Homological mirror symmetry without corrections

Abstract

Let $X$ be a closed symplectic manifold equipped a Lagrangian torus fibration over a base $Q$. A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space $Y$, which can be considered as a variant of the $T$-dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of tautologically unobstructed graded Lagrangians in $X$ embeds fully faithfully in the derived category of (twisted) coherent sheaves on $Y$, under the technical assumption that ${\pi }_2(Q)$ vanishes (all known examples satisfy this assumption). The main new tool is the construction and computation of Floer cohomology groups of Lagrangian fibres equipped with topological infinite rank local systems that correspond, under mirror symmetry, to the affinoid rings introduced by Tate, equipped with their natural topologies as Banach algebras.

116 pages, 13 figures. Final version prior to publication

References in corpus (2)

• Family Floer cohomology and mirror symmetry
• Rabinowitz Floer homology and mirror symmetry

Cited by in corpus (7)

• The canonical wall structure and intrinsic mirror symmetry
• Family Floer program and non-archimedean SYZ mirror construction
• Involutions, obstructions and mirror symmetry
• On the Complex Affine Structures of SYZ Fibration of Del Pezzo Surfaces
• Complex K-theory of mirror pairs
• The wrapped Fukaya category for semi-toric SYZ fibrations
• Cyclic group actions on Fukaya categories and mirror symmetry