Quasitriangular WZW model

TL;DR

This paper introduces a q-deformed WZW model derived from affine Kac-Moody groups, featuring a classical integrable structure with Poisson-Lie symmetry and connections to elliptic quantum groups and dynamical r-matrices.

Contribution

It constructs a new quasitriangular WZW model using Drinfeld doubles, revealing its classical integrability, Poisson-Lie symmetry, and links to elliptic dynamical r-matrices and quantum conformal field theory.

Findings

The deformed model admits chiral decomposition.

Its symplectic structure involves elliptic dynamical r-matrices.

Connections to quantum WZW theory on elliptic curves are established.

Abstract

A dynamical system is canonically associated to every Drinfeld double of any affine Kac-Moody group. The choice of the affine Lu-Weinstein-Soibelman double gives a smooth one-parameter deformation of the standard WZW model. In particular, the worldsheet and the target of the classical version of the deformed theory are the ordinary smooth manifolds. The quasitriangular WZW model is exactly solvable and it admits the chiral decomposition.Its classical action is not invariant with respect to the left and right action of the loop group, however it satisfies the weaker condition of the Poisson-Lie symmetry. The structure of the deformed WZW model is characterized by several ordinary and dynamical r-matrices with spectral parameter. They describe the q-deformed current algebras, they enter the definition of q-primary fields and they characterize the quasitriangular exchange (braiding) relations. Remarkably, the symplectic structure of the deformed chiral WZW theory is cocharacterized by the same elliptic dynamical r-matrix that appears in the Bernard generalization of the Knizhnik-Zamolodchikov equation, with q entering the modular parameter of the Jacobi theta functions. This reveals a tantalizing connection between the classical q-deformed WZW model and the quantum standard WZW theory on elliptic curves and opens the way for the systematic use of the dynamical Hopf algebroids in the rational q-conformal field theory.