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

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.

💡 Research Summary

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The paper presents a completely elementary, combinatorial construction of the sl(n)‑WZNW fusion ring (also known as the Verlinde algebra) and shows that this ring is isomorphic to a specific quotient of the small quantum cohomology ring of the Grassmannian Gr(k,k+n). The authors start by fixing a level k and consider the set 𝔓ₖ⁺ of dominant integral weights of sl(n) that can be identified with Young diagrams whose first row does not exceed k. For any two such diagrams λ and μ they define a “fusion product” ⊙ₖ by first expanding the ordinary product of the corresponding Schur functions s_λ·s_μ using the Littlewood‑Richardson rule, and then applying a level‑k reduction: any term s_ν with ν₁>k is either eliminated or replaced by a term involving the quantum parameter q, which encodes the effect of wrapping around the affine Dynkin diagram. The resulting structure constants N_{λμ}^{ν}(k) are precisely the fusion coefficients of the sl(n) WZNW model at level k.

Next, the paper recalls the presentation of the (small) quantum cohomology ring QH⁎(Gr(k,k+n)). In the language of symmetric functions this ring is Λ/(I_{n,k}), where Λ is the algebra of symmetric functions, and I_{n,k}=⟨h_{n+1},…,h_{n+k-1}, h_{n+k}+(-1)^{k}q⟩ is the ideal generated by the complete symmetric functions beyond degree n together with the quantum relation involving q. The Schur function s_λ maps to the quantum Schubert class σ_λ, and the quantum product σ_λ*σ_μ is given by a quantum Pieri rule that also produces q‑terms when a box is added beyond the allowed rectangle.

The central theorem establishes a ring isomorphism Φ: 𝔓ₖ⁺ → QH⁎(Gr(k,k+n))/I_{n,k} defined by Φ(λ)=