Fukaya categories of plumbings and multiplicative preprojective algebras

Given an arbitrary graph г and non-negative integers g_v for each vertex v of г, let X_г 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_г (with bulk parameters) using Legendrian surgery extending our previous work [14] where it was assumed that g_v = 0 for all v and г 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.