Quantization on the Virasoro group
The group approach to quantization developed by the authors is applied to quantize the Virasoro group. This approach differs from the standard method of geometric quantization by taking as a starting point the entire group manifold rather than the co-adjoint orbit. First, the formal group law of the centrally extended group \(diff^{\sim} S^ 1\) is found. The next step is to find the quantization form, i.e. the vertical component of the left-invariant canonical 1-form. This is done using duality with respect to the left- invariant vector fields which in turn can be quite simply calculated from the composition law. Finally, the quantization is accomplished by defining the wave functions as the U(1)-equivariant functions on \(diff^{\sim} S^ 1\) satisfying certain polarization conditions and finding the group representations by calculating the action of left translations on polarized functions. Some properties of the representations obtained in such a way are discussed. In particular, the absence of null vectors is established.
scientific article; zbMATH DE number 4152738