A quantization of moduli spaces of 3-dimensional gravity

Abstract

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 Teichmüller 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 $\mathcal{X}$-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 $\mathcal{X}$-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 $\Lambda \ge 0$ the representations are new.

64 pages, 2 figures. revised version accepted for publication in Commun. Math. Phys

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