Multiple cluster algebra structures for TCD maps I: theoretical framework

TL;DR

This paper introduces a unified theoretical framework for TCD maps, connecting various geometric structures and integrable systems through cluster algebra structures and local moves governed by the dSKP equation.

Contribution

It develops two distinct cluster structures on TCD maps, linking diverse geometric configurations and integrable systems within a comprehensive algebraic framework.

Findings

Established multi-dimensional consistency of local moves

Constructed projective and affine cluster structures on TCD maps

Unified various geometric and integrable systems under a common framework

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).