Vertex Operators, Grassmannians, and Hilbert Schemes

TL;DR

This paper connects geometric constructions of vertex operators on the Sato Grassmannian with their algebraic counterparts on Hilbert schemes, demonstrating limits and correspondences in equivariant cohomology.

Contribution

It establishes a geometric limit approach to vertex operators, linking finite-dimensional approximations with infinite-dimensional algebraic structures on Hilbert schemes.

Findings

Vertex operators are limits of geometric correspondences.

Locality and algebraic relations are preserved in the limit.

Identifies geometric operators with Hilbert scheme vertex operators for \\mathbb{C}^2.

Abstract

We describe a well-known collection of vertex operators on the infinite wedge representation as a limit of geometric correspondences on the equivariant cohomology groups of a finite-dimensional approximation of the Sato grassmannian, by cutoffs in high and low degrees. We prove that locality, the boson-fermion correspondence, and intertwining relations with the Virasoro algebra are limits of the localization expression for the composition of these operators. We then show that these operators are, almost by definition, the Hilbert scheme vertex operators defined by Okounkov and the author in \cite{CO} when the surface is $\mathbb{C}^2$ with the torus action $z\cdot (x,y) = (zx,z^{-1}y)$.