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Exts and Vertex Operators

Erik Carlsson, Andrei Okounkov

TL;DR

The paper constructs a geometric vertex-operator $\mathsf{W}(\mathcal{L})$ acting on the Nakajima Fock space of Hilbert schemes, derived from Ext groups between two ideal sheaves twisted by a line bundle. It proves an explicit factorization of $\mathsf{W}(\mathcal{L})$ into exponentials of Nakajima operators, revealing a direct connection between Hilbert scheme geometry and vertex operator algebras. In the $T$-equivariant setting, the trace of $\mathsf{W}(\mathcal{L})$ recovers Nekrasov partition functions, enabling corollaries for Chern data and leading to a Pieri-type formula for Jack polynomials via localization. The results provide practical recipes to compute Chern classes of $T\mathrm{Hilb}_n S$ and establish deep links between Nekrasov theory, Jack polynomials, and the geometry of Hilbert schemes, with potential extensions to broader moduli spaces of sheaves.

Abstract

The direct product of two Hilbert schemes of the same surface has natural K-theory classes given by the alternating Ext groups between the two ideal sheaves in question, twisted by a line bundle. We express the Chern classes of these virtual bundles in terms of Nakajima operators.

Exts and Vertex Operators

TL;DR

The paper constructs a geometric vertex-operator acting on the Nakajima Fock space of Hilbert schemes, derived from Ext groups between two ideal sheaves twisted by a line bundle. It proves an explicit factorization of into exponentials of Nakajima operators, revealing a direct connection between Hilbert scheme geometry and vertex operator algebras. In the -equivariant setting, the trace of recovers Nekrasov partition functions, enabling corollaries for Chern data and leading to a Pieri-type formula for Jack polynomials via localization. The results provide practical recipes to compute Chern classes of and establish deep links between Nekrasov theory, Jack polynomials, and the geometry of Hilbert schemes, with potential extensions to broader moduli spaces of sheaves.

Abstract

The direct product of two Hilbert schemes of the same surface has natural K-theory classes given by the alternating Ext groups between the two ideal sheaves in question, twisted by a line bundle. We express the Chern classes of these virtual bundles in terms of Nakajima operators.

Paper Structure

This paper contains 7 sections, 7 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Applications
  3. Nekrasov theory
  4. Chern classes of $T\mathop{\mathrm{Hilb}}\nolimits$
  5. Proof of corollaries \ref{['c_str']} and \ref{['c_c1']}
  6. Pieri-type formulas for Jack polynomials
  7. Proof of Theorem \ref{['t_main']}

Key Result

Theorem 1

Theorems & Definitions (9)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Lemma 4
  • proof
  • Corollary 5
  • Lemma 6
  • proof