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A New Gauge-Theoretic Construction of 4-Dimensional Hyperkähler ALE Spaces

Jiajun Yan

TL;DR

The paper develops a gauge-theoretic framework to realize all $4$-dimensional hyperkähler ALE spaces as moduli spaces of pairs $(B,\Theta)$ on orbifold bundles over $S^2/\Gamma$, modulo a $\tau$-invariant gauge group. Building on Kronheimer's finite-dimensional hyperkähler reduction, it introduces an infinite-dimensional hyperkähler structure on $C^{\infty}(S^2/\Gamma,E(\Gamma))$, with three moment maps $\tilde{\mu}_1,\tilde{\mu}_2,\tilde{\mu}_3$ and corresponding reductions, yielding moduli spaces that reproduce the Kronheimer spaces. Key contributions include a detailed gauge-theoretic construction, a proof that the resulting moduli spaces are smooth, finite-dimensional, and isometric to the Kronheimer ALE spaces, as well as uniqueness and freeness criteria that ensure no extraneous solutions arise. Collectively, the work provides a gauge-theoretic interpretation of Kronheimer’s construction and strengthens the link between hyperkähler geometry, gauge theory, and the McKay Correspondence in the ADE setting.

Abstract

Non-compact hyperkähler spaces arise frequently in gauge theory. The 4-dimensional hyperkähler ALE spaces are a special class of non-compact hyperkähler spaces. They are in one-to-one correspondence with the finite subgroups of SU(2) and have interesting connections with representation theory and singularity theory, captured by the McKay Correspondence. The 4-dimensional hyperkähler ALE spaces are first classified by Peter Kronheimer via a finite-dimensional hyperkähler reduction. In this paper, we give a new gauge-theoretic construction of these spaces. More specifically, we realize each 4-dimensional hyperkähler ALE space as a moduli space of solutions to a system of equations for a pair consisting of a connection and a section of a vector bundle over an orbifold Riemann surface, modulo a gauge group action. The construction given in this paper parallels Kronheimer's original construction and hence can also be thought of as a gauge-theoretic interpretation of Kronheimer's construction of these spaces.

A New Gauge-Theoretic Construction of 4-Dimensional Hyperkähler ALE Spaces

TL;DR

The paper develops a gauge-theoretic framework to realize all -dimensional hyperkähler ALE spaces as moduli spaces of pairs on orbifold bundles over , modulo a -invariant gauge group. Building on Kronheimer's finite-dimensional hyperkähler reduction, it introduces an infinite-dimensional hyperkähler structure on , with three moment maps and corresponding reductions, yielding moduli spaces that reproduce the Kronheimer spaces. Key contributions include a detailed gauge-theoretic construction, a proof that the resulting moduli spaces are smooth, finite-dimensional, and isometric to the Kronheimer ALE spaces, as well as uniqueness and freeness criteria that ensure no extraneous solutions arise. Collectively, the work provides a gauge-theoretic interpretation of Kronheimer’s construction and strengthens the link between hyperkähler geometry, gauge theory, and the McKay Correspondence in the ADE setting.

Abstract

Non-compact hyperkähler spaces arise frequently in gauge theory. The 4-dimensional hyperkähler ALE spaces are a special class of non-compact hyperkähler spaces. They are in one-to-one correspondence with the finite subgroups of SU(2) and have interesting connections with representation theory and singularity theory, captured by the McKay Correspondence. The 4-dimensional hyperkähler ALE spaces are first classified by Peter Kronheimer via a finite-dimensional hyperkähler reduction. In this paper, we give a new gauge-theoretic construction of these spaces. More specifically, we realize each 4-dimensional hyperkähler ALE space as a moduli space of solutions to a system of equations for a pair consisting of a connection and a section of a vector bundle over an orbifold Riemann surface, modulo a gauge group action. The construction given in this paper parallels Kronheimer's original construction and hence can also be thought of as a gauge-theoretic interpretation of Kronheimer's construction of these spaces.
Paper Structure (20 sections, 31 theorems, 173 equations)

This paper contains 20 sections, 31 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kronheimer's construction of ALE spaces
  4. Basic setup for the gauge-theoretic construction
  5. The $\Gamma$-action and orbifold vector bundles
  6. The gauge-theoretic framework
  7. An Overview of the Gauge-Theoretic Construction
  8. Symplectic reduction
  9. Further reduction
  10. Proof of Proposition \ref{['sympred']}
  11. Hyperkähler structure on $C^\infty(S^2/\Gamma,E(\Gamma))$
  12. The quaternions
  13. Quaternionic structures on associated bundles and spaces of sections
  14. Hyperkähler structure on the space of sections $C^\infty(E(\Gamma))$
  15. Uniqueness theorems
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\tilde{\zeta}=(\tilde{\zeta}_1,\tilde{\zeta}_2,\tilde{\zeta}_3)$, where for all $i$, $\tilde{\zeta}_i\in Z$. Let Then for a suitable choice of $\tilde{\zeta}$, $\mathcal{X}_{\tilde{\zeta}}$ is diffeomorphic to the resolution of singularity $\widetilde{\mathbb C^2/\Gamma}$. Furthermore, $\mathcal{X}_{\tilde{\zeta}}$ is isometric to $X_{\zeta}$ in kronheimer.

Theorems & Definitions (74)

  • Theorem 1.1
  • Proposition 2.1: cf. Proposition 2.1. in kronheimer
  • Proposition 2.2: cf. Proposition 2.8. in kronheimer
  • Theorem 2.3: cf. Theorem 1.1. in kronheimer
  • Theorem 2.4: cf. Theorem 1.2. in kronheimer
  • Theorem 2.5: cf. Theorem 1.3. in kronheimer
  • Definition 2.6
  • Example 2.7: The hyperplane bundle and the tautological bundle
  • Example 2.8
  • Definition 2.9
  • ...and 64 more