Cluster ensembles, quantization and the dilogarithm

TL;DR

This paper explores the mathematical structure of cluster ensembles, their symmetries, and connections to the motivic dilogarithm, proposing a framework that links cluster theory with higher Teichmuller theory and quantum deformations.

Contribution

It develops general properties of cluster ensembles, introduces a q-deformation, and formulates duality conjectures, advancing the theoretical understanding of these structures.

Findings

Defined the cluster modular group and its properties.

Constructed the canonical pairing in finite type cases.

Proposed a framework for higher quantum Teichmuller theory.

Abstract

Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its group of symmetries - the cluster modular group, and a relation with the motivic dilogarithm. We define a q-deformation of the X-space. Formulate general duality conjectures regarding canonical bases in the cluster ensemble context. We support them by constructing the canonical pairing in the finite type case. Interesting examples of cluster ensembles are provided the higher Teichmuller theory, that is by the pair of moduli spaces corresponding to a split reductive group G and a surface S defined in math.AG/0311149. We suggest that cluster ensembles provide a natural framework for higher quantum Teichmuller theory.