Noncommutative homological mirror functor

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.

arXiv comment114 pages, 21 figures; Final version published in Memoirs of the American Mathematical Society on June 25, 2021

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• Mirror symmetry, Langlands duality, and the Hitchin system
• Compact generators in categories of matrix factorizations
• Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves
• Computation of Superpotentials for D-Branes
• Homogeneous coordinate rings and mirror symmetry for toric varieties
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• Matrix factorizations via Koszul duality
• Lagrangian Floer superpotentials and crepant resolutions for toric orbifolds
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Cited by in corpus (7)

• Immersed two-spheres and SYZ with Application to Grassmannians
• LG/CY Correspondence for Elliptic Orbifold Curves via Modularity
• Pairings in mirror symmetry between a symplectic manifold and a Landau-Ginzburg $B$-model
• Moduli of Lagrangian immersions with formal deformations
• Maurer-Cartan deformation of Lagrangians
• Affine elliptic surfaces with type-A singularities and orbi-conifolds
• Kodaira-Spencer map, Lagrangian Floer theory and orbifold Jacobian algebras