Quantization of the theory of half-differentiable strings

The problem of quantizing the space $\Omega_d$ of smooth loops taking values in the $d$-dimensional vector space can be solved in the framework of the standard Dirac approach. But a natural symplectic form on $\Omega_d$ can be extended to the Hilbert completion of $\Omega_d$ coinciding with the Sobolev space $V_d:=H_0^{1/2}(\mathbb S^1,\mathbb R^d)$ of half-differentiable loops with values in $\mathbb R^d$. We regard $V_d$ as the phase space of the theory of half-differentiable strings. This theory can be quantized using ideas from noncommutative geometry.