WZW models of general simple groups

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.