Floer potentials, cluster algebras and quiver representations

TL;DR

This work establishes a precise correspondence between Floer potentials of monotone Lagrangian tori in toric del Pezzo surfaces and cluster characters of representations of quivers with potentials. By building a mutation-compatible bridge via a comparison map $\Phi_{\bf s}$ between two-variable Laurent polynomials and cluster algebras, the authors show that initial Landau-Ginzburg seeds give rise to potentials whose mutations align with quiver mutations. They construct a virtual representation $P(X)$ for each surface $X$ so that its cluster character matches the Floer potential under $\Phi_{\bf s(X)}$, and prove that this correspondence persists under all subsequent mutations, yielding a unified algebraic framework for Floer disk counts. The results illuminate a deep link between symplectic geometry and representation theory, enabling Floer-theoretic data to be studied through cluster algebra techniques and offering explicit realizations in terms of quivers with potentials.

Abstract

We use cluster algebras to interpret Floer potentials of monotone Lagrangian tori in toric del Pezzo surfaces as cluster characters of quiver representations.

Theorems & Definitions (17)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 2.4
• Theorem 2.5
• Corollary 2.6
• Lemma 2.7
• Definition 2.8
• Lemma 2.9
• proof