Fukaya categories of plumbings and multiplicative preprojective algebras

Given an arbitrary graph $\mathrm{\Gamma }$ and non-negative integers $g_v$ for each vertex $v$ of $\mathrm{\Gamma }$, let $X_{\mathrm{\Gamma }}$ be the Weinstein $4$-manifold obtained by plumbing copies of $T^{\ast }{\mathrm{\Sigma }}_v$ according to this graph, where ${\mathrm{\Sigma }}_v$ is a surface of genus $g_v$. We compute the wrapped Fukaya category of $X_{\mathrm{\Gamma }}$ (with bulk parameters) using Legendrian surgery extending our previous work arXiv:1502.07922 where it was assumed that $g_v=0$ for all $v$ and $\mathrm{\Gamma }$ 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 arXiv:math/0404186. Along the way, we find a smaller model for the internal DG-algebra of Ekholm-Ng arXiv:1307.8436 associated to $1$-handles in the Legendrian surgery presentation of Weinstein $4$-manifolds which might be of independent interest.