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

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 $q=e^{2\pi i\hslash }$, 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 Teichmüller 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.

76 pages (double-spaced), 41 figures + appended computer code; 115 pages in total. Version 4: Final version after publication

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• Higher laminations, webs and N=2 line operators
• Continuous tensor categories from quantum groups I: algebraic aspects
• Triangular decomposition of $S{L}_3$ skein algebras