A Dynamical Framework for the McKay Correspondence via Gauge-Theoretic Morse Flow
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912820081197056 |
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| author | Yan, Jiajun |
| author_facet | Yan, Jiajun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08195 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Dynamical Framework for the McKay Correspondence via Gauge-Theoretic Morse Flow Yan, Jiajun Differential Geometry Geometric Topology Representation Theory Symplectic Geometry 53D20, 53D30, 14B05, 20C15, 14J60, 32L05 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. |
| title | A Dynamical Framework for the McKay Correspondence via Gauge-Theoretic Morse Flow |
| topic | Differential Geometry Geometric Topology Representation Theory Symplectic Geometry 53D20, 53D30, 14B05, 20C15, 14J60, 32L05 |
| url | https://arxiv.org/abs/2601.08195 |