WZW fusion rings in the limit of infinite level

TL;DR

This paper constructs a mathematically rigorous 'classical limit' of WZW fusion rings by taking a projective limit over finite levels, revealing new structural insights and an analogue of the Verlinde formula.

Contribution

It introduces a novel projective system approach to define the infinite-level limit of WZW fusion rings, providing a new perspective on their classical limit.

Findings

Defined a projective system of WZW fusion rings based on divisibility of the level.

Constructed the projective limit as a well-defined classical limit.

Derived an analogue of the Verlinde formula for the limit.

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 role 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.