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

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 $\hat{I}›\in \{-1,0,1\}$. The invariance of this quantization under the action of the mapping class group $\mathrm{MCG}(\mathfrak{S})$ of $\mathfrak{S}$ yields families of unitary representations of $\mathrm{MCG}(\mathfrak{S})$ on a Hilbert space, with key ingredients being three versions of the quantum dilogarithm functions depending on $\hat{I}›$. In this survey article, we review and elaborate on this result.

32 pages; to appear in the proceedings of the conference in honor of Igor Frenkel's 70th birthday / v2: minor updates