A Field-theoretical Interpretation of the Holographic Renormalization Group

TL;DR

This paper provides a quantum-field theoretical interpretation of the holographic renormalization group, relating it to local RG equations and exploring its implications for C-theorems in various dimensions.

Contribution

It establishes a connection between holographic RG equations and field-theoretical local RG equations, clarifying the scheme dependence and extending the C-theorem discussion to four dimensions.

Findings

Relation between holographic and field-theoretical C functions.

Analysis of scheme dependence due to counterterms.

Extension of holographic RG interpretation to mass deformations.

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.