The Verlinde Algebra And The Cohomology Of The Grassmannian

TL;DR

This paper explains the connection between the Verlinde algebra of U(k) at a specific level and the cohomology of the Grassmannian using a quantum field theory approach, revealing a new equivalence with a gauged WZW model.

Contribution

It provides a quantum field theory derivation of the relationship between the Verlinde algebra and Grassmannian cohomology, showing their equivalence via a gauged WZW model.

Findings

The low energy effective action describes a theory with a mass gap.

The theory is equivalent to a gauged WZW model of U(k)/U(k).

This establishes a link between the Verlinde algebra and Grassmannian cohomology.

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.