insight - Scientific Computing - # Δa Invariants in Chern-Simons Theory

On the Δa Invariants in Non-Perturbative Complex Chern-Simons Theory: Analyzing the Structure and Properties of Δa Invariants for Weakly Negative-Definite Plumbed 3-Manifolds

Q: How can the relationship between Δa invariants and Heegaard Floer homology be further explored to uncover deeper connections between different topological invariants?

The observed congruence relation between the $\Delta_a$ invariants and the Heegaard Floer correction terms, $d(Y,a)$, hints at a potentially deeper connection that warrants further investigation. Here are some avenues for exploration: Refining the congruence: The current relation (Equation 2 in the text) is a modulo 1 congruence. A natural question is whether this can be strengthened. Are there families of manifolds where the difference $\Delta_a - (1/2 - d(Y,a))$ takes on specific values or exhibits interesting properties? Exploring this could reveal finer connections between these invariants. Beyond weakly negative-definite plumbings: The current results primarily focus on weakly negative-definite plumbed 3-manifolds. It would be highly valuable to investigate if similar relationships hold for broader classes of 3-manifolds. This would require understanding how to define or compute $\Delta_a$ invariants in more general settings. Connections to other Heegaard Floer invariants: Heegaard Floer homology is a rich theory with various flavors and related invariants (e.g., $\tau$ invariants, knot Floer homology). Investigating potential relationships between $\Delta_a$ and these other invariants could provide a more comprehensive picture of the interplay between complex Chern-Simons theory and Heegaard Floer homology. Geometric interpretations: A deeper understanding of the geometric meaning of both $\Delta_a$ invariants (potentially as some kind of "dimension" in a suitable sense) and the correction terms could provide insights into why this relationship exists.

Q: Could there be a modified or refined version of the Δa invariant that does exhibit homology cobordism invariance, perhaps by incorporating additional topological data?

The fact that $\Delta_a$ invariants are not homology cobordism invariants is an important observation. However, it doesn't rule out the possibility of constructing related invariants that are. Here are some potential strategies: Difference invariants: One could explore defining invariants based on differences of $\Delta_a$ invariants for different Spinc structures on the same manifold. It's conceivable that certain linear combinations or differences might become homology cobordism invariants. Incorporating homology cobordism information: Instead of directly modifying $\Delta_a$, one could try to construct a new invariant that combines $\Delta_a$ with additional data that captures homology cobordism information. This data could come from various sources, such as the linking form, the torsion part of Heegaard Floer homology, or other homology cobordism invariants. Higher order terms: The $\Delta_a$ invariant arises from the leading order term of the $bZ_a$ q-series. It's possible that information contained in higher-order terms of the q-series could be used to construct homology cobordism invariants. Generalizations of $bZ_a$ invariants: Exploring generalizations or variations of the $bZ_a$ invariants themselves, perhaps by considering different gauge theories or different types of "analytic continuations," might lead to new invariants with desired properties.

Q: Considering the intricate relationship between algebraic invariants and geometric structures, how can the study of Δa invariants shed light on the classification and properties of more general 3-manifolds beyond the weakly negative-definite plumbed case?

The study of $\Delta_a$ invariants, even though currently restricted to a specific class of 3-manifolds, can potentially offer valuable insights into the broader landscape of 3-manifold topology. Here's how: Testing ground for conjectures: Weakly negative-definite plumbed 3-manifolds provide a rich and tractable class to test conjectures about more general 3-manifolds. New relationships and properties discovered in this setting can inspire conjectures for broader classes. Building blocks: Many 3-manifolds can be constructed by gluing together simpler pieces, some of which might be weakly negative-definite plumbings. Understanding the behavior of invariants like $\Delta_a$ under such gluing operations could provide a way to extend their applicability. Connections to other theories: The connection between $\Delta_a$ invariants and Heegaard Floer homology suggests that these invariants might be related to other powerful tools in 3-manifold topology, such as Seiberg-Witten theory and quantum invariants. Exploring these connections could lead to new insights into the structure of 3-manifolds. New perspectives on classical invariants: Even for more general 3-manifolds, the ideas and techniques used to define and study $\Delta_a$ invariants might offer new perspectives on classical topological invariants or lead to the discovery of new ones. In summary, while the study of $\Delta_a$ invariants is still in its early stages, its connections to physics, q-series, and Heegaard Floer homology make it a promising area of research with the potential to significantly advance our understanding of 3-manifolds.

Core Concepts

This paper investigates the Δa invariants, a family of topological invariants associated with weakly negative-definite plumbed 3-manifolds, demonstrating that they are not homology cobordism invariants and exploring their relationship with the correction terms in Heegaard Floer homology.

Abstract

Bibliographic Information: Harichurn, S. (2024). On the Δa invariants in non-perturbative complex Chern-Simons theory. arXiv:2306.11298v2 [math.GT].
Research Objective: This paper aims to analyze the structure of the Δa invariants, a set of q-series invariants for weakly negative-definite plumbed 3-manifolds, and investigate their potential as homology cobordism invariants.
Methodology: The author employs mathematical tools from knot theory and low-dimensional topology, focusing on the properties of Brieskorn spheres and their associated invariants. They derive a formula for ∆0 on Brieskorn spheres and analyze its implications for homology cobordism.
Key Findings: The paper establishes that Δa invariants are not homology cobordism invariants, contrary to an open question in the field. It provides a complete description of the ∆0 invariants for Brieskorn spheres and explores the relationship between Δa invariants and the correction terms (d) in Heegaard Floer homology.
Main Conclusions: The study concludes that Δa invariants, while not homology cobordism invariants, offer valuable insights into the structure of 3-manifolds. The relationship between Δa and d suggests potential connections to other topological invariants and opens avenues for further research.
Significance: This research contributes to the understanding of topological invariants and their relationships in low-dimensional topology. The findings regarding Δa invariants and their connection to Heegaard Floer homology provide new tools and perspectives for studying 3-manifolds.
Limitations and Future Research: The paper primarily focuses on a subclass of integer homology spheres, leaving room for extending the analysis of Δa invariants to a broader class of 3-manifolds. The conjecture that Δa is not a Spinc homology cobordism invariant also presents an area for future investigation.

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Stats

bZ0(Y ; q) = −2q^(p−3)/4 for Y = L(p, 1) and Spinc structure 0.
bZ1(Y ; q) = q^(p−3)/4 (2q^(1/p)) for Y = L(p, 1) and Spinc structure 1.
∆0(Y ) = (p−3)/4 for Y = L(p, 1).
∆1(Y ) = (p^2−3p+1)/(4p) for Y = L(p, 1).
bZ0(S^3; q) = q^(-1/2) (2q −2).
∆0(S^3) = −1/2.

Quotes

Key Insights Distilled From

On the $\Delta_a$ invariants in non-perturbative complex Chern-Simons theory

by Shimal Haric... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2306.11298.pdf

$On the $\Delta_a$ invariants in non-perturbative complex Chern-Simons theory$

Deeper Inquiries

How can the relationship between Δa invariants and Heegaard Floer homology be further explored to uncover deeper connections between different topological invariants?

The observed congruence relation between the $\Delta_a$ invariants and the Heegaard Floer correction terms, $d(Y,a)$, hints at a potentially deeper connection that warrants further investigation. Here are some avenues for exploration:

Refining the congruence:  The current relation (Equation 2 in the text) is a modulo 1 congruence.  A natural question is whether this can be strengthened.  Are there families of manifolds where the difference $\Delta_a - (1/2 - d(Y,a))$ takes on specific values or exhibits interesting properties?  Exploring this could reveal finer connections between these invariants.

Beyond weakly negative-definite plumbings: The current results primarily focus on weakly negative-definite plumbed 3-manifolds. It would be highly valuable to investigate if similar relationships hold for broader classes of 3-manifolds. This would require understanding how to define or compute  $\Delta_a$ invariants in more general settings.

Connections to other Heegaard Floer invariants: Heegaard Floer homology is a rich theory with various flavors and related invariants (e.g.,  $\tau$ invariants,  knot Floer homology).  Investigating potential relationships between $\Delta_a$ and these other invariants could provide a more comprehensive picture of the interplay between complex Chern-Simons theory and Heegaard Floer homology.

Geometric interpretations:  A deeper understanding of the geometric meaning of both $\Delta_a$ invariants (potentially as some kind of "dimension" in a suitable sense) and the correction terms could provide insights into why this relationship exists.

Could there be a modified or refined version of the Δa invariant that does exhibit homology cobordism invariance, perhaps by incorporating additional topological data?

The fact that $\Delta_a$ invariants are not homology cobordism invariants is an important observation. However, it doesn't rule out the possibility of constructing related invariants that are. Here are some potential strategies:

Difference invariants: One could explore defining invariants based on differences of $\Delta_a$ invariants for different Spinc structures on the same manifold. It's conceivable that certain linear combinations or differences might become homology cobordism invariants.

Incorporating homology cobordism information:  Instead of directly modifying $\Delta_a$, one could try to construct a new invariant that combines $\Delta_a$ with additional data that captures homology cobordism information. This data could come from various sources, such as the linking form, the torsion part of Heegaard Floer homology, or other homology cobordism invariants.

Higher order terms: The $\Delta_a$ invariant arises from the leading order term of the  $bZ_a$ q-series. It's possible that information contained in higher-order terms of the q-series could be used to construct homology cobordism invariants.

Generalizations of  $bZ_a$ invariants:  Exploring generalizations or variations of the $bZ_a$ invariants themselves, perhaps by considering different gauge theories or different types of "analytic continuations," might lead to new invariants with desired properties.

Considering the intricate relationship between algebraic invariants and geometric structures, how can the study of Δa invariants shed light on the classification and properties of more general 3-manifolds beyond the weakly negative-definite plumbed case?

The study of $\Delta_a$ invariants, even though currently restricted to a specific class of 3-manifolds, can potentially offer valuable insights into the broader landscape of 3-manifold topology. Here's how:

Testing ground for conjectures:  Weakly negative-definite plumbed 3-manifolds provide a rich and tractable class to test conjectures about more general 3-manifolds.  New relationships and properties discovered in this setting can inspire conjectures for broader classes.

Building blocks: Many 3-manifolds can be constructed by gluing together simpler pieces, some of which might be weakly negative-definite plumbings. Understanding the behavior of invariants like $\Delta_a$ under such gluing operations could provide a way to extend their applicability.

Connections to other theories: The connection between $\Delta_a$ invariants and Heegaard Floer homology suggests that these invariants might be related to other powerful tools in 3-manifold topology, such as Seiberg-Witten theory and quantum invariants.  Exploring these connections could lead to new insights into the structure of 3-manifolds.

New perspectives on classical invariants:  Even for more general 3-manifolds, the ideas and techniques used to define and study $\Delta_a$ invariants might offer new perspectives on classical topological invariants or lead to the discovery of new ones.
In summary, while the study of $\Delta_a$ invariants is still in its early stages, its connections to physics, q-series, and Heegaard Floer homology make it a promising area of research with the potential to significantly advance our understanding of 3-manifolds.