Holographic RG from ERG: Locality and General Coordinate Invariance in the Bulk

Abstract: In earlier papers it was shown that the correct kinetic term for scalar, vector gauge field and the spin two field in $AdS_{D+1}$ space is obtained starting from the ERG equation for a $CFT_D$ perturbed by scalar composite, conserved vector current and conserved traceless energy momentum tensor respectively. In this paper interactions are studied and it is shown that a flipped version of Polchinski ERG equation that evolves towards the UV can be written down and is useful for making contact with the usual AdS/CFT prescriptions for correlation function calculations. The scalar-scalar-spin-2 interaction in the bulk is derived from the ERG equation in the large $N$ semiclasical approximation. It is also shown that after mapping to AdS the interaction is local on a scale of the bare cutoff rather than the moving cutoff (which would have corresponded to the AdS scale). The map to $AdS_{D+1}$ plays a crucial role in this locality. The local nature of the coupling ensures that this interaction term in the bulk action is obtained by gauge fixing a general coordinate invariant scalar kinetic term in the bulk action. A wave function renormalization of the scalar field is found to be required for a mutually consistent map of the two fields to $AdS_{D+1}$.