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Noncommutative Homological Mirror Functor
About this Title
Cheol-Hyun Cho, Hansol Hong and Siu-Cheong Lau
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 271, Number 1326
ISBNs: 978-1-4704-4761-8 (print); 978-1-4704-6630-5 (online)
DOI: https://doi.org/10.1090/memo/1326
Published electronically: June 25, 2021
Keywords: Fukaya category,
homological mirror symmetry,
noncommutative algebra,
Landau-Ginzburg model,
deformation quantization
Table of Contents
Chapters
- 1. Introduction
- 2. A-infinity algebra over a noncommutative base
- 3. Noncommutative mirror from a single Lagrangian and the centrality theorem
- 4. The mirror functor from a single Lagrangian
- 5. Elliptic curves and deformation quantizations
- 6. Mirror construction using several Lagrangians and quiver algebras
- 7. Finite group symmetry and graded mirror functors
- 8. 4-punctured spheres and the pillowcase
- 9. Extended mirror functor
- 10. Mirror construction for punctured Riemann surface
- 11. Mirrors of Calabi-Yau threefolds associated with quadratic differentials
- A. Theta function calculations
Abstract
We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results in terms of mirror symmetry and symplectic geometry. As an interesting application, we construct an explicit global deformation quantization of an affine del Pezzo surface as a noncommutative mirror to an elliptic orbifold.- Mohammed Abouzaid, Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov, and Dmitri Orlov, Homological mirror symmetry for punctured spheres, J. Amer. Math. Soc. 26 (2013), no. 4, 1051–1083. MR 3073884, DOI 10.1090/S0894-0347-2013-00770-5
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