A Lagrangian filling for every cluster seed

TL;DR

This paper proves that every cluster seed corresponds to an embedded exact Lagrangian filling, using quiver with potential techniques and Lagrangian disk surgeries, advancing understanding of the augmentation variety.

Contribution

It introduces a novel method to construct Lagrangian fillings for all cluster seeds via quiver with potential and disk surgeries, establishing surjectivity.

Findings

Every cluster seed has an associated embedded exact Lagrangian filling.

The deformation space of the quiver with potential is shown to be trivial.

Lagrangian disk surgeries can manipulate fillings with $\

Abstract

We show that each cluster seed in the augmentation variety is inhabited by an embedded exact Lagrangian filling. This resolves the matter of surjectivity of the map from Lagrangian fillings to cluster seeds. The main new technique to produce these Lagrangian fillings is the construction and study of a quiver with potential associated to curve configurations. We prove that its deformation space is trivial and show how to use it to manipulate Lagrangian fillings with $\mathbb{L}$-compressing systems via Lagrangian disk surgeries.