Deformations of cluster mutations and invariant presymplectic forms

TL;DR

This paper studies deformations of cluster mutation sequences that preserve presymplectic forms but lose the Laurent property, leading to integrable maps and their Laurentification, with applications to affine Dynkin quivers and the discrete sine-Gordon equation.

Contribution

It introduces a framework for deformations of cluster mutations that maintain presymplectic structures and explores their integrability and Laurentification, extending cluster algebra theory.

Findings

Deformations of cluster mutations can produce integrable symplectic maps.

Laurentification lifts these maps to higher-dimensional spaces with restored Laurent property.

Connections established between deformed mutations and affine Dynkin quivers, including reductions of the discrete sine-Gordon equation.

Abstract

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type $A_2$: this deforms to the Lyness family of integrable symplectic maps in the plane. For types $A_3$ and $A_4$ we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions of the discrete sine-Gordon equation.