Quantization of the universal Teichmüller space

Abstract

In the first part of this article we describe the complex geometry of the universal Teichmuller space, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The universal Teichmuller space contains classical Teichmuller spaces T(G), where G is a Fuchsian group, as complex submanifolds. The quotient Diff+(S 1)/Mob(S 1) of the diffeomorphism group of the unit circle modulo Mobius transformations can be considered as a ‘smooth’ part of the universal Teichmuller space. The second part is devoted to the quantization of the universal Teichmuller space. The smooth part Diff+(S 1)/Mob(S 1) may be quantized, using its embedding into the Hilbert–Schmidt Grassmannian. However, this quantization method does not apply to the whole universal Teichmuller space, to which the ‘quantized calculus’ of Connes and Sullivan may be applied.