String theory and quasiconformal maps

Abstract

The phase space of the closed string theory may be identified with the space of smooth loops. This reduces the problem of quantization of string theory to the quantization of the space of smooth loops. In this paper we describe the solution of the latter problem obtained in a series of papers. But the symplectic form of string theory is correctly defined not only on the space of smooth loops but also on its Hilbert completion coinciding with the Sobolev space of half-differentiable functions. So it is reasonable to consider this space as the phase manifold of non-smooth string theory. There is a natural group associated with this Sobolev space, namely the group of quasisymmetric homeomorphisms of the circle acting by change of variable. Unfortunately, this action is not smooth. However, we are able to quantize the Sobolev space of half-differentiable functions provided with the action of the group of quasisymmetric homeomorphisms using methods of noncommutative geometry.