Quantum Mirror Symmetry for Legendrian Surfaces Associated to Cubic Planar Graphs

Core Concepts

This research paper proves a quantum mirror symmetry relation for a class of Legendrian surfaces in the five-sphere associated with cubic planar graphs, showing that certain skein-valued operator equations annihilate the holomorphic curve invariants of any filling.

Abstract

• Bibliographic Information: Scharitzer, M., & Shende, V. (2024). Quantum Mirrors of Cubic Planar Graph Legendrians. arXiv:2304.01872v2 [math.SG].
• Research Objective: This paper aims to determine the quantum mirrors for a class of asymptotically conic Lagrangians in C3, specifically those defined by taking boundary connect sums of Harvey-Lawson branes as constructed by Treumann and Zaslow.
• Methodology: The authors utilize the framework of skein-valued holomorphic curve invariants and the relationship between open topological string theory and Chern-Simons theory. They analyze Morse flow trees associated with the Legendrian surfaces and employ techniques from symplectic geometry and contact topology.
• Key Findings: The authors prove that the skein-valued holomorphic curve invariants of any Lagrangian filling of the Legendrian boundary are annihilated by certain explicit skein-valued operator equations. These operator equations, referred to as "face relations," are defined in terms of the skein module of the Legendrian surface.
• Main Conclusions: The paper establishes a quantum mirror symmetry relation for the considered class of Legendrian surfaces. This result provides a mathematical framework for understanding the correspondence between the A-model and B-model sides of mirror symmetry in the context of open Gromov-Witten theory.
• Significance: This research contributes to the understanding of quantum mirror symmetry and its connections to skein theory, contact geometry, and open Gromov-Witten invariants. It extends previous work on the Harvey-Lawson brane and provides a framework for studying more general classes of Legendrian surfaces.
• Limitations and Future Research: The authors acknowledge that they do not solve the derived operator equations in this paper. They suggest that a potential approach for solving these equations in the full skein module could involve developing a skein-valued analogue of cluster algebras. Additionally, they highlight the need for an explicit algebraic expression for the action of the skein module of the boundary on the skein module of the filling.

Stats

Quotes

"By quantum mirror symmetry, we mean the phenomenon where the all genus partition function on the A-model side of mirror symmetry is a ‘wave function for’, i.e. is annihilated by, certain operators which quantize the moduli space of the B-model mirror."
"The fundamental observation is that the boundaries of the moduli space of holomorphic maps from curves-with-boundary can be grouped together in such a way that in each group, we meet a collection of curves-with-boundary whose boundaries themselves satisfy the HOMFLYPT skein relation."

Key Insights Distilled From

Quantum mirrors of cubic planar graph Legendrians

by Matthias Sch... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2304.01872.pdf

Deeper Inquiries

How might the skein-valued operator equations derived in this paper be utilized to compute other topological invariants of the associated Legendrian surfaces or their fillings?

The skein-valued operator equations, encapsulating the quantum mirror symmetry properties of these Legendrian surfaces, offer a powerful tool for computing various topological invariants. Here's how:
1. Recursively Determining Invariants:

Skein-Valued Partition Function: The operator equations annihilate the skein-valued partition function, which encodes a wealth of information about the holomorphic curves ending on the Legendrian. By systematically solving these equations, one can recursively determine the coefficients of the partition function.
Extracting Other Invariants: These coefficients, in turn, are related to other topological invariants. For instance, specializing the HOMFLYPT skein to the Jones polynomial by setting a = q^2 allows us to extract the colored Jones polynomials of the Legendrian knot. Similarly, other specializations or manipulations of the skein relations can lead to other invariants like the Alexander polynomial or the Khovanov homology.
2. Bridging to Other Theories:

Quantum Cluster Algebra: The connection to quantum cluster algebra, as hinted in the paper, suggests a potential avenue for computing invariants. The operator equations, when translated into the language of quantum cluster algebra, might provide new combinatorial or algebraic tools for their computation.
Augmentation Variety and Wrapped Fukaya Category: The operators are essentially quantizations of the augmentation variety of the Legendrian DGA. This connection opens the door to leveraging techniques from symplectic geometry and the Fukaya category to study the invariants. The wrapped Fukaya category, in particular, provides a powerful framework for understanding the topology of Lagrangian fillings.
3. Exploring New Relationships:

Skein-Valued Cluster Algebra: The paper suggests the possibility of a "skein-valued cluster algebra." If such a structure exists, the operator equations could provide defining relations or key properties, leading to new methods for computing invariants within this framework.
Action of Skein Modules: Understanding the explicit algebraic action of the skein module of the boundary on the skein module of the filling could unveil further relationships between the operator equations and other invariants.
In essence, the skein-valued operator equations provide a bridge between the world of holomorphic curves and the world of topological invariants. By carefully navigating this bridge, we can gain deeper insights into the topology of Legendrian surfaces and their fillings.

Could there be alternative geometric constructions or interpretations of these Legendrian surfaces that provide further insights into their quantum mirror symmetry properties?

Yes, exploring alternative geometric constructions or interpretations of these Legendrian surfaces could unlock a deeper understanding of their quantum mirror symmetry. Here are some potential avenues:
1. Legendrian Mutations and Cobordisms:

Mutations: Investigating how these Legendrian surfaces behave under Legendrian mutations, which are specific types of surgeries, could reveal interesting transformations of the corresponding operator equations and their solutions. This could provide a combinatorial handle on the quantum mirror symmetry.
Cobordisms: Constructing Legendrian cobordisms between different Legendrian surfaces in this class could lead to relations between their corresponding skein-valued partition functions. This could provide a geometric interpretation of the operator equations and their solutions in terms of cobordism invariants.
2. Connections to Contact Topology:

Open Book Decompositions: These Legendrian surfaces are naturally associated with open book decompositions of the 5-sphere. Studying the relationship between the properties of the open book and the operator equations could provide new insights. For example, the monodromy of the open book might be reflected in the structure of the operators.
Contact Homology: Exploring potential connections between the skein-valued curve counting and contact homology invariants, such as cylindrical contact homology, could offer a different perspective on the quantum mirror symmetry.
3. Generalizations and Variations:

Higher Dimensions: Generalizing the construction of these Legendrian surfaces to higher-dimensional contact manifolds could lead to new classes of Legendrian submanifolds with interesting quantum mirror symmetry properties.
Different Ambient Spaces: Instead of the 5-sphere, embedding these Legendrian surfaces in different contact manifolds could lead to variations in their quantum mirror symmetry, potentially revealing new aspects of the theory.
4. Mirror Symmetry Perspective:

B-Model Geometry: Investigating the mirror B-model geometry corresponding to these Legendrian surfaces could provide a completely different perspective on the operator equations. The operators might have a natural interpretation in terms of the mirror geometry.
Mirror Symmetry Functor: Understanding how the skein-valued curve counting, and hence the operator equations, transform under the mirror symmetry functor could provide deeper insights into the nature of the correspondence.
By exploring these alternative constructions and interpretations, we can hope to uncover hidden symmetries, relationships, and structures that shed further light on the fascinating interplay between Legendrian geometry and quantum mirror symmetry.

How does the concept of "framing" in the context of quantum tori and skein modules relate to the broader notion of framing in other areas of mathematics and physics, such as knot theory or quantum field theory?

The concept of "framing" in the context of quantum tori and skein modules is deeply intertwined with the broader notion of framing in other areas of mathematics and physics. Let's draw some connections:
1. Knot Theory:

Framing of Knots: In knot theory, a framing of a knot is a choice of a nonzero normal vector field along the knot. This framing is crucial for defining invariants like the framing-dependent version of the Jones polynomial.
Skein Relations and Framing: The skein relations themselves, which define the skein module, inherently encode information about framing. The relation
- = a - a^-1 z
reflects how the knot invariant changes under a change of framing.
2. Quantum Tori and Linking Number:

Framing and Linking: The framing lines in the skein module, as described in the paper, directly affect the linking number between knots. Crossing a framing line changes the framing of a knot, thereby altering the linking number.
Quantum Torus Representation: This change in linking number is precisely captured by the commutation relations in the quantum torus. The relation
A_i B_j = q B_j A_i
reflects how the generators of the quantum torus, representing loops on a surface, interact with each other, mirroring the change in linking number under crossing.
3. Quantum Field Theory:

Wilson Lines and Framing: In Chern-Simons theory, a quantum field theory whose partition function is closely related to knot invariants, Wilson lines are fundamental objects. The framing of a knot corresponds to a choice of framing for the Wilson line, which affects the value of the Wilson line observable.
Skein Modules and Quantization: The skein module can be viewed as a quantization of the algebra of functions on the moduli space of flat connections. In this context, framing appears as a choice of polarization, which is necessary for the quantization procedure.
4. Common Threads:

Ambiguity in Self-Intersection: The common theme across these areas is the need to resolve the ambiguity in defining self-intersection. In knot theory, framing resolves the ambiguity of a knot intersecting itself. In quantum tori, it clarifies how loops on a surface intersect. In quantum field theory, it specifies how to compute self-linking of Wilson lines.
Global vs. Local Information: Framing bridges the gap between local and global information. Locally, we have the skein relations governing crossings. Globally, framing determines how these local relations combine to give a well-defined invariant.
In summary, the concept of "framing" in the context of quantum tori and skein modules is not merely an artifact of the formalism but rather a reflection of a fundamental concept that permeates various areas of mathematics and physics. It highlights the subtle interplay between local and global information, providing a framework for understanding and computing topological invariants.