$\mathfrak{sl}_3$ Matrix Dilogarithm as a $6j$-Symbol

TL;DR

This paper constructs new quantum invariants for 3-manifolds using an $rak{sl}_3$ matrix dilogarithm, extending Kashaev's invariants and linking them to quantum groups, offering a higher-rank analogue of established invariants.

Contribution

It demonstrates that the $rak{sl}_3$ matrix dilogarithm functions as a 6j-symbol for quantum group modules, enabling the definition of $rak{sl}_3$ quantum invariants of 3-manifolds.

Findings

$rak{sl}_3$ matrix dilogarithm acts as a 6j-symbol.

New $rak{sl}_3$ quantum invariants of 3-manifolds are constructed.

Extension of Kashaev's invariants to higher rank $rak{sl}_3$ case.

Abstract

We construct quantum invariants of 3-manifolds based on a $\mathfrak{sl}_3$ matrix dilogarithm proposed by Kashaev. This matrix dilogarithm is an $\mathfrak{sl}_3$ analogue of the (cyclic) quantum dilogarithm used to define Kashaev's invariants as well as Baseilhac and Benedetti's quantum hyperbolic invariants. % In this article, we show that the $\mathfrak{sl}_3$ matrix dilogarithm can be considered as a 6$j$-symbol associated to modules of a quantum group related to $U_q(\mathfrak{sl}_3)$. Moreover, we show that the quantum invariants aforementioned allow to define a $\mathfrak{sl}_3$ version of Kashaev's invariants, opening a route to define a $\mathfrak{sl}_3$ version of Baseilhac and Benedetti's quantum hyperbolic invariants.