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A Dynamical Framework for the McKay Correspondence via Gauge-Theoretic Morse Flow

Jiajun Yan

TL;DR

This work introduces a gauge-theoretic Morse-flow framework to realize the McKay correspondence dynamically on Kronheimer's ALE spaces, recasting the static basis–representation bijection as a flow-based identification between $H^2(X)$ generators and irreducible representations of the finite subgroup $oldsymbol{\Gamma} o SU(2)$. By constructing an $S^1$-invariant Morse-Bott function on the ALE moduli space and analyzing gradient trajectories of flat connections, the authors associate 1-parameter holonomy representations to flow lines and show convergence to irreducible representations at infinity; they prove the mechanism for cyclic groups $oldsymbol{\Gamma} o Z_n$ in the paper and provide detailed verifications for $oldsymbol{\Gamma}=Z_2$ and $Z_3$. The approach unifies Kronheimer’s gauge-theoretic ALE construction with Morse theory, offering a dynamic interpretation of the McKay correspondence and highlighting a bridge to potential connections with Floer-theoretic and symplectic viewpoints. These results suggest a broader framework in which cohomology generators are recovered as asymptotic projectors along gradient flows, with explicit computations validating the conjecture in key low-rank cyclic cases.

Abstract

The McKay correspondence establishes a bijection between the cohomology of a minimal resolution and the irreducible representations of a finite subgroup $Γ\subset \text{SU}(2)$. While traditional proofs rely on static algebraic isomorphisms, we propose a dynamical framework grounded in gauge theory and Morse-Bott theory. We analyze an $S^1$-invariant Morse-Bott function on the minimal resolution, interpreting its gradient flow lines as $1$-parameter families of holonomy representations of flat connections from $Γ$ to $GL(R)$. We conjecture that the flow emanating from a critical submanifold converges asymptotically at the boundary to a specific irreducible representation of $Γ$. This dynamical process explicitly constructs the identification between the cohomology basis and the irreducible representations of $Γ$ prescribed by the McKay correspondence. We prove this conjecture for cyclic cases.

A Dynamical Framework for the McKay Correspondence via Gauge-Theoretic Morse Flow

TL;DR

This work introduces a gauge-theoretic Morse-flow framework to realize the McKay correspondence dynamically on Kronheimer's ALE spaces, recasting the static basis–representation bijection as a flow-based identification between generators and irreducible representations of the finite subgroup . By constructing an -invariant Morse-Bott function on the ALE moduli space and analyzing gradient trajectories of flat connections, the authors associate 1-parameter holonomy representations to flow lines and show convergence to irreducible representations at infinity; they prove the mechanism for cyclic groups in the paper and provide detailed verifications for and . The approach unifies Kronheimer’s gauge-theoretic ALE construction with Morse theory, offering a dynamic interpretation of the McKay correspondence and highlighting a bridge to potential connections with Floer-theoretic and symplectic viewpoints. These results suggest a broader framework in which cohomology generators are recovered as asymptotic projectors along gradient flows, with explicit computations validating the conjecture in key low-rank cyclic cases.

Abstract

The McKay correspondence establishes a bijection between the cohomology of a minimal resolution and the irreducible representations of a finite subgroup . While traditional proofs rely on static algebraic isomorphisms, we propose a dynamical framework grounded in gauge theory and Morse-Bott theory. We analyze an -invariant Morse-Bott function on the minimal resolution, interpreting its gradient flow lines as -parameter families of holonomy representations of flat connections from to . We conjecture that the flow emanating from a critical submanifold converges asymptotically at the boundary to a specific irreducible representation of . This dynamical process explicitly constructs the identification between the cohomology basis and the irreducible representations of prescribed by the McKay correspondence. We prove this conjecture for cyclic cases.
Paper Structure (20 sections, 8 theorems, 204 equations, 9 figures)

This paper contains 20 sections, 8 theorems, 204 equations, 9 figures.

Table of Contents

  1. Introduction
  2. A review of a gauge-theoretic construction of the $4$-dimensional hyperkähler ALE spaces
  3. Kronheimer's construction
  4. A gauge-theoretic construction of ALE spaces
  5. Step 1: Introducing the principal and vector bundles
  6. Step 2: Introducing the gauge group and the configuration space
  7. Step 3: A sketch of the gauge-theoretic construction of ALE spaces and the main theorem
  8. Hopf fibration and the pullback bundle $E(\Gamma)^h$
  9. Calculation of the holonomy representations
  10. An $S^1$-invariant Morse-Bott function on the moduli space and its critical points
  11. $S^1$-fixed points of $A_n$-type ALE spaces
  12. $n=2m+1$:
  13. $n=2m$:
  14. Proof of Proposition \ref{['prop']}
  15. Irreducible representations as limits of gradient flow lines
  16. ...and 5 more sections

Key Result

Proposition 2.5

There are three symplectic forms on $C^\infty(S^2/\Gamma, E(\Gamma))$ compatible with complex structures $I$, $J$, $K$, respectively: and a hyperkähler metric $g_h$ such that together giving rise to a hyperkähler structure on $C^\infty(S^2/\Gamma, E(\Gamma))$.

Figures (9)

  • Figure 1: $(\alpha,\beta)=((\alpha_1,...,\alpha_n),(\beta_1,...,\beta_n))$
  • Figure 2: $f(\varphi)$-action
  • Figure 3: $\mathbb Z_5$: $S^1$-fixed points / critical points
  • Figure 4: $\mathbb Z_5$: $S^1$-fixed points / critical points
  • Figure 5: $\mathbb Z_4$: $S^1$-fixed points / critical points
  • ...and 4 more figures

Theorems & Definitions (21)

  • Conjecture 1.2
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: Symplectic structure on $\mathcal{A}^F_\tau \times C^\infty(S^2/\Gamma, E(\Gamma))$
  • Proposition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Lemma 3.1
  • Lemma 3.2
  • ...and 11 more