Recollements of partially wrapped Fukaya categories and surface cuts

TL;DR

This paper explores how cutting surfaces affects the structure of partially wrapped Fukaya categories using recollements, linking geometric surface modifications to algebraic category decompositions.

Contribution

It demonstrates that surface cuts induce recollements in Fukaya categories and characterizes the existence of full exceptional sequences and silting objects within this framework.

Findings

Cutting surfaces yields recollements of Fukaya categories.

Characterized when full exceptional sequences exist.

Provided an example lacking silting objects.

Abstract

In this paper we use recollements to investigate partially wrapped Fukaya categories of surfaces with marked points. In particular, we show that cutting surfaces gives rise to recollements of the corresponding partially wrapped Fukaya categories. Our approach is based on the fact that the partially wrapped Fukaya category of a surface with marked points is triangle equivalent to the perfect derived category of a homologically smooth and proper graded gentle algebra with zero differential as shown by Haiden, Katzarkov and Kontsevich. Using this, we study particular generators of partially wrapped Fukaya categories, namely full exceptional sequences, silting objects and simple-minded collections. In particular, we fully characterise the existence of full exceptional sequences and we give an example of a partially wrapped Fukaya category which does not admit a silting object, that is a generator with no positive self-extensions.