A trilogy of mapping class group representations from three-dimensional quantum gravity

TL;DR

This paper reviews a recent construction of three-dimensional quantum gravity representations of the mapping class group derived from quantized moduli spaces of Lorentzian metrics, highlighting the role of quantum dilogarithm functions.

Contribution

It elaborates on the construction of three families of unitary representations of the mapping class group from quantized Lorentzian 3-manifold moduli spaces, connecting quantum gravity and group representations.

Findings

Constructed three families of unitary MCG representations

Linked quantum dilogarithm functions to these representations

Provided a comprehensive review of the quantization approach

Abstract

For a punctured surface $\mathfrak{S}$, the author and Scarinci (arXiv:2112.13329) have recently constructed a quantization of a moduli space of Lorentzian metrics on the 3-manifold $\mathfrak{S} \times \mathbb{R}$ of constant sectional curvature $\Lambda \in \{-1,0,1\}$. The invariance of this quantization under the action of the mapping class group ${\rm MCG}(\mathfrak{S})$ of $\mathfrak{S}$ yields families of unitary representations of ${\rm MCG}(\mathfrak{S})$ on a Hilbert space, with key ingredients being three versions of the quantum dilogarithm functions depending on $\Lambda$. In this survey article, we review and elaborate on this result.