A Field-theoretical Interpretation of the Holographic Renormalization Group

TL;DR

This work reframes the holographic renormalization group as Osborn's local RG, linking Weyl-anomaly structures in holography to field-theoretical consistency conditions. It derives a holographic C theorem in two dimensions from the Zamolodchikov metric and analyzes four-dimensional cases, highlighting scheme dependence via finite counterterms. By extending PBH-type diffeomorphisms to deformed AdS spaces and computing anomaly coefficients in a minimal subtraction-like scheme, it demonstrates that the holographic C function aligns with field-theoretical expectations under Weyl consistency in certain limits. The results provide a unified, holographically grounded perspective on C-functions and renormalization group flows with potential implications for holographic duals beyond standard AdS/CFT setups.

Abstract

A quantum-field theoretical interpretation is given to the holographic RG equation by relating it to a field-theoretical local RG equation which determines how Weyl invariance is broken in a quantized field theory. Using this approach we determine the relation between the holographic C theorem and the C theorem in two-dimensional quantum field theory which relies on the Zamolodchikov metric. Similarly we discuss how in four dimensions the holographic C function is related to a conjectured field-theoretical C function. The scheme dependence of the holographic RG due to the possible presence of finite local counterterms is discussed in detail, as well as its implications for the holographic C function. We also discuss issues special to the situation when mass deformations are present. Furthermore we suggest that the holographic RG equation may also be obtained from a bulk diffeomorphism which reduces to a Weyl transformation on the boundary.