Beyond CFT: Deformed Virasoro and Elliptic Algebras

Abstract

In this lecture we discuss 'beyond CFT' from symmetry point of view. After reviewing the Virasoro algebra, we introduce deformed Virasoro algebras and elliptic algebras. These algebras appear in solvable lattice models and we study them by free field approach.

On matrix models and their q-deformations

“…Here, the notion of "quantum" will be related to the introduction of a special class of q-deformations with respect to an un-deformed model. In the classical case, we will be considering the β-deformed Hermitean matrix model, in other words a one parameter deformation of the standard Hermitean matrix model [6,7]. On the quantum side, we will explore the 3d N = 2 supersymmetric theory with U(N ) gauge group on both D 2 × q S 1 [8,9] and also S 3 b [10,11] with one adjoint chiral and an arbitrary number of anti-chiral fundamental multiplets.…”

“…It is common at this point to introduce a 1-parameter deformation of the model by generalizing the usual Vandermonde term as follows (for details we refer to [7])…”

Holomorphic blocks and the 5d AGT correspondence

“…Deformed Virasoro and W N algebras were introduced independently in the mid 90s by various groups [76,77], [78,79,80], [81,82], see [83] for a review. The Vir q,t algebra is a deformation of the Virasoro algebra, it depends on two complex parameters q, t and is generated by an infinite set of generators T n with n ∈ Z, satisfying the commutation relations…”

Free Oscillator Realization of the Laguerre Polynomial

“…are studied by free field realization, see e.g. [1]. Roughly speaking, a free field is an infinite collection of free oscillators, which are the creation/annihilation operators of the harmonic oscillator.…”