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.
Categories: math.GT math-ph math.MP math.RT