A quantization of moduli spaces of 3-dimensional gravity

TL;DR

This paper develops a novel quantization framework for moduli spaces of 3D gravity with constant curvature, utilizing cluster varieties and generalized complex numbers, leading to new representations of the mapping class group.

Contribution

It introduces a $ ext{Lambda}$-dependent quantization of moduli spaces of 3D gravity using cluster $ ext{X}$-varieties and generalized complex numbers, extending existing theories.

Findings

Constructed $ ext{Lambda}$-dependent quantum theories for moduli spaces.

Established $ ext{R}_ ext{Lambda}$-versions of quantum dilogarithm functions.

Derived new projective unitary representations of the mapping class group for $ ext{Lambda} extgreater=0$.

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 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.