Proper modules over Ginzburg dg algebras and compact Fukaya categories of plumbings

TL;DR

This paper explores the relationship between Ginzburg dg algebras and the compact Fukaya category of plumbing spaces, providing generation results and categorical equivalences in symplectic topology.

Contribution

It introduces a generation result for proper modules over Ginzburg dg algebras without Jacobi-finiteness and establishes equivalences with Fukaya categories and microlocal sheaves.

Findings

Proper modules generate all proper modules over Ginzburg dg algebras.

The compact Fukaya category is generated by immersed objects.

Equivalence between the Fukaya category and categories of proper modules and microlocal sheaves.

Abstract

We study Ginzburg dg algebras which appear at the intersection of representation theory and symplectic topology. First, we provide a collection of proper modules that generates all proper modules over a Ginzburg dg algebra, without assuming the Jacobi-finite condition. Using this generation result, we study the immersed compact Fukaya category of a general plumbing space. In particular, we prove a generation result for the compact Fukaya category and show that it is equivalent to the category of proper modules over the wrapped Fukaya category, and hence to the category of microlocal sheaves on the Lagrangian skeleton.