Mathematics – Geometric Topology
Scientific paper
2009-02-08
SIGMA 5 (2009), 015, 20 pages
Mathematics
Geometric Topology
Scientific paper
10.3842/SIGMA.2009.015
In the first part of the paper we describe the complex geometry of the universal Teichm\"uller space $\mathcal T$, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The quotient $\mathcal S$ of the diffeomorphism group of the circle modulo M\"obius transformations may be treated as a smooth part of $\mathcal T$. In the second part we consider the quantization of universal Teichm\"uller space $\mathcal T$. We explain first how to quantize the smooth part $\mathcal S$ by embedding it into a Hilbert-Schmidt Siegel disc. This quantization method, however, does not apply to the whole universal Teichm\"uller space $\mathcal T$, for its quantization we use an approach, due to Connes.
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