Conformal Blocks and Bilocal Vertex Operator Transition Amplitudes

TL;DR

This paper explores the construction of 2D conformal blocks as bilocal vertex correlators, providing new interpretations and connecting various formalisms in conformal field theory and related theories.

Contribution

It introduces an additional path integral interpretation of conformal blocks and bridges different theoretical frameworks in 2D CFT and related gauge theories.

Findings

New path integral interpretation of conformal blocks

Connection between Virasoro coadjoint orbits and CFT

Unified view of reparametrization formalism

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 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.