Conformal blocks and bilocal vertex operator transition amplitudes

Abstract

We revisit the construction of the 2d conformal blocks of primary operator four-point functions as bilocal vertex operator correlators. We find an additional interpretation as a path integral over the reparametrizations of an intermediate cylinder. As a consequence we bridge the gap between the Kähler quantization of virasoro coadjoint orbits, SL(2, ℝ) Chern-Simons theory and the reparametrization formalism of 2d CFT that has made an appearance in recent literature.