  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 K\"ahler
quantization of virasoro coadjoint orbits, $SL(2,\mathbb{R})$ Chern-Simons
theory and the reparametrization formalism of 2d CFT that has made an
appearance in recent literature.
