Naturality of ${\rm SL}_n$ quantum trace maps for surfaces

TL;DR

This paper demonstrates the naturality of ${ m SL}_n$ quantum trace maps for surfaces, showing their compatibility across different ideal triangulations through quantum coordinate change isomorphisms, extending Fock-Goncharov theory.

Contribution

It establishes the compatibility of ${ m SL}_n$ quantum trace maps for various triangulations via a balanced root version of quantum coordinate change isomorphisms, extending existing quantum cluster theory.

Findings

Quantum trace maps are compatible across triangulations.

Extension of Fock-Goncharov's quantum coordinate change isomorphism.

Avoidance of heavy computations using splitting homomorphisms.

Abstract

The ${\rm SL}_n$-skein algebra of a punctured surface $\mathfrak{S}$, studied by Sikora, is an algebra generated by isotopy classes of $n$-webs living in the thickened surface $\mathfrak{S} \times (-1,1)$, where an $n$-web is a union of framed links and framed oriented $n$-valent graphs satisfying certain conditions. For each ideal triangulation $\lambda$ of $\mathfrak{S}$, L\^e and Yu constructed an algebra homomorphism, called the ${\rm SL}_n$-quantum trace, from the ${\rm SL}_n$-skein algebra of $\mathfrak{S}$ to a so-called balanced subalgebra of the $n$-root version of Fock and Goncharov's quantum torus algebra associated to $\lambda$. We show that the ${\rm SL}_n$-quantum trace maps for different ideal triangulations are related to each other via a balanced $n$-th root version of the quantum coordinate change isomorphism, which extends Fock and Goncharov's isomorphism for quantum cluster varieties. We avoid heavy computations in the proof, by using the splitting homomorphisms of L\^e and Sikora, and a network dual to the $n$-triangulation of $\lambda$ studied by Schrader and Shapiro.