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Correlation Functions in Holographic RG Flows

Ioannis Papadimitriou, Kostas Skenderis

TL;DR

The paper advances holographic renormalization by adopting a Hamiltonian framework that treats the radial coordinate as time, enabling renormalized correlators to be obtained from linearized fluctuations and only the relevant counterterm contributions. This method makes Ward identities and anomalies manifest while avoiding a full near-boundary analysis, significantly reducing computational effort across RG-flow backgrounds. It applies to both flat Poincaré domain walls and AdS-sliced walls, and includes a detailed treatment of the Janus solution, where vevs and wall-conformal two-point functions are shown to satisfy defect conformal symmetry. The results provide a consistent, efficient toolkit for computing n-point functions in holographic RG flows and offer a path to exploring non-AdS geometries and defect field theories.

Abstract

We discuss the computation of correlation functions in holographic RG flows. The method utilizes a recently developed Hamiltonian version of holographic renormalization and it is more efficient than previous methods. A significant simplification concerns the treatment of infinities: instead of performing a general analysis of counterterms, we develop a method where only the contribution of counterterms to any given correlator needs to be computed. For instance, the computation of renormalized 2-point functions requires only an analysis at the linearized level. We illustrate the method by discussing flat and AdS-sliced domain walls. In particular, we discuss correlation functions of the Janus solution, a recently discovered non-supersymmetric but stable AdS-sliced domain wall.

Correlation Functions in Holographic RG Flows

TL;DR

The paper advances holographic renormalization by adopting a Hamiltonian framework that treats the radial coordinate as time, enabling renormalized correlators to be obtained from linearized fluctuations and only the relevant counterterm contributions. This method makes Ward identities and anomalies manifest while avoiding a full near-boundary analysis, significantly reducing computational effort across RG-flow backgrounds. It applies to both flat Poincaré domain walls and AdS-sliced walls, and includes a detailed treatment of the Janus solution, where vevs and wall-conformal two-point functions are shown to satisfy defect conformal symmetry. The results provide a consistent, efficient toolkit for computing n-point functions in holographic RG flows and offer a path to exploring non-AdS geometries and defect field theories.

Abstract

We discuss the computation of correlation functions in holographic RG flows. The method utilizes a recently developed Hamiltonian version of holographic renormalization and it is more efficient than previous methods. A significant simplification concerns the treatment of infinities: instead of performing a general analysis of counterterms, we develop a method where only the contribution of counterterms to any given correlator needs to be computed. For instance, the computation of renormalized 2-point functions requires only an analysis at the linearized level. We illustrate the method by discussing flat and AdS-sliced domain walls. In particular, we discuss correlation functions of the Janus solution, a recently discovered non-supersymmetric but stable AdS-sliced domain wall.

Paper Structure

This paper contains 16 sections, 156 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hamiltonian Holographic Renormalization
  3. Poincaré Domain Walls
  4. Deformations vs VEVs
  5. Linearized Equations
  6. Examples
  7. AdS-sliced Domain Walls
  8. Janus solution
  9. VEVs
  10. Two-point functions
  11. Janus two-point functions vs boundary CFT
  12. Conclusions
  13. Conformal compactification of the Janus solution
  14. Fefferman-Graham coordinates for Janus metric
  15. Two-point functions for Janus from scalar momentum
  16. ...and 1 more sections

Figures (2)

  • Figure 1: The qualitatively different radial geodesics of the Janus geometry: (i) $a_2<u_o$ with $a_1>0$ (solid arrow) or $a_1<0$ (broken arrow), (ii) $a_2=u_o$ with $a_1>0$ (solid arrow) or $a_1<0$ (broken arrow) and (iii) $a_1=0$. These qualitative features are insensitive to the value of $b$.
  • Figure 2: A radial geodesic in the Fefferman-Graham coordinates (left) is not defined beyond $u=1$ due to the failure of the coordinate system at this point. In the coordinate system which extends beyond $u=1$ this geodesic bounces back at $u=1$. Only the branch before the bounce corresponds to the geodesic on the left.