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Title:
A deformation quantization of moduli spaces of 3-dimensional gravity
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Abstract:
In this talk, I will present a deformation quantization of the moduli space 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 $S$ regardless of the value of $\Lambda$, the deformation quantization we provide depends on $\Lambda$. Using special coordinate systems, this moduli space can be viewed as the set of points of a cluster X-variety valued in the ring of generalized complex numbers $\mathbb{R}_\Lambda = \mathbb{R}[\ell]/(\ell^2+\Lambda)$. We first review quantum theory of Teichmüller spaces and cluster X-varieties, and then develop an $\mathbb{R}_\Lambda$-version of it by establishing $\mathbb{R}_\Lambda$ versions of the quantum dilogarithm function, which yields a sought-for quantization. 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 seem to be new. This is based on 2112.13329, joint with C. Scarinci.
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