Chern-Simons propagators in AdS$_3$

TL;DR

This work develops a parity-violating framework for AdS$_3$ by constructing parity-odd spin-1 harmonics and a Chern-Simons operator that maps parity-even to parity-odd structures, enabling simultaneous diagonalization with the Laplacian. It builds an embedding-formalism extension to accommodate parity-odd contributions, derives explicit bulk-to-bulk and bulk-to-boundary propagators for Abelian CS, Proca, and Maxwell–CS theories, and verifies consistency with boundary current correlators. A split representation is established, expressing CS propagators as integrals over products of bulk-to-boundary propagators, which is poised to simplify loop computations in CS QFTs on AdS$_3$. These results provide a covariant, isometry-preserving toolkit for perturbative studies of parity-violating QFTs on AdS$_3$ and their holographic boundary data, with broader implications for higher-spin and non-Abelian generalizations.

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$_3$.