QUASITRIANGULAR WZW MODEL

Abstract

A dynamical system is canonically associated to every Drinfeld double of any affine Kac–Moody group. In particular, the choice of the affine Lu–Weinstein double gives a smooth one-parameter deformation of the standard WZW model. The deformed 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 theory is characterized by several ordinary and dynamical r-matrices with spectral parameter. They describe the q-deformed current algebras, appear in the definition of q-primary fields and characterize the quasitriangular exchange (braiding) relations. 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 remarkable connection between the classical q-deformed WZW model and the quantum standard WZW theory on elliptic curves.