A Note on CFT Correlators in Three Dimensions

TL;DR

This paper introduces a straightforward method to construct three-point functions in 3D conformal field theories, revealing new parity-violating structures and clarifying the realization of these structures by free fields.

Contribution

It provides a simple construction method for conformally invariant three-point functions in three dimensions, including new parity-violating structures and their relation to free field realizations.

Findings

All parity-preserving structures are realized by free fields.

There is at most one parity-violating structure per set of spins.

New parity-violating structures are identified for certain currents.

Abstract

In this note we present a simple method of constructing general conformally invariant three point functions of operators of various spins in three dimensions. Upon further imposing current conservation conditions, we find new parity violating structures for the three point functions involving either the stress-energy tensor, spin one currents, or higher spin currents. We find that all parity preserving structures for conformally invariant three point functions of higher spin conserved currents can be realized by free fields, whereas there is at most one parity violating structure for three point functions for each set of spins, which is not realized by free fields.