On the combinatorics of exact Lagrangian surfaces

TL;DR

This paper explores the combinatorial and geometric properties of Lagrangian surfaces within Weinstein 4-manifolds, revealing how surgeries on skeleta induce cluster transformations and connect symplectic geometry with cluster algebra theory.

Contribution

It introduces a new framework for understanding Lagrangian skeleta via curve surgeries and links these to cluster transformations in Fukaya categories, expanding the geometric-combinatorial dictionary.

Findings

Surgeries preserve Weinstein structures while altering skeleta.

Cluster transformations relate local systems on surfaces to skeleta modifications.

Connections established between symplectic geometry, cluster algebras, and character varieties.

Abstract

We study Weinstein 4-manifolds which admit Lagrangian skeleta given by attaching disks to a surface along a collection of simple closed curves. In terms of the curves describing one such skeleton, we describe surgeries that preserve the ambient Weinstein manifold, but change the skeleton. The surgeries can be iterated to produce more such skeleta --- in many cases, infinitely many more. Each skeleton is built around a Lagrangian surface. Passing to the Fukaya category, the skeletal surgeries induce cluster transformations on the spaces of rank one local systems on these surfaces, and noncommutative analogues of cluster transformations on the spaces of higher rank local systems. In particular, the problem of producing and distinguishing such Lagrangians maps to a combination of combinatorial-geometric questions about curve configurations on surfaces and algebraic questions about exchange graphs of cluster algebras. Conversely, this expands the dictionary relating the cluster theory of character varieties, positroid strata, and related spaces to the symplectic geometry of Lagrangian fillings of Legendrian knots, by incorporating cluster charts more general than those associated to bicolored surface graphs.