Spectral Floer Theory and Tangential Structures for Constructing Donaldson-Fukaya Categories over Ring Spectra

Core Concepts

This research paper introduces a novel framework for constructing Donaldson-Fukaya categories over ring spectra, leveraging the concepts of spectral Floer theory and tangential structures to study Lagrangian submanifolds within Liouville manifolds.

Abstract

• Bibliographic Information: Porcelli, N., & Smith, I. (2024). Spectral Floer theory and tangential structures. arXiv preprint arXiv:2411.03257.

• Research Objective: This paper aims to develop a spectral Donaldson-Fukaya category for any 'graded tangential pair' Θ → Φ of spaces, generalizing previous work that focused on stably framed Liouville manifolds. This category encompasses Lagrangians with classifying maps for their tangent bundles that lift to Θ → Φ.

• Methodology: The authors employ techniques from symplectic geometry, algebraic topology, and homotopy theory. They construct spaces of abstract discs with tangential structures, providing an adaptable model for Bott periodicity. They then utilize flow categories, enriched with tangential structures, to define and study the spectral Donaldson-Fukaya category.

• Key Findings:

  • The authors successfully define a spectral Donaldson-Fukaya category F(X; (Θ, Φ)) for Liouville manifolds X with a Φ-structure, where objects are geometrically bounded exact Lagrangian Θ-branes.
  • They extend their previous obstruction theory to this general setting, providing conditions for lifting quasi-isomorphisms from the Fukaya category over Z to F(X; (Θ, Φ)).
  • The authors demonstrate applications of their framework, including results on Lagrangian cobordism and connections to sheaf quantization.

• Main Conclusions: This work significantly advances the understanding of spectral Floer theory and its applications to the study of Lagrangian submanifolds. The introduction of tangential structures provides a powerful tool for constructing and analyzing Donaldson-Fukaya categories over various ring spectra, leading to new insights into Lagrangian cobordism and connections with other areas of mathematics.

• Significance: This research has substantial implications for symplectic geometry, algebraic topology, and related fields. It opens new avenues for investigating the topology and geometry of Lagrangian submanifolds using tools from homotopy theory and stable homotopy theory.

• Limitations and Future Research: The paper primarily focuses on theoretical aspects of spectral Floer theory. Further research could explore computational aspects and specific examples in more detail. Additionally, investigating the connections between the spectral Fukaya category and other categorical structures arising in symplectic geometry and mirror symmetry would be a fruitful direction for future work.

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Spectral Floer theory and tangential structures

by Noah Porcell... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.03257.pdf

Deeper Inquiries

How can the computational aspects of spectral Floer theory be further developed to facilitate concrete calculations and applications?

This is a central question in the field. Here are some potential avenues for progress:
1. Leveraging Existing Computational Tools:

Exploit Relations with Classical Floer Theory: The paper emphasizes connections between spectral Fukaya categories (e.g., F(X; (Θ, Φ))) and their classical counterparts (F(X; Z)). We can leverage computational techniques from classical Floer theory, such as:

Morse-Bott Techniques: When Lagrangians intersect cleanly, these methods simplify computations.
Symmetry Arguments: Symmetries of the symplectic manifold can be used to reduce the complexity of calculations.

Spectral Sequence Arguments: Spectral sequences are powerful tools in algebraic topology. We can develop spectral sequences relating spectral Floer homology groups to more computable invariants.
2. Developing New Computational Techniques:

Equivariant Transversality: The paper discusses tangential structures related to various bordism theories. Developing robust equivariant transversality results for moduli spaces with these structures would be crucial.
Combinatorial Models: Analogous to the use of Morse theory in classical Floer theory, we could seek combinatorial models for spectral Floer homology, potentially drawing inspiration from algebraic topology (e.g., cellular decompositions of moduli spaces).
Computer-Assisted Proofs: For specific examples, computer-assisted proofs might be feasible, especially as computational power increases.
3. Focusing on Tractable Cases:

Low-Dimensional Examples: Start with symplectic manifolds of low dimension, where the moduli spaces of holomorphic curves are easier to understand.
Special Tangential Structures: Focus on tangential pairs (Θ, Φ) that lead to more computable bordism theories. For example, the paper highlights cases related to framed, oriented, and complex cobordism.
Key Point: Progress in computational aspects will likely involve a combination of adapting existing tools, developing new techniques, and focusing on specific cases where computations are more tractable.

Could there be alternative constructions of Donaldson-Fukaya categories over ring spectra that do not rely on tangential structures, and if so, how would they compare to the framework presented in this paper?

Yes, alternative constructions are conceivable. Here are some possibilities and comparisons:
1. K-Theoretic Methods:

Idea: Instead of tangential structures, one could try to directly construct categories enriched over spectra using K-theoretic techniques. This might involve:

Geometric K-Theory: Representing objects as families of Fredholm operators over Lagrangian submanifolds.
Algebraic K-Theory: Working with categories of perfect modules over suitable rings associated with the symplectic manifold.

Comparison: This approach would be more algebraic and less geometric than the tangential structure framework. It might offer different computational tools but could be more abstract.
2. Homotopy-Theoretic Constructions:

Idea: One could attempt to construct spectral Fukaya categories using more abstract homotopy-theoretic machinery, potentially drawing inspiration from:

Derived Algebraic Geometry: Viewing Fukaya categories as categories of sheaves on some derived moduli space.
∞-Categories: Working directly with ∞-categorical enhancements of Fukaya categories.

Comparison: This approach would be the most abstract and would likely require significant technical machinery. However, it might offer deeper insights into the homotopy-theoretic nature of Floer theory.
3. Alternative Geometric Structures:

Idea: Instead of tangential structures on the tangent bundle, one could explore other geometric structures on Lagrangians or the symplectic manifold that naturally lead to enrichments over ring spectra. Examples might include:

Spin^c Structures: These are refinements of orientations and are closely related to index theory.
Geometric Quantization Data: If the symplectic manifold admits a prequantization line bundle, one could consider Lagrangians equipped with compatible structures.

Comparison: The feasibility and properties of these constructions would depend heavily on the specific geometric structure chosen.
Key Point: While alternative constructions are possible, the tangential structure framework presented in the paper offers a concrete and geometrically intuitive approach, leveraging the well-established theory of bordism.

What are the implications of this research for the study of mirror symmetry, particularly in the context of relating symplectic geometry to algebraic geometry through categorical equivalences?

This research has the potential to significantly impact the study of mirror symmetry by providing a richer categorical framework:
1. Refined Invariants:

Beyond the Hochschild Homology Level: The paper hints at the possibility of open-closed maps landing in topological Hochschild homology (THH) with more general coefficients than the sphere spectrum. This suggests that spectral Fukaya categories might capture finer invariants than their classical counterparts, potentially revealing deeper connections between mirror pairs.
New Connections to Algebraic Geometry: Bordism theories, which are naturally encoded in the tangential structure framework, have intriguing connections to algebraic geometry through tools like algebraic cobordism. This could lead to new ways of relating symplectic and algebraic invariants.
2. Extended Range of Mirror Pairs:

Beyond Calabi-Yau Manifolds: The paper discusses spectral Fukaya categories for Liouville manifolds with vanishing first Chern class, which is a broader class than Calabi-Yau manifolds. This raises the possibility of extending mirror symmetry to a wider range of geometric settings.
Non-Geometric Mirror Symmetry: The use of general ring spectra and tangential structures suggests connections to more abstract and potentially "non-geometric" versions of mirror symmetry, where the mirror might not be a conventional geometric object.
3. Deeper Understanding of Homotopical Aspects:

Homotopy Coherence: Working with categories enriched over spectra naturally incorporates higher homotopical information, which is often crucial for understanding mirror symmetry phenomena.
Connections to Derived Geometry: The paper mentions potential connections to derived algebraic geometry, which is a powerful framework for studying mirror symmetry.
Key Point: This research provides a more refined and flexible categorical language for studying mirror symmetry, potentially leading to new insights, connections, and a broader understanding of the relationship between symplectic and algebraic geometry.