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

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.