Non-Abelian Bosonization and Higher Spin Symmetries

TL;DR

This paper explores the complex higher spin symmetries in non-abelian bosonization, revealing how classical and quantum algebras differ and require extended current spaces for closure.

Contribution

It uncovers the structure of higher spin symmetries in non-abelian bosonization, highlighting the necessity of extended current spaces and the differences between classical and quantum algebras.

Findings

Classical symmetry and current algebras are linear Gel'fand-Dickey types.

Quantum current algebra in WZNW models contains nonlinear terms.

Closure of the algebra requires an extended current space including higher isotopic spin currents.

Abstract

The higher spin properties of the non-abelian bosonization in the classical theory are investigated. Both the symmetry transformation algebra and the classical current algebra for the non-abelian free fermionic model are linear Gel'fand-Dickey type algebras. However, for the corresponding WZNW model these algebras are different. There exist symmetry transformations which algebra remains the linear Gel'fand-Dickey algebra while in the corresponding current algebra nonlinear terms arised. Moreover, this algebra is closed (in Casimir form) only in an extended current space in which nonlinear currents are included. In the affine sector, it is necessary to be included higher isotopic spin current too. As result we have a triple extended algebra.