Note on character varieties and cluster algebras

TL;DR

This paper explores the connection between character varieties of punctured surfaces and cluster algebras, revealing how cluster mutations induce automorphisms and examining quantizations via quantum cluster algebra.

Contribution

It clarifies the relationship between character varieties and cluster algebras, showing how Poisson structures and automorphisms are interconnected, and extends to quantizations.

Findings

Goldman Poisson algebra is recovered from cluster algebra Poisson structure

Cluster mutations induce automorphisms of character varieties

Quantizations are achieved through quantum cluster algebra

Abstract

We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recovered from the Poisson structure of cluster algebra. It is also shown that cluster mutations give the automorphism of the character varieties. Motivated by a work of Chekhov-Mazzocco-Rubtsov, we revisit confluences of punctures on sphere from cluster algebraic viewpoint, and we obtain associated affine cubic surfaces constructed by van der Put-Saito based on the Riemann-Hilbert correspondence. Further studied are quantizations of character varieties by use of quantum cluster algebra.