Topological algebra of symplectic geometry of symmetric powers

TL;DR

This paper constructs an open 2D topological field theory from Fukaya categories of symmetric powers of noncompact surfaces, extending Heegaard-Floer theory without relying on Lagrangian specifics.

Contribution

It introduces a novel topological field theory framework linking Fukaya categories and symmetric powers of surfaces, extending prior Heegaard-Floer results.

Findings

Associates Fukaya categories to symmetric powers of surfaces

Constructs sectorial covers with bar resolution combinatorics

Extends Heegaard-Floer theory to an interval

Abstract

To a noncompact orientable surface with no closed boundary, we associate the sum of Fukaya categories of (Liouville sectors associated to) its symmetric powers. We construct sectorial covers with the combinatorics of the bar resolution to show this association extends to an open 2d topological field theory -- without naming a Lagrangian, let alone a holomorphic disk. In particular, we recover results of Rouquier and Manion on extending Heegaard-Floer theory down to an interval.