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Instantons and affine Lie algebras

Hiraku Nakajima

TL;DR

The work develops geometric realizations of affine Lie algebra actions on the homology of instanton moduli spaces, using Hecke-type correspondences on the Hilbert scheme of points to model point-twists and elementary transformations along embedded curves on ALE spaces to capture affine generators. It demonstrates that these operators satisfy the affine algebra relations, yielding integrable highest-weight representations with the middle-degree homology capturing a canonical irreducible component, and clarifies the role of mass/local-global considerations in guiding the construction. While the Hilbert-scheme construction yields irreducible Heisenberg/Clifford actions, extending the framework to general $4$-manifolds remains challenging due to dependence on the complex structure, motivating further development of correspondences and a broader homological framework. The results form a bridge between geometric representation theory, gauge-theoretic invariants, and the modular properties predicted by S-duality, offering a pathway to generalize to projective surfaces and beyond.

Abstract

Various constructions of the affine Lie algebra action on the homology group of moduli spaces of instantons on 4-manifolds are discussed. The analogy between the local-global principle and the role of mass is also explained. The detailed proofs are given in separated papers \cite{Na-algebra,Na-Hilbert}.

Instantons and affine Lie algebras

TL;DR

The work develops geometric realizations of affine Lie algebra actions on the homology of instanton moduli spaces, using Hecke-type correspondences on the Hilbert scheme of points to model point-twists and elementary transformations along embedded curves on ALE spaces to capture affine generators. It demonstrates that these operators satisfy the affine algebra relations, yielding integrable highest-weight representations with the middle-degree homology capturing a canonical irreducible component, and clarifies the role of mass/local-global considerations in guiding the construction. While the Hilbert-scheme construction yields irreducible Heisenberg/Clifford actions, extending the framework to general -manifolds remains challenging due to dependence on the complex structure, motivating further development of correspondences and a broader homological framework. The results form a bridge between geometric representation theory, gauge-theoretic invariants, and the modular properties predicted by S-duality, offering a pathway to generalize to projective surfaces and beyond.

Abstract

Various constructions of the affine Lie algebra action on the homology group of moduli spaces of instantons on 4-manifolds are discussed. The analogy between the local-global principle and the role of mass is also explained. The detailed proofs are given in separated papers \cite{Na-algebra,Na-Hilbert}.

Paper Structure

This paper contains 4 sections, 3 theorems, 10 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Hilbert scheme of points and the Heisenberg algebra: Twist around points
  3. Elementary transformation: Twist along embedded submanifolds
  4. Local-Global Principle and the Mass

Key Result

Theorem 1

. The following relations hold as operators on $\bigoplus_{n} H_*({X}^{\lbrack{n}\rbrack})$. where $c_i$ is a nonzero integer depending only on $i$(independent of $X$).

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3