Fukaya categories of plumbings and multiplicative preprojective algebras · arXivDesk
1703.04515Mar 13, 201729 pages, 10 figures. Clarifications and minor corrections following comments by referees. Notably, the proofs of Theorem 12 and Proposition 14 are made explicit, the connection to deformation quantization is spelled out in Example 6, and Remark 19 about negative plumbings is included
Fukaya categories of plumbings and multiplicative preprojective algebras
Given an arbitrary graph Γ and non-negative integers gv for each vertex v of Γ, let XΓ
Nearby in the stack
be the Weinstein
4
-manifold obtained by plumbing copies of
T∗Σv
according to this graph, where
Σv
is a surface of genus
gv
. We compute the wrapped Fukaya category of
XΓ
(with bulk parameters) using Legendrian surgery extending our previous work arXiv:1502.07922 where it was assumed that
gv=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 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.