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

TL;DR

This paper introduces a gauge-theoretic Morse flow framework to understand the McKay correspondence, providing a dynamic perspective that links cohomology of resolutions with irreducible representations of finite groups.

Contribution

It develops a novel dynamical approach using Morse-Bott theory and gauge theory to realize the McKay correspondence, proving the conjecture for cyclic groups.

Findings

Flow lines correspond to holonomy representations

Conjecture proven for cyclic groups

Explicit construction of the correspondence

Abstract

The McKay correspondence establishes a bijection between the cohomology of a minimal resolution and the irreducible representations of a finite subgroup $\Gamma \subset \text{SU}(2)$. 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 $\Gamma$ to $GL(R)$. We conjecture that the flow emanating from a critical submanifold converges asymptotically at the boundary to a specific irreducible representation of $\Gamma$. This dynamical process explicitly constructs the identification between the cohomology basis and the irreducible representations of $\Gamma$ prescribed by the McKay correspondence. We prove this conjecture for cyclic cases.