The sl(n)-WZNW Fusion Ring: a combinatorial construction and a realisation as quotient of quantum cohomology

TL;DR

This paper provides a combinatorial construction of the sl(n)-WZNW fusion ring, establishing an isomorphism with a quotient of quantum cohomology, and introduces new recursion formulas for structure constants and invariants.

Contribution

It introduces a novel combinatorial method to construct the fusion ring and links it to quantum cohomology, including new recursion formulas for key invariants.

Findings

Established an isomorphism between the fusion ring and a quotient of quantum cohomology.

Developed new recursion formulas for fusion coefficients and Gromov-Witten invariants.

Connected combinatorial descriptions of fusion rings and quantum cohomology rings.

Abstract

A simple, combinatorial construction of the sl(n)-WZNW fusion ring, also known as Verlinde algebra, is given. As a byproduct of the construction one obtains an isomorphism between the fusion ring and a particular quotient of the small quantum cohomology ring of the Grassmannian Gr(k,k+n). We explain how our approach naturally fits into known combinatorial descriptions of the quantum cohomology ring, by establishing what one could call a `Boson-Fermion-correspondence' between the two rings. We also present new recursion formulae for the structure constants of both rings, the fusion coefficients and the Gromov-Witten invariants.