Lagrangian sections on mirrors of toric Calabi-Yau 3-folds

Abstract

We construct Lagrangian sections of a Lagrangian torus fibration on a 3-dimensional conic bundle, which are SYZ dual to holomorphic line bundles over the mirror toric Calabi-Yau 3-fold. We then demonstrate a ring isomorphism between the wrapped Floer cohomology of the zero-section and the regular functions on the mirror toric Calabi-Yau 3-fold. Furthermore, we show that in the case when the Calabi-Yau 3-fold is affine space, the zero section generates the wrapped Fukaya category of the mirror conic bundle. This allows us to complete the proof of one direction of homological mirror symmetry for toric Calabi-Yau orbifold quotients of the form $\mathbb{C}^3/\Check{G}$. We finish by describing some elementary applications of our computations to symplectic topology.

62 pages. v2: corrected errors in various formulas; added the proof of homological mirror symmetry for C^3/G and some applications to Lagrangian embeddings

References in corpus (1)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor

Cited by in corpus (6)

• A Log PSS morphism with applications to Lagrangian embeddings
• Aspects of functoriality in homological mirror symmetry for toric varieties
• Intrinsic mirror symmetry and categorical crepant resolutions
• Tropical Lagrangian multi-sections and toric vector bundles
• Reconstruction of $T_{{\mathbb{P}}^2}$ via tropical Lagrangian multi-section
• The wrapped Fukaya category for semi-toric SYZ fibrations