# Gauged non-compact WZNW models, revisited

## TL;DR

This paper investigates the unitarity conditions of non-compact gauged WZNW models, highlighting the complexity and case-by-case nature of the problem, and providing specific results for certain Lie algebras.

## Contribution

It offers detailed analysis of unitarity conditions, character identities, and branching functions for non-compact gauged WZNW models, especially for su(n,1) and related algebras.

## Key findings

- Necessary conditions for unitarity are complex and case-dependent.
- Explicit character identities and branching functions are derived.
- Nearly complete conditions are provided for su(n,1) and parts of su(p,q).

## Abstract

The purpose of the present paper is to investigate the necessary conditions for unitarity of the spectrum of non-compact gauged WZNW models to some depth. In particular, we would like to investigate the necessity of integer weights and level. We will learn that the problem is very complex and we have not found any simple and general way to formulate the necessary conditions. Instead one must resort to studying the problem almost case by case. The only nearly complete conditions that we will find, is for the case g = su(n,1). Furthermore, the horizontal part of the case g = su(p,q) is nearly completed as well. In other cases, we will find conditions associated with certain subalgebras and nodes in the Dynkin diagram close to the one corresponding to the non-compact root. In these examples we can give conditions for the horizontal part of the algebra. As a by-product of our investigation we will prove some nice formulae of character identities and explicit branching functions for these representations.