  Using the recently developed notion of a fractional Virasoro algebra, we
explore the implied operator product expansions in nonlocal conformal field
theories and their geometric meaning. We probe the interplay between classical
nonlocality in the functional-analytic sense and quantization in a
two-dimensional setting and find that nonlocal quantum dynamics realize this
fractional Virasoro algebra exclusively with a state dependent central charge.
Notably, we prove that the widely studied free Gaussian fixed points with a
fractional Laplacian kinetic term does not fit this criterion but that the RG
flow associated non-Gaussian fixed points do.
