Holographic RG from an exact RG: Locality and general coordinate invariance in the bulk

In earlier papers it was shown that the correct kinetic term for scalar, vector gauge field and the spin two field in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>AdS</a:mi></a:mrow><a:mrow><a:mi>D</a:mi><a:mo>+</a:mo><a:mn>1</a:mn></a:mrow></a:msub></a:mrow></a:math> space is obtained starting from the exact renormalization group (ERG) equation for a <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>CFT</c:mi><c:mi>D</c:mi></c:msub></c:math> 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 the Polchinski ERG equation that evolves toward 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>N</e:mi></e:math> semiclassical 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 anti–de Sitter scale). The map to <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>AdS</g:mi><g:mrow><g:mi>D</g:mi><g:mo>+</g:mo><g:mn>1</g:mn></g:mrow></g:msub></g:math> 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 <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>AdS</i:mi><i:mrow><i:mi>D</i:mi><i:mo>+</i:mo><i:mn>1</i:mn></i:mrow></i:msub></i:math>.