Dynamical r-matrices and the chiral WZNW phase space

TL;DR

This paper reviews the dynamical generalization of the classical Yang-Baxter equation relevant to the Poisson structures on chiral WZNW fields, connecting it to the CDYB equation and exploring solutions with various monodromy conditions.

Contribution

It clarifies the relationship between dynamical r-matrices, the CDYB equation, and Poisson structures in the chiral WZNW model, including new solutions with different monodromy constraints.

Findings

Reduction of the dynamical Yang-Baxter equation to the CDYB equation for specific Poisson brackets

Derivation of interesting dynamical r-matrices for generic monodromy

Construction of solutions via Dirac constraints on monodromy

Abstract

The dynamical generalization of the classical Yang-Baxter equation that governs the possible Poisson structures on the space of chiral WZNW fields with generic monodromy is reviewed. It is explained that for particular choices of the chiral WZNW Poisson brackets this equation reduces to the CDYB equation recently studied by Etingof--Varchenko and others. Interesting dynamical r-matrices are obtained for generic monodromy as well as by imposing Dirac constraints on the monodromy.