Quantization of the universal Teichmüller space (Q5895120)

The universal Teichmüller space, introduced by L. Ahlfors and L. Bers, plays a key role in the theory of quasiconformal maps and Riemann surfaces. This space has a natural Kähler structure and the first part of the paper is devoted to a description of it. The second part presents the geometric quantization of the homogeneous space \(\text{Diff}_{+}(S^1)/\text{Möb}(S^1)\), which is the quotient of the diffeomorphism group of the unit circle modulo Möbius transformations. Since the previous quantization does not apply to the whole universal Teichmüller space the last section gives applications of the quantized calculus of Connes and Sullivan towards this aim.