Fukaya A-infinity structure near infinity and the categorical formal completion

TL;DR

This paper constructs a symplectic analogue of the formal neighborhood of a stop in Liouville manifolds using Floer theory, and proves its equivalence to a categorical formal completion, advancing understanding in homological mirror symmetry.

Contribution

It establishes the equivalence between Floer-theoretic and categorical formal neighborhoods in Liouville sectors, connecting geometric and algebraic approaches.

Findings

Proves the equivalence of two formal neighborhood constructions.

Demonstrates computability of Floer theory via categorical methods.

Contributes to computations in homological mirror symmetry.

Abstract

For a stopped Liouville manifold arising from a Liouville sector, we construct a symplectic analogue of the formal neighborhood of the stop on the level of Fukaya categories. This geometric construction is performed via Floer-theoretic methods by allowing wrappings in the negative direction. On the other hand, inspired by homological mirror symmetry for pairs, where the mirror is the formal neighborhood of a divisor in an ambient projective variety, there is a different approach by taking a `categorical formal completion' introduced by Efimov. Our main results establishes equivalence of these two approaches, confirms computability of this new type of Floer theory by categorical and algebraic means, and indicates contributions from and to computations in homological mirror symmetry.