WZW Fusion Rings in the Limit of Infinite Level

Abstract:

We show that the WZW fusion rings at finite levels form a projective system with respect to the partial ordering provided by divisibility of the height, i.e. the level shifted by a constant. From this projective system we obtain WZW fusion rings in the limit of infinite level. This projective limit constitutes a mathematically well-defined prescription for the “classical limit” of WZW theories which replaces the naive idea of “sending the level to infinity.” The projective limit can be endowed with a natural topology, which plays an important rôle for studying its structure. The representation theory of the limit can be worked out by considering the associated fusion algebra; this way we obtain in particular an analogue of the Verlinde formula.