A quantization of moduli spaces of 3-dimensional gravity

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