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Multiple cluster algebra structures for TCD maps I: theoretical framework

Niklas Affolter, Terrence George, Max Glick, Sanjay Ramassamy

TL;DR

This work develops triple crossing diagram maps (TCD maps) as a unifying framework for projective configurations encoded by BTB graphs, introducing a dual (projective) and a hyperplane-fixed (affine) cluster structure. A geometric section operation connects these two structures, showing that the affine cluster structure of a map equals the projective cluster structure of its section; iterated sections yield a hierarchy of cluster structures. The paper establishes multi-dimensional consistency for TCD maps via Desargues’ theorem, linking to Desargues maps and dSKP lattices, and situates TCD maps within Grassmannian parametrizations and dimer embeddings. A companion paper then develops concrete geometric systems (e.g., Q-nets, pentagram map, t-embeddings) as special cases of the TCD framework, highlighting its broad unifying potential.

Abstract

We introduce triple crossing diagram (TCD) maps, which encode projective configurations of points and lines, as a unified framework for constructions arising in various areas of geometry, such as discrete differential geometry, discrete geometric dynamics and hyperbolic geometry. We define two types of local moves for TCD maps, one of which is governed by the discrete Schwarzian KP (dSKP) equation, and establish their multi-dimensional consistency. We construct two distinct cluster structures on the space of TCD maps, called projective and affine cluster structures, and show that they are related via an operation called section. This framework organizes and unifies a wide range of examples, including Q-nets, Darboux maps, line complexes, T-graphs, t-embeddings, triangulations and geometric discrete integrable systems such as the pentagram map and cross-ratio dynamics, which are further developed in a companion paper and in (arXiv:2108.12692).

Multiple cluster algebra structures for TCD maps I: theoretical framework

TL;DR

This work develops triple crossing diagram maps (TCD maps) as a unifying framework for projective configurations encoded by BTB graphs, introducing a dual (projective) and a hyperplane-fixed (affine) cluster structure. A geometric section operation connects these two structures, showing that the affine cluster structure of a map equals the projective cluster structure of its section; iterated sections yield a hierarchy of cluster structures. The paper establishes multi-dimensional consistency for TCD maps via Desargues’ theorem, linking to Desargues maps and dSKP lattices, and situates TCD maps within Grassmannian parametrizations and dimer embeddings. A companion paper then develops concrete geometric systems (e.g., Q-nets, pentagram map, t-embeddings) as special cases of the TCD framework, highlighting its broad unifying potential.

Abstract

We introduce triple crossing diagram (TCD) maps, which encode projective configurations of points and lines, as a unified framework for constructions arising in various areas of geometry, such as discrete differential geometry, discrete geometric dynamics and hyperbolic geometry. We define two types of local moves for TCD maps, one of which is governed by the discrete Schwarzian KP (dSKP) equation, and establish their multi-dimensional consistency. We construct two distinct cluster structures on the space of TCD maps, called projective and affine cluster structures, and show that they are related via an operation called section. This framework organizes and unifies a wide range of examples, including Q-nets, Darboux maps, line complexes, T-graphs, t-embeddings, triangulations and geometric discrete integrable systems such as the pentagram map and cross-ratio dynamics, which are further developed in a companion paper and in (arXiv:2108.12692).
Paper Structure (29 sections, 36 theorems, 115 equations, 22 figures)

This paper contains 29 sections, 36 theorems, 115 equations, 22 figures.

Table of Contents

  1. Introduction
  2. TCD maps and their projective cluster structures
  3. Affine cluster structures
  4. Sections and the correspondence between cluster structures
  5. Multi-dimensional consistency
  6. Applications and related work
  7. Background
  8. Projective and affine geometry
  9. Triple crossing diagrams
  10. Black-trivalent bipartite graphs
  11. Vector-relation configurations
  12. Triple crossing diagram maps and vector-relation configurations
  13. The space of TCD maps
  14. Moves for TCD maps
  15. The resplit
  16. ...and 14 more sections

Key Result

Lemma 2.4

Let ${\mathsf{P}}_1,{\mathsf{P}}_{1,2},{\mathsf{P}}_2,{\mathsf{P}}_{2,3},\dots, {\mathsf{P}}_m,{\mathsf{P}}_{m,1}$ be $2m$ points in $\mathbb{C}\mathsf{P}^d$ such that every ${\mathsf{P}}_{i,i+1}$ lies on the line ${\mathsf{P}}_i {\mathsf{P}}_{i+1}$ (indices taken modulo $m$). Then the multi-ratio $

Figures (22)

  • Figure 1: A BTB graph ${G}$ (dotted) with its projective quiver (left), its affine quiver (center), and the projective quiver of a section $\sigma({G})$ (right).
  • Figure 2: Local moves for BTB graphs: resplit (left) and spider move (right).
  • Figure 3: Local configurations in the definition of the projective (left) and affine (right) cluster variables.
  • Figure 4: The 2-2 move: for a clockwise bigon (left) and a counterclockwise bigon (right).
  • Figure 5: A black-trivalent bipartite graph ${G}$.
  • ...and 17 more figures

Theorems & Definitions (126)

  • Definition 2.1: Central projection
  • Definition 2.2: Oriented length ratio
  • Definition 2.3: Multi-ratio
  • Lemma 2.4
  • Definition 2.5: Triple crossing diagrams
  • Definition 2.6: Strand permutation of a TCD
  • Definition 2.7: Minimal TCDs
  • Theorem 2.8: thurstontriple
  • Theorem 2.9: thurstontriple
  • Definition 2.10: 2-2 move
  • ...and 116 more