Holographic currents in first order Gravity and finite Fefferman-Graham expansions

TL;DR

This work analyzes holographic currents in Chern-Simons theories, first illustrating chiral anomalies in a 2D boundary via a doubled CS setup, then embedding a five-dimensional first-order gravity where the spin connection is dynamical. It shows that finite Fefferman-Graham expansions persist in these CS frameworks, with boundary sources given by the vielbein and spin connection, and that bulk constraints encode the boundary stress tensor and spin current along with their Ward identities. The Abelian and non-Abelian 2D cases reproduce the expected chiral anomalies and their covariant vs. Bardeen-Zumino structures, while the 5D analysis derives explicit expressions for holographic currents $ au$ and $oldsymbol{ au}$ and the Weyl (Euler) anomaly, clarifying the role of torsion in holography. The results highlight how torsion enters the boundary theory through curvature couplings in CS gravity, discuss potential torsion-dependent chiral anomalies, and establish a framework applicable to odd-dimensional CS gravities as a robust holographic setting.

Abstract

We study the holographic currents associated to Chern-Simons theories. We start with an example in three dimensions and find the holographic representations of vector and chiral currents reproducing the correct expression for the chiral anomaly. In five dimensions, Chern-Simons theory for AdS group describes first order gravity and we show that there exists a gauge fixing leading to a finite Fefferman-Graham expansion. We derive the corresponding holographic currents, namely, the stress tensor and spin current which couple to the metric and torsional degrees of freedom at the boundary, respectively. We obtain the correct Ward identities for these currents by looking at the bulk constraint equations.