Partially wrapped Fukaya categories of orbifold surfaces

TL;DR

This paper provides a comprehensive description of partially wrapped Fukaya categories for graded orbifold surfaces, establishing local-to-global properties and explicit algebraic models, thus extending Fukaya category theory to orbifold settings.

Contribution

It introduces a new global construction for orbifold Fukaya categories via cosheaves and orbit categories, and classifies their generators, connecting to Weinstein sectorial descent.

Findings

Equivalence of cosheaf and orbit category constructions.

Orbifold surfaces admit Weinstein sectors of type D.

Fukaya categories can be described as derived categories of graded algebras.

Abstract

We give a complete description of partially wrapped Fukaya categories of graded orbifold surfaces with stops. We show that a construction via global sections of a natural cosheaf of A$_\infty$ categories on a Lagrangian core of the surface is equivalent to a global construction via the (equivariant) orbit category of a smooth cover. We therefore establish the local-to-global properties of partially wrapped Fukaya categories of orbifold surfaces closely paralleling a proposal by Kontsevich for Fukaya categories of smooth Weinstein manifolds. From the viewpoint of Weinstein sectorial descent in the sense of Ganatra, Pardon and Shende, our results show that orbifold surfaces also have Weinstein sectors of type $\mathrm D$ besides the type $\mathrm A$ or type $\widetilde{\mathrm A}$ sectors on smooth surfaces. We describe the global sections of the cosheaf explicitly for any generator given by an admissible dissection of the orbifold surface and we give a full classification of the formal generators which arise in this way. This shows in particular that the partially wrapped Fukaya category of an orbifold surface can always be described as the perfect derived category of a graded associative algebra. We conjecture that associative algebras obtained from dissections of orbifold surfaces form a new class of associative algebras closed under derived equivalence.