Quantization of Sobolev space of half-differentiable functions

We shall construct the quantization of the Sobolev space $V=H_0^{1/2}(S^1,\mathbb R)$ of half-differentiable functions on the circle closely related to the string theory. The group $\text{QS}(S^1)$ of quasisymmetric homeomorphisms of the circle acts on this space by reparameterization but this action is not smooth. However, we can introduce a quantized infinitesimal action of $\text{QS}(S^1)$ on the Sobolev space $V$ which allows us to construct a quantum algebra of observables associated with the classical system $(V,\text{QS}(S^1))$.