Poisson geometry, monoidal Fukaya categories, and commutative Floer cohomology rings

TL;DR

This paper explores the relationship between Poisson geometry and Fukaya categories, revealing how symplectic groupoids induce monoidal structures and explaining the commutativity of certain Floer cohomology rings.

Contribution

It establishes a connection between symplectic groupoids and monoidal structures on Fukaya categories, providing insights into the algebraic properties of Floer cohomology rings.

Findings

Fukaya categories of symplectic groupoids are monoidal

Monoidal structures on Fukaya categories originate from symplectic groupoids

Some Lagrangian Floer cohomology rings are proven to be commutative

Abstract

We describe connections between concepts arising in Poisson geometry and the theory of Fukaya categories. The key concept is that of a symplectic groupoid, which is an integration of a Poisson manifold. The Fukaya category of a symplectic groupoid is monoidal, and it acts on the Fukaya categories of the symplectic leaves of the Poisson structure. Conversely, we consider a wide range of known monoidal structures on Fukaya categories and observe that they all arise from symplectic groupoids. We also use the picture developed to resolve a conundrum in Floer theory: why are some Lagrangian Floer cohomology rings commutative?