The Verlinde Algebra And The Cohomology Of The Grassmannian

TL;DR

The paper provides a conceptual bridge between the quantum cohomology of Grassmannians and the Verlinde algebra by showing that the low-energy limit of a supersymmetric Grassmannian sigma model reduces to a gauged WZW theory G/G, whose correlation functions reproduce Verlinde data. Central to this is the identification of non-abelian theta functions with sections of a prequantum line bundle and the construction of a kernel implementing the gauge action, connecting genus-zero correlators to fusion coefficients. A detailed abelianization analysis and a Landau-Ginzburg formulation of the quantum cohomology reveal explicit, computable relations: the quantum product on H^*(G(k,N)) matches the Verlinde algebra of U(k) at level (N−k,N), with corrections encoded by a Landau–Ginzburg potential and a determinant factor relating metrics. The work also develops a robust framework for extending these ideas to other symplectic quotients and highlights the mass-gap mechanism that underpins the long-distance/topological reduction to the G/G model, yielding a topological field theory whose invariants encode geometric and representation-theoretic data.

Abstract

The article is devoted to a quantum field theory explanation of the relationship (noticed some years ago by Gepner) between the Verlinde algebra of the group $U(k)$ at level $N-k$ and the cohomology of the Grassmannian. The argument proceeds by starting with the two dimensional sigma model whose target space is the Grassmannian and integrating out some fields in a standard way. It has long been known that the resulting low energy effective action describes a theory with a mass gap; the novelty here is that this theory in fact is equivalent at long distances to a gauged WZW model of $U(k)/U(k)$, and hence is related to the Verlinde algebra.