A quantization of moduli spaces of 3-dimensional gravity

We construct a quantization of the moduli space $\mathcal{GH}_\Lambda(S\times\mathbb{R})$ of maximal globally hyperbolic Lorentzian metrics on $S\times \mathbb{R}$ with constant sectional curvature $\Lambda$, for a punctured surface $S$. Although this moduli space is known to be symplectomorphic to the cotangent bundle of the Teichm\"uller space of $S$ independently of the value of $\Lambda$, we define geometrically natural classes of observables leading to $\Lambda$-dependent quantizations. Using special coordinate systems, we first view $\mathcal{GH}_\Lambda(S\times\mathbb{R})$ as the set of points of a cluster $\mathscr{X}$-variety valued in the ring of generalized complex numbers $\mathbb{R}_\Lambda = \mathbb{R}[\ell]/(\ell^2+\Lambda)$. We then develop an $\mathbb{R}_\Lambda$-version of the quantum theory for cluster $\mathscr{X}$-varieties by establishing $\mathbb{R}_\Lambda$-versions of the quantum dilogarithm function. As a consequence, we obtain three families of projective unitary representations of the mapping class group of $S$. For $\Lambda <0$ these representations recover those of Fock and Goncharov, while for $\Lambda\geq 0$ the representations are new.