Birational maps conjugate to the rank 2 cluster mutations of affine types and their geometry

TL;DR

This paper explores the geometric structures of birational maps related to cluster mutations of affine types, revealing their integrability, invariants, and connections to elliptic curves and Toda lattices.

Contribution

It demonstrates the conjugacy of birational maps from different affine cluster types and links their geometry to elliptic curves and integrable systems.

Findings

Invariant curves are singular quartic curves with resolutions leading to integrable systems.

Birational maps from different cluster types are conjugate and commute on the invariant conic.

Connections established between cluster mutations, elliptic curves, and Toda lattice spectral curves.

Abstract

Mutations of the cluster variables generating the cluster algebra of type $A^{(2)}_2$ reduce to a two-dimensional discrete integrable system given by a quartic birational map. The invariant curve of the map is a singular quartic curve, and its resolution of the singularity induces a discrete integrable system on a conic governed by a cubic birational map conjugate to the cluster mutations of type $A^{(2)}_2$. Moreover, it is shown that the conic is also the invariant curve of the quadratic birational map arising from the cluster mutations of type $A^{(1)}_1$ and the two birational maps on the conic are commutative. Finally, the commutative birational maps are reduced as singular limits of additions of points on an elliptic curve arising as the spectral curve of the discrete Toda lattice of type $A^{(1)}_1$.