A Quantization of Moduli Spaces of 3-Dimensional Gravity

We construct a quantization of the moduli space \(\mathcal{G}\mathcal{H}_\Lambda (S\times \mathbb {R})\) $\mathcal{G}{\mathcal{H}}_{\mathrm{\Lambda }}(S\times \mathbb{R})$ of maximal globally hyperbolic Lorentzian metrics on \(S\times \mathbb {R}\) $S\times \mathbb{R}$ with constant sectional curvature \(\Lambda \) $\mathrm{\Lambda }$ , 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 \(\Lambda \) $\mathrm{\Lambda }$ , we define geometrically natural classes of observables leading to \(\Lambda \) $\mathrm{\Lambda }$ -dependent quantizations. Using special coordinate systems, we first view \(\mathcal{G}\mathcal{H}_\Lambda (S\times \mathbb {R})\) $\mathcal{G}{\mathcal{H}}_{\mathrm{\Lambda }}(S\times \mathbb{R})$ as the set of points of a cluster \(\mathscr {X}\) $\mathcal{X}$ -variety valued in the ring of generalized complex numbers \(\mathbb {R}_\Lambda = \mathbb {R}[\ell ]/(\ell ^2+\Lambda )\) ${\mathbb{R}}_{\mathrm{\Lambda }}=\mathbb{R}[\ell ]/({\ell }^2+\mathrm{\Lambda })$ . We then develop an \(\mathbb {R}_\Lambda \) ${\mathbb{R}}_{\mathrm{\Lambda }}$ -version of the quantum theory for cluster \(\mathscr {X}\) $\mathcal{X}$ -varieties by establishing \(\mathbb {R}_\Lambda \) ${\mathbb{R}}_{\mathrm{\Lambda }}$ -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 \(\Lambda <0\) $\mathrm{\Lambda }<0$ these representations recover those of Fock and Goncharov, while for \(\Lambda \ge 0\) $\mathrm{\Lambda }\ge 0$ the representations are new.