Computing the Cylindrical Contact Homology of Links of Simple Singularities via Orbifold Morse Theory

Core Concepts

This paper establishes a "Floer theoretic McKay correspondence" by computing the cylindrical contact homology of links of simple singularities, revealing a relationship between the ranks of this homology and the number of conjugacy classes of associated finite subgroups of SU(2).

Abstract

• Bibliographic Information: Digiosia, L., & Nelson, J. (2024). A contact McKay correspondence for links of simple singularities. arXiv preprint arXiv:2107.07102v5.
• Research Objective: This paper aims to compute the cylindrical contact homology of links of simple singularities, which are 3-dimensional contact manifolds contactomorphic to S³/G for finite subgroups G of SU(2).
• Methodology: The authors employ a method of perturbing the degenerate contact form on S³/G using a Morse function invariant under the corresponding symmetry group in SO(3). This perturbation achieves nondegeneracy up to an action threshold, allowing for the computation of cylindrical contact homology as a direct limit of action-filtered homology groups.
• Key Findings: The main result demonstrates that the ranks of the cylindrical contact homology of these links are determined by the number of conjugacy classes of the group G. This finding establishes a connection between contact geometry and representation theory, mirroring the classical McKay correspondence.
• Main Conclusions: The paper concludes that the cylindrical contact homology of links of simple singularities can be explicitly computed and expressed in terms of the number of conjugacy classes of the associated finite group. This result provides a new perspective on the geometry of these manifolds and suggests deeper connections between contact geometry, singularity theory, and representation theory.
• Significance: This research significantly contributes to the understanding of contact geometry and its relationship to other mathematical fields. The explicit computation of cylindrical contact homology for this family of manifolds offers a valuable testing ground for further exploration of Floer-theoretic invariants and their applications.
• Limitations and Future Research: The paper focuses specifically on links of simple singularities. Exploring similar computations for more general singularities or higher-dimensional analogues would be a natural extension of this work. Additionally, investigating potential applications of these results to related areas like symplectic geometry and gauge theory could yield further insights.

Stats

Quotes

"We compute the cylindrical contact homology of the links of the simple singularities. These manifolds are contactomorphic to S³/G for finite subgroups G ⊂SU(2)."
"Our computation realizes a contact Floer theoretic McKay correspondence result, namely that the ranks of the cylindrical contact homology of the links of simple singularities are given in terms of the number of conjugacy classes of the group G."

Key Insights Distilled From

A contact McKay correspondence for links of simple singularities

by Leo Digiosia... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2107.07102.pdf

Deeper Inquiries

How do the techniques used in this paper generalize to the study of contact homology in higher dimensions?

While the specific computations in this paper focus on dimension three, several underlying techniques and concepts have analogues in higher-dimensional contact homology. However, significant challenges arise when generalizing to higher dimensions:
Generalizations:

Perturbing degenerate contact forms: The idea of perturbing a degenerate contact form using a Morse function on the base of a fibration can be extended to higher dimensions. This requires suitable generalizations of the notions of Reeb vector fields, Hamiltonian vector fields, and their lifts to the total space of the fibration.
Action filtration and direct limits: The concept of action filtration and the use of direct limits to define contact homology remain valid in higher dimensions. However, the analysis of Reeb dynamics and the structure of moduli spaces of holomorphic curves become significantly more intricate.
Relationship to orbifold Morse homology: The analogy between bad Reeb orbits and non-orientable critical points, as well as the structural similarities between the differentials in contact and orbifold Morse homology, can be explored in higher dimensions. This requires developing appropriate notions of orbifold Morse theory for higher-dimensional orbifolds.
Challenges:

Transversality: Achieving transversality for moduli spaces of holomorphic curves in higher dimensions is a major technical hurdle. Standard techniques like perturbing the almost complex structure might not be sufficient, and more sophisticated methods, such as polyfolds or virtual techniques, are often required.
Grading: Defining a well-behaved grading on contact homology in higher dimensions is more involved. The Conley-Zehnder index, which provides a natural grading in dimension three, has more complicated generalizations in higher dimensions.
Computational complexity: The computational complexity of contact homology increases dramatically with dimension. The moduli spaces of holomorphic curves become higher-dimensional and more challenging to analyze, making explicit computations difficult.

Could there be alternative geometric interpretations of the "Floer theoretic McKay correspondence" established in this paper, perhaps outside the realm of contact geometry?

Yes, the "Floer theoretic McKay correspondence" established in this paper, relating the contact homology of links of simple singularities to the representation theory of the corresponding finite groups, hints at deeper connections that could manifest in other geometric contexts. Here are some potential avenues for exploration:

Symplectic geometry: The link of a singularity can also be viewed as the boundary of a neighborhood of the singularity. This suggests a potential connection to the symplectic cohomology of the resolution of the singularity. Exploring this relationship could lead to a symplectic interpretation of the McKay correspondence.
Gauge theory: The finite subgroups of SU(2) also play a crucial role in gauge theory, particularly in the study of instantons on four-manifolds. It would be interesting to investigate if the contact Floer theoretic McKay correspondence has an analogue in the context of gauge-theoretic invariants, such as Donaldson invariants or Seiberg-Witten invariants.
Mirror symmetry: Mirror symmetry predicts surprising relationships between symplectic geometry and algebraic geometry. It is conceivable that the McKay correspondence, which has both algebraic and symplectic aspects, could be understood from a mirror symmetry perspective. This could involve relating the contact homology of the link to some invariants of the mirror object.
Geometric representation theory: The McKay correspondence itself has deep connections to geometric representation theory, particularly through the study of quiver varieties. It is plausible that the contact Floer theoretic perspective could provide new insights into the geometry of these varieties and their relation to representation theory.

What are the implications of this research for understanding the topology and geometry of resolutions of singularities in algebraic geometry?

This research offers a novel perspective on resolutions of singularities by connecting the contact geometry of their links to their algebraic and topological properties. Here are some potential implications:

New invariants of singularities: Cylindrical contact homology, as computed in this paper, provides new invariants for simple singularities. These invariants could potentially distinguish different resolutions of the same singularity or provide finer information about the singularity itself.
Understanding exceptional divisors: The computation of contact homology highlights the role of exceptional fibers in the resolution. The correspondence between these fibers and the generators of certain homology groups could lead to a better understanding of the geometry and intersection theory of exceptional divisors.
Connections to other invariants: The observed relationship between contact homology and orbifold Morse homology suggests deeper connections between different geometric invariants associated with singularities. Exploring these connections could lead to a more unified understanding of the topology and geometry of resolutions.
Generalizations to higher dimensions: While this paper focuses on simple singularities in dimension three, the techniques and ideas could potentially be extended to study resolutions of more general singularities in higher dimensions. This could provide new tools for investigating the topology and geometry of these resolutions.
Overall, this research opens up new avenues for studying resolutions of singularities by bringing to bear the powerful tools and techniques of contact geometry. The explicit computations and the established connections to other areas, such as orbifold Morse theory and representation theory, provide a promising starting point for further investigations.