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Commuting integrable maps from a deformed D$_4$ cluster algebra

A. N. W. Hone, W. Kim, T. Mase

TL;DR

This work analyzes a two-parameter deformation of the D$_4$ cluster map, showing that while Zamolodchikov periodicity and the Laurent property fail in the original coordinates, the system preserves a presymplectic form and admits a planar Liouville-integrable reduction $\hat{\varphi}$. By lifting to an extended cluster algebra, the authors construct a second commuting map $\hat{\chi}$ and develop tau-function dynamics on a $\mathbb{Z}^2$ lattice, revealing quadratic degree growth with distinct leading coefficients and independence. The geometry of the first integral's elliptic fibration yields a rational elliptic surface of rank $2$ (Mordell–Weil group), enabling a Laurentification that preserves integrability and allows a QRT-type reformulation in a suitable chart. The results connect cluster mutation dynamics, elliptic surface arithmetic, and integrable mappings, and point to deep links with q-Painlevé equations and higher-rank deformations, with several open directions for general Dynkin types. $

Abstract

In this paper we revisit an integrable map of the plane which we obtained recently as a two-parameter family of deformed mutations in the cluster algebra of type D$_4$. The rational first integral for this map defines an invariant foliation of the plane by level curves, and we explain how this corresponds to a rational elliptic surface of rank 2. This leads us to construct another (independent) integrable map, commuting with the first, such that both maps lift to compositions of mutations in an enlarged cluster algebra, whose underlying quiver is equivalent to the one found by Okubo for the $q$-Painlevé VI equation. The degree growth of the two commuting maps is calculated in two different ways: firstly, from the tropical (max-plus) equations for the d-vectors of the cluster variables; and secondly, by constructing the minimal space of initial conditions for the two maps, via blowing up $\mathbb{P}^1 \times \mathbb{P}^1$.

Commuting integrable maps from a deformed D$_4$ cluster algebra

TL;DR

This work analyzes a two-parameter deformation of the D cluster map, showing that while Zamolodchikov periodicity and the Laurent property fail in the original coordinates, the system preserves a presymplectic form and admits a planar Liouville-integrable reduction . By lifting to an extended cluster algebra, the authors construct a second commuting map and develop tau-function dynamics on a lattice, revealing quadratic degree growth with distinct leading coefficients and independence. The geometry of the first integral's elliptic fibration yields a rational elliptic surface of rank (Mordell–Weil group), enabling a Laurentification that preserves integrability and allows a QRT-type reformulation in a suitable chart. The results connect cluster mutation dynamics, elliptic surface arithmetic, and integrable mappings, and point to deep links with q-Painlevé equations and higher-rank deformations, with several open directions for general Dynkin types. $

Abstract

In this paper we revisit an integrable map of the plane which we obtained recently as a two-parameter family of deformed mutations in the cluster algebra of type D. The rational first integral for this map defines an invariant foliation of the plane by level curves, and we explain how this corresponds to a rational elliptic surface of rank 2. This leads us to construct another (independent) integrable map, commuting with the first, such that both maps lift to compositions of mutations in an enlarged cluster algebra, whose underlying quiver is equivalent to the one found by Okubo for the -Painlevé VI equation. The degree growth of the two commuting maps is calculated in two different ways: firstly, from the tropical (max-plus) equations for the d-vectors of the cluster variables; and secondly, by constructing the minimal space of initial conditions for the two maps, via blowing up .

Paper Structure

This paper contains 15 sections, 12 theorems, 114 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Deformation of the $\mathrm{D}_{4}$ cluster map
  4. Deformed $\mathrm{D}_4$ symplectic map $\hat{\varphi}$ in case (2)
  5. Rational elliptic surface
  6. Cluster structure and commuting map
  7. Extended cluster algebra associated with the map $\hat{\varphi}$
  8. The commuting map $\hat{\chi}$
  9. Tau functions on the ${\mathbb Z}^2$ lattice
  10. Degree growth of cluster maps
  11. Elliptic surface and degree growth for deformed D$_4$ map
  12. Space of initial conditions
  13. Degree growth
  14. QRT map
  15. Conclusions

Key Result

Theorem 2.2

hkm24 For the deformed symplectic map $\hat{\varphi}$ to admit the first integral firstintD4, it is necessary and sufficient for the parameters $\mathrm{b}_{i}$ to satisfy one of the following sets of conditions: In each of these cases, subject to the given conditions on the parameters $\mathrm{b}_i$, the deformed map $\hat{\varphi}$ given by newphihatmapD4 is Liouville integrable, preserving the

Figures (4)

  • Figure 1: Extended quiver associated with the cluster map $\psi$
  • Figure 2: Initial stencil of tau functions on ${\mathbb Z}^2$.
  • Figure 3: The surface ${\cal X}_1$ obtained by blowing up the indeterminacies of the pencil defined by $K$ and the minimal space of initial conditions ${\cal X}_0$.
  • Figure 4: Blow-ups needed to obtain the surface ${\cal X}$ (upper) from ${\cal X}_1$ (lower).

Theorems & Definitions (21)

  • Remark 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.4
  • Proposition 3.5
  • Lemma 3.6
  • Theorem 3.7
  • ...and 11 more