Holographic $SO(2,d)$ anomaly

In the $SO(2,d)$ gauge theory formalism of AdS gravity established in arXiv:1811.05286, the dynamics of bulk gravity is emergent from the vanishing of the boundary covariant anomaly for the $SO(2,d)$ conservation law. Parallel with the known results of chiral anomalies, we establish the descendent structure of the holographic $SO(2,d)$ anomaly. The corresponding anomaly characteristic class, bulk Chern-Simons like action as well as the boundary effective action are constructed systematically. The anomalous conservation law is presented both in terms of the covariant and consistent formalisms. Due to the existence of the ruler field, not only the Bardeen-Zumino polynomial, but also the covariant and consistent currents are explicitly constructed.