[Paper Review] The Verlinde Algebra And The Cohomology Of The Grassmannian

Edward Witten|arXiv (Cornell University)|Dec 13, 1993

Algebraic structures and combinatorial models37 references219 citations

TL;DR

This paper establishes a conceptual link between the quantum cohomology of the Grassmannian $G(k,N)$ and the Verlinde algebra of $U(k)$ at level $N-k$ via a 2D ${\cal N}=2$ supersymmetric $U(k)$ gauge theory with $N$ fundamental chiral multiplets. By analyzing the low-energy effective theory, the authors show that the topological correlation functions of the sigma model on $G(k,N)$ map to those of the $U(k)/U(k)$ gauged WZW model, which computes the Verlinde algebra, thereby proving the equivalence of the two rings at the quantum level.

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.

Motivation & Objective

• To provide a physical, quantum field theory explanation for the conjectured isomorphism between the quantum cohomology of the Grassmannian $G(k,N)$ and the Verlinde algebra of $U(k)$ at level $N-k$.
• To demonstrate that the low-energy effective theory of a ${\cal N}=2$ $U(k)$ gauge theory with $N$ fundamental chiral multiplets is the $U(k)/U(k)$ gauged Wess-Zumino-Witten model.
• To show that correlation functions in the sigma model on $G(k,N)$, which compute quantum cohomology, are equivalent to those in the $G/G$ model, which compute the Verlinde algebra.
• To verify the isomorphism explicitly for $k=2$ by comparing relations, dimensions, and metrics in both rings.

Proposed method

• Realize the Grassmannian $G(k,N)$ as a symplectic quotient of a linear space via a $U(k)$ gauge theory with $N$ fundamental chiral superfields in ${\cal N}=2$ superspace.
• Use path integral techniques to derive the low-energy effective action after integrating out the matter multiplets, showing it becomes a $U(k)/U(k)$ gauged WZW model.
• Apply abelianization to reduce the $G/G$ model to a theory with the maximal torus and Weyl group, simplifying the computation of Verlinde algebra structure.
• Compute topological correlation functions in both the sigma model and the $G/G$ model, showing they match via the Verlinde formula.
• Use the superpotential $W(c_1, c_2) = Σ_{i=1}^N (λ_i^{N+1} + λ_i)$ to define the quantum cohomology relations via $dW = 0$, and compare with Verlinde algebra relations.
• Verify metric agreement between quantum cohomology and Verlinde algebra by computing $g_{{\sigma}}(f_r,1)$ and matching it to $g_V(V_s\eta^t,1)$, confirming normalization and structure.

Experimental results

Research questions

• RQ1 How can the quantum cohomology of the Grassmannian $G(k,N)$ be derived from a physical quantum field theory?
• RQ2 What is the low-energy effective theory of a ${\cal N}=2$ $U(k)$ gauge theory with $N$ fundamental chiral multiplets?
• RQ3 Does the $U(k)/U(k)$ gauged WZW model compute the same algebraic structure as the quantum cohomology of $G(k,N)$?
• RQ4 Is the Verlinde algebra of $U(k)$ at level $N-k$ isomorphic to the quantum cohomology ring of $G(k,N)$, and if so, via what physical mechanism?
• RQ5 How do the ring relations, dimensions, and metrics of the two algebras match?

Key findings

• The quantum cohomology ring of $G(k,N)$ is isomorphic to the Verlinde algebra of $U(k)$ at level $(N-k, N)$, with the $u(1)$ factor contributing level $N$.
• For $k=2$, the quantum cohomology is defined by relations $\frac{\lambda_1^N - \lambda_2^N}{\lambda_1 - \lambda_2} = 0$ and $\frac{\lambda_1^{N+1} - \lambda_2^{N+1}}{\lambda_1 - \lambda_2} + 1 = 0$, which match the Verlinde algebra relations.
• The dimension of both the quantum cohomology and Verlinde algebra is $N(N-1)/2$, confirming the isomorphism at the level of vector spaces.
• The metric on the quantum cohomology, computed via the classical limit of the residue formula, matches the Verlinde metric up to a normalization, with $g_\sigma(f_r,1) = \delta_{r,0}$, confirming the isomorphism of bilinear forms.
• The normalization constant $c$ in the $\lambda_i \leftrightarrow \tilde{\lambda}_i$ map is fixed by invariance under scaling, and the constant $a$ in the metric is fixed by agreement at $r=0$, confirming consistency.
• The isomorphism extends beyond $k=2$ via similar arguments, suggesting the result holds for general $k$ and $N$.