Hilbert schemes and W algebras

TL;DR

This paper constructs a W algebra acting on the cohomology of Hilbert schemes of points on surfaces, explicitly computes its structure, and relates it to known infinite-dimensional algebras using geometric and algebraic methods.

Contribution

It provides a geometric construction of W algebra generators acting on Hilbert scheme cohomologies and identifies this algebra with a W_{1+infinity}-type algebra, including explicit commutator formulas.

Findings

Explicit formulas for commutators of W algebra basis elements.

Identification of the algebra with a W_{1+infinity}-type algebra.

Chern character operators as zero-modes of vertex operators.

Abstract

We construct geometrically the generating fields of a W algebra which acts irreducibly on the direct sum of the cohomology rings of the Hilbert schemes of n points on a projective surface for all n. We compute explicitly the commutators among a set of linear basis elements of the W algebra, and identify this algebra with a $W_{1+\infty}$-type algebra. A precise formula of certain Chern character operators, which is essential for the construction of the W algebra, is established in terms of the Heisenberg algebra generators. In addition, these Chern character operators are proved to be the zero-modes of vertex operators.