The family Floer functor is faithful

TL;DR

This paper proves that the family Floer functor, connecting the Fukaya category to perfect complexes on the mirror space, is faithful using a degeneration argument involving annuli moduli spaces.

Contribution

It establishes the faithfulness of the family Floer functor, a key step in understanding mirror symmetry for symplectic manifolds with Lagrangian torus fibrations.

Findings

The functor is faithful, preserving morphisms accurately.

Degeneration techniques involving annuli moduli spaces are effective.

Supports the mirror symmetry conjecture for certain symplectic manifolds.

Abstract

Family Floer theory yields a functor from the Fukaya category of a symplectic manifold admitting a Lagrangian torus fibration to a (twisted) category of perfect complexes on the mirror rigid analytic space. This functor is shown to be faithful by a degeneration argument involving moduli spaces of annuli.