Lattice Wess-Zumino-Witten model and quantum groups

Abstract

Quantum groups play a role of symmetries of integrable theories in two dimensions. They may be detected on the classical level as Poisson-Lie symmetries of the corresponding phase spaces. We discuss specifically the Wess-Zumino-Witten conformally invariant quantum field model combining two chiral parts which describe the left-and right-moving degrees of freedom. On one hand side, the quantum group plays the role of the symmetry of the chiral components of the theory. On the other hand, the model admits a lattice regularization (in the Minkowski space) in which the current algebra symmetry of the theory also becomes quantum, providing the simplest example of a quantum group symmetry coupling space-time and internal degrees of freedom. We develop a free field approach to the representation theory of the lattice sl(2)-based current algebra and show how to use it to rigorously construct an exact solution of the quantum SL(2) WZW model on lattice.

Chiral extensions of the WZNW phase space, Poisson–Lie symmetries and groupoids

“…The problem now arises from the well-known fact that no globally defined smooth 2-form exists on G that would satisfy this condition for all M C ∈ G. There are two rather different wayouts from the above difficulty [14]. The first is to restrict the possible domain of the monodromy matrix M C to some open submanifold in G on which an appropriate 2-form ρ may be found.…”

“…(3.34) has a smooth, unique solution), and we shall describe the general features of the PBs onM chir associated with this symplectic form. We will then consider examples, in particular the choices of ρ introduced in [14] that lead to Poisson-Lie symmetry on the full M chir .…”

“…The problem of the chiral separation can be described as follows [14]. First, recall that the chiral currents J C (C = L, R) generate two commuting copies of the nontwisted affine KacMoody (KM) algebra of G and the WZNW field (2.8) is a KM primary field under the Poisson bracket defined by the symplectic form on M sol .…”

“…For chiral fields with generic monodromy, it has been argued in [11,14] that the possible exchange r-matrices should correspond to certain local differential 2-forms ρ on open domainš G ⊂ G, whose exterior derivative is the 3-form that occurs in the WZNW action. The precise connection between ρ andr has not been elaborated, and in most papers dealing with generic monodromy actually only those very special cases were considered for whichr is a monodromy independent constant.…”

“…• First, we establish the relationship between the 2-form ρ introduced in [11,14] and the corresponding exchange r-matrix in the general case. The result is given by eq.…”