A Note on CFT Correlators in Three Dimensions

TL;DR

The paper develops a systematic framework to classify conformally invariant three-point functions of higher-spin currents in three-dimensional CFTs using a generating-polynomial formalism with invariants $P_i$, $Q_i$, and parity-odd structures $S_i$. Upon imposing current conservation, the authors propose a universal decomposition into parity-even and a unique parity-odd piece, with the parity-even sector reproducing free-scalar and free-fermion results and the parity-odd sector arising in parity-violating theories such as Chern-Simons-matter systems. They provide explicit conserved-structure examples across various spins, reveal how triangle inequalities control the existence of parity-odd terms, and discuss scalar operator insertions that further delineate parity-preserving versus parity-violating dynamics. The work connects to holography by identifying bulk parity-violating couplings in AdS$_4$ that can generate the boundary parity-odd structures, illuminating links to Vasiliev-like higher-spin theories. Overall, the paper offers a concrete, testable classification of 3d CFT correlators with higher-spin currents and clarifies when parity violation constrains or enables novel structures, with implications for holographic duals.

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.