Quantization of the Sobolev space of half-differentiable functions

A quantization of the Sobolev space $V=H_0^{1/2}(S^1,\mathbb R)$ of half-differentiable functions on the circle, which is closely connected with string theory, is constructed. The group $\mathrm{QS}(S^1)$ of quasisymmetric circle homeomorphisms acts on $V$ by reparametrizations, but this action is not smooth. Nevertheless, a quantum infinitesimal action of $\mathrm{QS}(S^1)$ on $V$ can be defined, which enables one to construct a quantum algebra of observables which is associated with the system $(V,\mathrm{QS}(S^1))$.
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