The wrapped Fukaya category for semi-toric SYZ fibrations

TL;DR

This paper constructs the wrapped Fukaya category for semi-toric SYZ fibrations, computes its Floer cohomology, and establishes a key step towards proving homological mirror symmetry for certain Calabi-Yau manifolds.

Contribution

It introduces the wrapped Donaldson-Fukaya category for semi-toric SYZ fibrations and computes its Floer cohomology, linking it to mirror symmetry.

Findings

Wrapped Floer cohomology equals the algebra of functions on the mirror

Provides a framework for constructing wrapped Fukaya categories on open Calabi-Yau manifolds

Advances understanding of algebraic and analytic mirror symmetry aspects

Abstract

We introduce the wrapped Donaldson-Fukaya category of a (generalized) semi-toric SYZ fibration with Lagrangian section satisfying a tameness condition at infinity. Examples include the Gross fibration on the complement of an anti-canonical divisor in a toric Calabi-Yau 3-fold. We compute the wrapped Floer cohomology of a Lagrangian section and find that it is the algebra of functions on the Hori-Vafa mirror. The latter result is the key step in proving homological mirror symmetry for this case. The techniques developed here allow the construction in general of the wrapped Fukaya category on an open Calabi-Yau manifold carrying an SYZ fibration with nice behavior at infinity. We discuss the relation of this to the algebraic vs analytic aspects of mirror symmetry.