AdS/CFT and local renormalization group with gauge fields

TL;DR

This paper explores the local renormalization group in the context of AdS/CFT, deriving flow equations with gauge fields, and establishing a holographic c-theorem through systematic scheme fixing and anomaly analysis.

Contribution

It provides a systematic derivation of local RG equations with gauge fields in AdS/CFT and proves a holographic c-theorem by fixing scheme ambiguities.

Findings

Derived local RG equations incorporating gauge fields.

Determined coefficients in the stress tensor trace for d=4.

Established a holographic c-theorem with a family of schemes.

Abstract

We revisit a study of local renormalization group (RG) with background gauge fields incorporated using the AdS/CFT correspondence. Starting with a $(d+1)$-dimensional bulk gravity coupled to scalars and gauge fields, we derive a local RG equation from a flow equation by working in the Hamilton-Jacobi formulation of the bulk theory. The Gauss's law constraint associated with gauge symmetry plays an important role. RG flows of the background gauge fields are governed by vector $\beta$-functions, and some interesting properties of them are known to follow. We give a systematic rederivation of them on the basis of the flow equation. Fixing an ambiguity of local counterterms in such a manner that is natural from the viewpoint of the flow equation, we determine all the coefficients uniquely appearing in the trace of the stress tensor for $d=4$. A relation between a choice of schemes and a Virial current is discussed. As a consistency check, these are found to satisfy the integrability conditions of local RG transformations. From these results, we are led to a proof of a holographic $c$-theorem by finding out a full family of schemes where a trace anomaly coefficient is related with a holographic $c$-function.