Fukaya categories of plumbings and multiplicative preprojective algebras

Abstract

Given an arbitrary graph $\Gamma$ and non-negative integers $g_v$ for each vertex $v$ of $\Gamma$, let $X_{\Gamma }$ be the Weinstein 4-manifold obtained by plumbing copies of $T^{\ast }{\Sigma }_v$ according to this graph, where ${\Sigma }_v$ is a surface of genus $g_v$. We compute the wrapped Fukaya category of $X_{\Gamma }$ (with bulk parameters) using Legendrian surgery extending our previous work [14] where it was assumed that $g_v=0$ for all $v$ and $\Gamma$ 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.