Zero modes' fusion ring and braid group representations for the extended chiral su(2) WZNW model

TL;DR

This paper explores the algebraic structure of zero modes in the extended chiral su(2) WZNW model, revealing connections to quantum groups, fusion rings, and braid group representations, with implications for conformal field theory.

Contribution

It demonstrates how the zero modes' Fock space realizes the Grothendieck fusion ring and relates fusion properties to braid group representations in the extended model.

Findings

Realization of the Grothendieck fusion ring in zero modes' Fock space

Derivation of the characteristic equation of the Casimir invariant

Connection between fusion ring properties and braiding of primary fields

Abstract

The zero modes' Fock space for the extended chiral $su(2)$ WZNW model gives room to a realization of the Grothendieck fusion ring of representations of the restricted $U_q sl(2)$ quantum universal enveloping algebra (QUEA) at an even ($2h$-th) root of unity, and of its extension by the Lusztig operators. It is shown that expressing the Drinfeld images of canonical characters in terms of Chebyshev polynomials of the Casimir invariant $C$ allows a streamlined derivation of the characteristic equation of $C$ from the defining relations of the restricted QUEA. The properties of the fusion ring of the Lusztig's extension of the QUEA in the zero modes' Fock space are related to the braiding properties of correlation functions of primary fields of the extended $su(2)_{h-2}$ current algebra model.