Fusion multiplicities as polytope volumes: N-point and higher-genus su(2) fusion

TL;DR

This paper derives explicit polytope volume formulas for affine fusion multiplicities in Wess-Zumino-Witten theories, extending previous tensor product results to higher points and genus, and identifies conditions for non-zero multiplicities.

Contribution

It introduces the first polytope volume formulas for affine fusion multiplicities, generalizing tensor product methods to higher-point and higher-genus cases in su(2).

Findings

Polytope volume formulas for fusion multiplicities are derived.

Necessary and sufficient conditions for non-vanishing fusion multiplicities are established.

The minimum level where fusion and tensor product multiplicities coincide is determined.

Abstract

We present the first polytope volume formulas for the multiplicities of affine fusion, the fusion in Wess-Zumino-Witten conformal field theories, for example. Thus, we characterise fusion multiplicities as discretised volumes of certain convex polytopes, and write them explicitly as multiple sums measuring those volumes. We focus on su(2), but discuss higher-point (N>3) and higher-genus fusion in a general way. The method follows that of our previous work on tensor product multiplicities, and so is based on the concepts of generalised Berenstein-Zelevinsky diagrams, and virtual couplings. As a by-product, we also determine necessary and sufficient conditions for non-vanishing higher-point fusion multiplicities. In the limit of large level, these inequalities reduce to very simple non-vanishing conditions for the corresponding tensor product multiplicities. Finally, we find the minimum level at which the higher-point fusion and tensor product multiplicities coincide.