WZW models of general simple groups

TL;DR

This paper explores different quantizations of WZW models for simple groups, analyzing their monodromy and modular invariance properties, and providing explicit formulas and examples for these theories.

Contribution

It introduces a classification of quantizations of WZW models based on the fundamental group of the group, and derives formulas for their monodromy properties and invariance conditions.

Findings

Multiple quantizations exist for WZW models, parametrized by homomorphisms involving the fundamental group.

Number of monodromy invariant theories depends on the fundamental group's structure.

Explicit examples demonstrate the diversity of quantizations and their invariance properties.

Abstract

It is shown that a WZW model corresponding to a general simple group possesses in general different quantisations which are parametrised by $Hom(\pi_1(G),Hom(\pi_1(G),U(1)))$. The quantum theories are generically neither monodromy nor modular invariant, but all the modular invariant theories of Felder et.al. are contained among them. A formula for the transformation of the Sugawara expression for $L_0$ under conjugation with respect to non-contractible loops in $LG$ is derived. This formula is then used to analyse the monodromy properties of the various quantisations. It turns out that for $\pi_1(G)\cong \Zop_N$, with $N$ even, there are $2$ monodromy invariant theories, one of which is modular invariant, and for $\pi_1(G)\cong \Zop_2\times\Zop_2$ there are $8$ monodromy invariant theories, two of which are modular invariant. A few specific examples are worked out in detail to illustrate the results.