Higher-genus su(N) fusion multiplicities as polytope volumes

TL;DR

This paper presents a novel geometric approach to compute higher-genus su(N) fusion multiplicities as volumes of specific polytopes, extending previous methods with new gluing diagrams, applicable to all higher-point and higher-genus cases.

Contribution

It introduces loop-gluing diagrams to characterize higher-genus fusions as polytope volumes, expanding the Berenstein-Zelevinsky framework.

Findings

Higher-genus su(N) fusion multiplicities can be computed as discretized polytope volumes.

Explicit analyses provided for su(3) and su(4) fusions.

Genus-2 0-point su(3) fusion multiplicity equals a simple binomial coefficient.

Abstract

We show how higher-genus su(N) fusion multiplicities may be computed as the discretized volumes of certain polytopes. The method is illustrated by explicit analyses of some su(3) and su(4) fusions, but applies to all higher-point and higher-genus su(N) fusions. It is based on an extension of the realm of Berenstein-Zelevinsky triangles by including so-called gluing and loop-gluing diagrams. The identification of the loop-gluing diagrams is our main new result, since they enable us to characterize higher-genus fusions in terms of polytopes. Also, the genus-2 0-point su(3) fusion multiplicity is found to be a simple binomial coefficient in the affine level.