Chern-Simons propagators in AdS$_3$

TL;DR

This paper introduces parity-odd harmonic functions in AdS$_3$, relates them to parity-even ones via a Chern-Simons operator, and constructs propagators for Chern-Simons and Maxwell-Chern-Simons theories, aiding studies of parity-violating QFTs.

Contribution

It presents a novel Chern-Simons operator relating parity-odd and even harmonics, and develops a formalism for constructing propagators and higher-spin structures in AdS$_3$.

Findings

Constructed parity-odd harmonic functions in AdS$_3$

Developed a split representation for these functions

Validated propagators against boundary current correlators

Abstract

We introduce parity-odd spin-1 harmonic functions in AdS$_3$ and study their properties. We demonstrate that such parity-odd harmonics are related to their parity-even counterparts through the action of a `Chern-Simons operator', which we present as a novelty in this paper. This relation leads to the construction of simultaneous eigen-functions of the Laplacian and the Chern-Simons operators. Subsequently, these harmonic functions are employed to construct propagators in pure abelian Chern-Simons theory as well as Maxwell-Chern-Simons theory in a covariant gauge. We demonstrate the consistency of the Chern-Simons propagator with the expected two-point function of the boundary currents. Our results are built upon the embedding formalism, which we modify suitably to incorporate parity-odd structures. This formalism also readily helps us write down parity odd structures for the propagators of higher-spin fields. Finally, we construct a split representation for the parity-odd harmonic functions, which may be useful to compute Witten diagrams with loops. Our results are expected to be useful in perturbative studies of parity violating QFTs on AdS$_3$.