Quantum traces for $\mathrm{SL}_n(\mathbb{C})$: The case $n=3$

TL;DR

This paper extends the quantum trace map from SL_2(C) to SL_3(C), associating Laurent polynomials to links in a surface, advancing the understanding of quantum invariants in higher Teichmüller theory.

Contribution

It generalizes Bonahon-Wong's quantum trace map to SL_3(C), introducing a new invariant for links in punctured surfaces and proposing a framework for SL_n(C).

Findings

Defined a quantum trace map for SL_3(C)

Connected quantum invariants to higher Teichmüller space coordinates

Proposed a generalization framework for SL_n(C)

Abstract

We generalize Bonahon-Wong's $\mathrm{SL}_2(\mathbb{C})$-quantum trace map to the setting of $\mathrm{SL}_3(\mathbb{C})$. More precisely, given a non-zero complex parameter $q=e^{2 \pi i \hbar}$, we associate to each isotopy class of framed oriented links $K$ in a thickened punctured surface $\mathfrak{S} \times (0, 1)$ a Laurent polynomial $\mathrm{Tr}_\lambda^q(K) = \mathrm{Tr}_\lambda^q(K)(X_i^q)$ in $q$-deformations $X_i^q$ of the Fock-Goncharov $\mathcal{X}$-coordinates for higher Teichm\"{u}ller space. This construction depends on a choice $\lambda$ of ideal triangulation of the surface $\mathfrak{S}$. Along the way, we propose a definition for a $\mathrm{SL}_n(\mathbb{C})$-version of this invariant.