Constrained current algebras and $g/u(1)^d$ conformal field theories

TL;DR

This paper explores the operator quantization of WZNW theories with constrained currents, showing how the energy-momentum tensor transforms into a coset construction and establishing equivalences with known conformal field theories.

Contribution

It introduces a method for quantizing constrained affine Kac-Moody algebras and demonstrates their relation to coset conformal field theories, expanding understanding of their structure.

Findings

Quantization replaces the energy-momentum tensor with a coset construction.

The $ ext{su}(2)$ WZNW theory with a constrained $ ext{u}(1)$ current is equivalent to the $ ext{su}(2)/ ext{u}(1)$ CFT.

Operator quantization using Dirac's procedure is successfully applied to constrained current algebras.

Abstract

Operator quantization of the WZNW theory invariant with respect to an affine Kac-Moody algebra $\hat g$ with constrained $\hat {u}(1)^d$ currents is performed using Dirac's procedure. Upon quantization the initial energy-momentum tensor is replaced by the $g/u(1)^d$ coset construction. The $\hat {su}(2)$ WZNW theory with a constrained $\hat u(1)$ current is equivalent to the $su(2)/u(1)$ conformal field theory.