The wrapped Fukaya category of plumbings

TL;DR

This paper provides a comprehensive formula for the wrapped Fukaya category of plumbings of cotangent bundles along quivers, using local-to-global methods, and describes its applications in symplectic topology.

Contribution

It introduces a general formula for the wrapped Fukaya category of plumbings with arbitrary grading, expanding understanding of symplectic invariants of Weinstein manifolds.

Findings

Computed wrapped Fukaya categories for plumbing sectors as local models.

Fully described wrapped Fukaya categories for 4-dimensional plumbing spaces.

Showed equivalence between Ginzburg dg algebras of graded quivers and wrapped Fukaya categories of plumbings.

Abstract

Plumbing spaces have drawn significant attention among symplectic topologists due to their natural occurrence as examples of Weinstein manifolds. In our paper, we provide a general formula for the wrapped Fukaya category of plumbings (with arbitrary grading structure) of cotangent bundles along any quiver. Our approach relies on "local-to-global" computations. Specifically, we compute the wrapped Fukaya category of "plumbing sectors" that serve as local models for the singularities of Lagrangian skeletons of plumbing spaces. As corollaries, we fully describe the wrapped Fukaya category of plumbing spaces in dimension $4$ and plumbings of $T^*S^n$ for $n \geq 3$. We show that any Ginzburg dg algebra/category of a graded quiver without potential is equivalent to the wrapped Fukaya category of a plumbing of $T^*S^n$ (with the corresponding grading structure).