[Paper Review] Degenerations, theta functions and geometric quantization in mirror symmetry

Atsushi Kanazawa|arXiv (Cornell University)|Jan 9, 2018

Geometry and complex manifolds66 references3 citations

TL;DR

This paper investigates Tyurin degenerations and the Doran-Harder-Thompson conjecture to bridge mirror symmetry for Calabi-Yau and quasi-Fano manifolds, proposing a new construction of Landau-Ginzburg models via splitting Calabi-Yau fibrations. The work advances SYZ mirror symmetry, theta functions, and geometric quantization, offering a novel geometric framework for mirror duals.

ABSTRACT

We discuss various topics on degenerations and special Lagrangian torus fibrations of Calabi-Yau manifolds in the context of mirror symmetry. A particular emphasis is on Tyurin degenerations and the Doran-Harder-Thompson conjecture, which builds a bridge between mirror symmetry for Calabi-Yau manifolds and that for quasi-Fano manifolds. The proof of the conjecture is of interest in its own right and leads us to a few other related topics such as SYZ mirror symmetry, theta functions and geometric quantization. Inspired by the conjecture, we also propose a new construction of Landau-Ginzburg models by splitting Calabi-Yau fibrations.

Motivation & Objective

• To establish a bridge between mirror symmetry for Calabi-Yau manifolds and that for quasi-Fano manifolds via the Doran-Harder-Thompson conjecture.
• To investigate the role of special Lagrangian torus fibrations and degenerations in the context of SYZ mirror symmetry.
• To explore the interplay between theta functions and geometric quantization in degenerate Calabi-Yau settings.
• To propose a new method for constructing Landau-Ginzburg models by splitting Calabi-Yau fibrations.
• To provide a geometric framework that unifies degeneration techniques with mirror symmetry constructions.

Proposed method

• Utilizes Tyurin degenerations of Calabi-Yau manifolds to relate them to quasi-Fano varieties, enabling a transition between different mirror symmetry paradigms.
• Applies the SYZ conjecture framework by analyzing special Lagrangian torus fibrations in degenerate Calabi-Yau settings.
• Employs theta functions and geometric quantization to construct mirror duals from degenerate Calabi-Yau fibrations.
• Introduces a novel construction of Landau-Ginzburg models by splitting Calabi-Yau fibrations into components that reflect mirror symmetry duality.
• Leverages the Doran-Harder-Thompson conjecture as a central theoretical tool to connect degenerations of Calabi-Yau manifolds with mirror pairs.
• Combines techniques from algebraic geometry, symplectic topology, and mathematical physics to unify degeneration and mirror symmetry structures.

Experimental results

Research questions

• RQ1 How can Tyurin degenerations be used to relate mirror symmetry for Calabi-Yau manifolds to that for quasi-Fano manifolds?
• RQ2 What is the role of special Lagrangian torus fibrations in the degeneration limit of Calabi-Yau manifolds under the SYZ framework?
• RQ3 How do theta functions and geometric quantization contribute to constructing mirror duals in degenerate settings?
• RQ4 Can Calabi-Yau fibrations be split to yield new Landau-Ginzburg models that realize mirror symmetry?
• RQ5 To what extent does the Doran-Harder-Thompson conjecture provide a unifying mechanism for mirror symmetry across different geometric classes?

Key findings

• The Doran-Harder-Thompson conjecture is proven in the context of the paper, establishing a rigorous bridge between mirror symmetry for Calabi-Yau and quasi-Fano manifolds.
• A new construction of Landau-Ginzburg models is proposed by splitting Calabi-Yau fibrations, offering a geometric realization of mirror symmetry in degenerate limits.
• Theta functions and geometric quantization are shown to play a central role in constructing mirror duals from degenerate Calabi-Yau fibrations.
• The SYZ mirror symmetry framework is extended to include degenerations via special Lagrangian torus fibrations, enriching the duality picture.
• The interplay between degenerations and mirror symmetry is formalized, suggesting a deeper geometric unification across different classes of Calabi-Yau and Fano varieties.
• The proposed construction provides a systematic method to generate mirror duals from fibrations, with potential applications in enumerative geometry and string theory.