Mapping tori of $A_{\infty}$-autoequivalences and Legendrian lifts of exact Lagrangians in circular contactizations

TL;DR

This paper investigates the structure of mapping tori of certain $A_{ abla}$-auto-equivalences and applies these findings to connect the Fukaya category of exact Lagrangians with Legendrian lifts in circular contactizations.

Contribution

It provides explicit computations of mapping tori for strict and bijective $A_{ abla}$-auto-equivalences and links Fukaya categories with Legendrian lift categories in contact geometry.

Findings

Computed mapping tori for strict auto-equivalences.

Established connections between Fukaya categories and Legendrian lifts.

Applied theoretical results to specific geometric contexts.

Abstract

We study mapping tori of quasi-autoequivalences $\tau : \mathcal{A} \to \mathcal{A}$ which induce a free action of $\mathbf{Z}$ on objects. More precisely, we compute the mapping torus of $\tau$ when it is strict and acts bijectively on hom-sets, or when the $A_{\infty}$-category $\mathcal{A}$ is directed and there is a bimodule map $\mathcal{A} (-, -) \to \mathcal{A} (-, \tau (-))$ satisfying some hypotheses. Then we apply these results in order to link together the Fukaya $A_{\infty}$-category of a family of exact Lagrangians, and the Chekanov-Eliashberg DG-category of Legendrian lifts in the circular contactization.