A Quantization of Moduli Spaces of 3-Dimensional Gravity

Abstract

We construct a quantization of the moduli space GHΛ(S×R) of maximal globally hyperbolic Lorentzian metrics on S×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 GHΛ(S×R) as the set of points of a cluster X-variety valued in the ring of generalized complex numbers RΛ=R[ℓ]/(ℓ2+Λ). We then develop an RΛ-version of the quantum theory for cluster X-varieties by establishing RΛ-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≥0 the representations are new. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.