0 and affine for d = 1). This is in nice correspondence with the recent results on two- and three-point correlators in the 1+1 Liouville model, where emergence of peculiar double periodicity is observed. The Whittaker wave functions of (d + 1)-dimensional nonaffine ("conformal") Toda type models are given by simple averages in the (d + 0)-dimensional theories of the affine Toda type. This phenomenon is in obvious parallel with representation of the free field wave functional, which was originally a Gaussian integral over the interior of a (d + 1)-dimensional disk with given boundary conditions, as a (nonlocal) quadratic integral over the d-dimensional boundary itself. In this paper we concentrate on the finite-dimensional case. The results for finite-dimensional "Iwasawa" Whittaker functions are known, and we present a survey. We also construct new "Gauss" Whittaker functions.

Liouville Type Models in the Group Theory Framework:. I. Finite-Dimensional Algebras

The Hopf algebra of Feynman diagrams, analyzed by A. Connes and D. Kreimer, is considered from the perspective of the theory of effective actions and generalized τ-functions, which describes the action of diffeomorphism and shift groups in the moduli space of coupling constants. These considerations provide additional evidence of the hidden group (integrable) structure behind the standard formalism of quantum field theory.

A. Gerasimov

Papers

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

Generalized Hirota Equations and Representation Theory. I. The case of $SL(2)$ and $SL_q(2)$"

GENERALIZED HIROTA EQUATIONS AND REPRESENTATION THEORY I: THE CASE OF SL(2) AND SLq(2)

Liouville Type Models in the Group Theory Framework:. I. Finite-Dimensional Algebras

Bogolubov's Recursion and Integrability of Effective Actions