insight - Scientific Computing - # Homological Mirror Symmetry

Homological Mirror Symmetry for a Class of Calabi-Yau Hypersurfaces in Toric Varieties

Q: Could there be alternative approaches to proving homological mirror symmetry for these Batyrev mirror pairs that circumvent the reliance on the specific assumptions about the relative Fukaya category?

Yes, there could be alternative approaches to proving HMS for Batyrev mirror pairs that bypass some of the assumptions made about the relative Fukaya category in this paper. Here are a few possibilities: Microlocal Sheaf Theory: Instead of the Fukaya category, one could work with the category of microlocal sheaves on the symplectic side. This approach has been successful in proving HMS for toric varieties [Kuw20, GS22] and doesn't require the same assumptions on the Fukaya category. However, it would still require carefully relating the microlocal sheaf category to the derived category of coherent sheaves on the mirror. SYZ Fibrations: One could try to construct an SYZ fibration for the Batyrev mirror pair. This would involve finding a special Lagrangian torus fibration on the Calabi-Yau manifold and relating its geometry to the mirror. While powerful, constructing SYZ fibrations is generally very difficult. Deformation Theory: One could try to deform the Batyrev mirror pair to a simpler case where HMS is already known, and then use deformation theory to deduce HMS for the original pair. This approach has been successful in other contexts, but it would require a good understanding of the deformation theory of both the A-side and B-side categories. Non-commutative Geometry: One could explore approaches based on non-commutative geometry, where the B-side category is replaced by a suitable non-commutative resolution of the singular mirror. This is a more abstract approach, but it might offer new insights. It's important to note that each of these alternative approaches comes with its own set of challenges. However, they represent promising directions for future research and could potentially lead to proofs of HMS in greater generality.

Core Concepts

The authors prove Kontsevich's homological mirror symmetry conjecture for a new class of mirror pairs of Calabi-Yau hypersurfaces in toric varieties, constructed by Batyrev from dual reflexive polytopes.

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arxiv.org

Ganatra, S., Hanlon, A., Hicks, J., Pomerleano, D., & Sheridan, N. (2024). Homological mirror symmetry for Batyrev mirror pairs. arXiv preprint arXiv:2406.05272v2.

This research paper aims to prove Kontsevich's homological mirror symmetry conjecture for a specific class of mirror pairs, namely Calabi-Yau hypersurfaces in toric varieties constructed by Batyrev using dual reflexive polytopes. The authors focus on establishing this equivalence in both characteristic zero and all but finitely many positive characteristics.

Key Insights Distilled From

Homological mirror symmetry for Batyrev mirror pairs

by Sheel Ganatr... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2406.05272.pdf

Homological mirror symmetry for Batyrev mirror pairs

Deeper Inquiries

How do the techniques used in this paper extend to other constructions of mirror pairs, such as Batyrev-Borisov mirror pairs or Gross-Siebert mirror pairs?

Extending the techniques of this paper to more general mirror pair constructions like Batyrev-Borisov or Gross-Siebert is a challenging but potentially fruitful endeavor. Here's a breakdown of the key obstacles and possible avenues:
Batyrev-Borisov Mirror Pairs:

Increased Complexity: Batyrev-Borisov pairs generalize to complete intersections in toric varieties, not just hypersurfaces. This introduces additional complexities on both the A-side (constructing appropriate Lagrangian submanifolds) and the B-side (dealing with derived categories of higher-dimensional varieties).
Nef Partitions:  Batyrev-Borisov duality relies on "nef partitions" of the polytope's vertices. Adapting the "arclike" Lagrangian and geometric cap functor constructions to this setting would be crucial.
Potential Approach: One could try to decompose the complete intersection into simpler pieces, prove HMS for each piece, and then glue the results together. This strategy has been successful in other contexts, but it's unclear if it's feasible here.
Gross-Siebert Mirror Pairs:

Toric Degenerations: Gross-Siebert mirror symmetry deals with degenerating families of Calabi-Yau varieties, not just individual ones. This requires understanding how the Fukaya category behaves under such degenerations, which is still an active area of research.
Tropical Geometry:  The Gross-Siebert program heavily utilizes tropical geometry.  Connecting the symplectic constructions in this paper to the tropical geometry of the base of the degeneration would be essential.
Potential Approach: One could try to adapt the techniques of "family Floer homology" to study how the Fukaya category changes along the degeneration. This is a highly technical area, but it might provide the necessary tools.
General Challenges:

Smoothness Assumptions: The paper relies heavily on smoothness assumptions for both the toric variety and the hypersurface. Relaxing these assumptions to handle more singular cases is a major challenge.
Characteristic Dependence: The proof works in characteristic zero and all but finitely many positive characteristics. Extending it to all characteristics would require new ideas.
In summary, while directly applying the techniques to Batyrev-Borisov or Gross-Siebert pairs is not straightforward, the paper provides a blueprint and highlights key areas where further development could lead to progress in these more general settings.

Could there be alternative approaches to proving homological mirror symmetry for these Batyrev mirror pairs that circumvent the reliance on the specific assumptions about the relative Fukaya category?

Yes, there could be alternative approaches to proving HMS for Batyrev mirror pairs that bypass some of the assumptions made about the relative Fukaya category in this paper. Here are a few possibilities:

Microlocal Sheaf Theory:  Instead of the Fukaya category, one could work with the category of microlocal sheaves on the symplectic side. This approach has been successful in proving HMS for toric varieties [Kuw20, GS22] and doesn't require the same assumptions on the Fukaya category. However, it would still require carefully relating the microlocal sheaf category to the derived category of coherent sheaves on the mirror.
SYZ Fibrations:  One could try to construct an SYZ fibration for the Batyrev mirror pair. This would involve finding a special Lagrangian torus fibration on the Calabi-Yau manifold and relating its geometry to the mirror. While powerful, constructing SYZ fibrations is generally very difficult.
Deformation Theory:  One could try to deform the Batyrev mirror pair to a simpler case where HMS is already known, and then use deformation theory to deduce HMS for the original pair. This approach has been successful in other contexts, but it would require a good understanding of the deformation theory of both the A-side and B-side categories.
Non-commutative Geometry:  One could explore approaches based on non-commutative geometry, where the B-side category is replaced by a suitable non-commutative resolution of the singular mirror. This is a more abstract approach, but it might offer new insights.
It's important to note that each of these alternative approaches comes with its own set of challenges. However, they represent promising directions for future research and could potentially lead to proofs of HMS in greater generality.

What are the potential implications of this research for related fields, such as string theory or mathematical physics, where mirror symmetry plays a crucial role?

This research on homological mirror symmetry for Batyrev mirror pairs holds significant potential implications for related fields where mirror symmetry plays a crucial role, particularly string theory and mathematical physics:
String Theory:

Predictions and Dualities: HMS provides a mathematical framework for understanding string theory dualities, where two seemingly different physical theories are actually equivalent. This research, by proving HMS for a new class of Calabi-Yau manifolds, could lead to new predictions about string dualities and a deeper understanding of the relationships between different string theories.
Enumerative Geometry: Mirror symmetry has been incredibly successful in making predictions in enumerative geometry, such as counting rational curves on Calabi-Yau manifolds. This research could lead to new techniques for performing such calculations and potentially new enumerative invariants.
Beyond Calabi-Yau: The techniques developed in this paper might be adaptable to study HMS for non-Calabi-Yau manifolds, which are relevant to certain string theory compactifications. This could open up new avenues for exploring string theory in more general settings.
Mathematical Physics:

Quantum Field Theories: Mirror symmetry has connections to the study of supersymmetric quantum field theories. This research could provide new insights into the structure of these theories and their moduli spaces.
Integrable Systems:  HMS is related to the theory of integrable systems, which are important in many areas of physics. This research could lead to new connections between integrable systems and algebraic geometry.
Derived Categories and Physics: The use of derived categories in this research highlights their importance in modern mathematical physics. This could inspire further applications of derived categories to other areas of physics.
Broader Impact:

New Mathematical Tools: The techniques developed in this research, such as the use of relative Fukaya categories and the geometric cap functor, are likely to have applications beyond HMS, enriching the toolkit for studying symplectic geometry and algebraic geometry.
Interdisciplinary Research: This research strengthens the bridge between mathematics and physics, fostering further collaboration and cross-fertilization of ideas between these fields.
In conclusion, this research on HMS for Batyrev mirror pairs has the potential to deepen our understanding of fundamental concepts in string theory and mathematical physics, leading to new predictions, connections, and mathematical tools with far-reaching implications.