XXVII Workshop on Geometric Methods in Physics 29.06-05.07.2008

Armen Sergeev

Quantization of the universal Teichmueller space

Universal Teichm\"uller space $\mathcal T$ consists of quasisymmetric homeomorphisms of the circle $S^1$ (i.e. homeomorphisms of $S^1$, preserving its orientation and extending to quasiconformal homeomorphisms of the disc), normalized modulo M\"obius transformations. It has a natural K\"ahler structure and contains all classical Teichm\"uller spaces as complex submanifolds. Moreover, the space $\mathcal T$ includes also the space $\text{Diff}_+(S^1)/\text{M\"ob}(S^1)$ of normalized diffeomorphisms of the circle, which may be considered as a "smooth" part of $\mathcal T$. The space $\text{Diff}_+(S^1)/\text{M\"ob}(S^1)$ may be quantized, using its embedding into an infinite-dimensional Siegel disc. However, this method does not apply to the whole universal Teichm\"uller space $\mathcal T$, for which quantization we use the "quantum calculus" of Connes--Sullivan.