A contact McKay correspondence for links of simple singularities

TL;DR

This paper computes the cylindrical contact homology of links of simple singularities, revealing a Floer theoretic McKay correspondence that relates the homology ranks to the conjugacy classes of finite subgroups of SU(2).

Contribution

It establishes a contact homology version of the McKay correspondence for links of simple singularities, connecting Floer theory with classical group theory.

Findings

Contact homology ranks are expressed in terms of the number of conjugacy classes of G.

The work demonstrates a Floer theoretic analogue of the McKay correspondence.

Perturbation techniques achieve nondegeneracy of the contact form on $S^3/G$.

Abstract

We compute the cylindrical contact homology of the links of the simple singularities. These manifolds are contactomorphic to $S^3/G$ for finite subgroups $G\subset SU(2)$. We perturb the degenerate contact form on $S^3/G$ with a Morse function, which is invariant under the corresponding $H\subset SO(3)$ action on $S^2$, to achieve nondegeneracy up to an action threshold. The cylindrical contact homology is recovered by taking a direct limit of the action filtered homology groups. The ranks of this homology are given in terms of $|\text{Conj}(G)|$, demonstrating a Floer theoretic McKay correspondence.