Affine su(3) and su(4) fusion multiplicities as polytope volumes

TL;DR

This paper introduces novel polytope-volume formulas for affine su(3) and su(4) fusion multiplicities, providing explicit volume calculations and sum formulas, and discusses threshold levels using a refined Gepner-Witten rule.

Contribution

It presents the first polytope-volume formulas for higher-rank fusion multiplicities in affine Lie algebras, expanding the mathematical tools available for these calculations.

Findings

Explicit volume formulas for su(3) and su(4) fusion multiplicities

Derived sum formulas for these multiplicities

Established an upper bound on threshold levels using a refined Gepner-Witten rule

Abstract

Affine su(3) and su(4) fusion multiplicities are characterised as discretised volumes of certain convex polytopes. The volumes are measured explicitly, resulting in multiple sum formulas. These are the first polytope-volume formulas for higher-rank fusion multiplicities. The associated threshold levels are also discussed. For any simple Lie algebra we derive an upper bound on the threshold levels using a refined version of the Gepner-Witten depth rule.