Quantization of the Universal Teichmüller Space

In the first part of the paper, we describe the Kähler geometry of the universal Teichmüller space, which can be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The universal Teichmüller space contains classical Teichmüller spaces $T(G)$, where $G$ is a Fuchsian group, as complex submanifolds. The quotient $\text{Diff}_+(S^1)/\text{M\"ob}(S^1)$ of the diffeomorphism group of the unit circle modulo Möbius transformations can be considered as a “smooth” part of the universal Teichmüller space. In the second part we describe how to quantize $\text{Diff}_+(S^1)/\text{M\"ob}(S^1)$ by embedding it in an infinite-dimensional Siegel disc. This quantization method does not apply to the whole universal Teichmüller space. However, this space can be quantized using the “quantized calculus” of A. Connes and D. Sullivan.