Quantization of universal Teichmüller space

The universal Teichmüller space was introduced in theory of quasiconformal maps as a unifying set for all classical Teichmüller spaces of compact Riemann surfaces of finite genuses. It consists of homeomorphisms of the unit circle, which can be extended to quasiconformal homeomorphisms of the unit disc, considered up to fractional linear transformations. Apart from classical Teichmüller spaces the universal Teichmüller space contains the subspace, consisting of smooth diffeomorphisms of the circle, considered up to fractional linear transformations.
The space of diffeomorphisms and the universal Teichmüller space play an important role in string theory where both spaces are interpreted as phase manifolds of this theory. Therefore we are faced with the problem of their quantization. The quantization of the space of diffeomorphisms may be constructed in frames of the classical Dirac approach. However, this approach fails when applied to the whole universal Teichmüller space. For its quantization we use another method based on ideas from noncommutative geometry.