# Fukaya categories of plumbings and multiplicative preprojective algebras

### Tolga Etgü

Koç University, Istanbul, Turkey### Yankı Lekili

King's College London, UK

## Abstract

Given an arbitrary graph $Γ$ and non-negative integers $g_{v}$ for each vertex $v$ of $Γ$, let $X_{Γ}$ be the Weinstein 4-manifold obtained by plumbing copies of $T_{∗}Σ_{v}$ according to this graph, where $Σ_{v}$ is a surface of genus $g_{v}$. We compute the wrapped Fukaya category of $X_{Γ}$ (with bulk parameters) using Legendrian surgery extending our previous work [14] where it was assumed that $g_{v}=0$ for all $v$ and $Γ$ was a tree. The resulting algebra is recognized as the (derived) multiplicative preprojective algebra (and its higher genus version) defined by Crawley-Boevey and Shaw [8]. Along the way, we find a smaller model for the internal DG-algebra of Ekholm and Ng [12] associated to 1-handles in the Legendrian surgery presentation of Weinstein 4-manifolds which might be of independent interest.

## Cite this article

Tolga Etgü, Yankı Lekili, Fukaya categories of plumbings and multiplicative preprojective algebras. Quantum Topol. 10 (2019), no. 4, pp. 777–813

DOI 10.4171/QT/131