Fukaya categories of plumbings and multiplicative preprojective algebras

Abstract

Given an arbitrary graph. 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*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.