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Holographic Correlators in a Flow to a Fixed Point

Marcus Berg, Henning Samtleben

TL;DR

This work computes two-point functions along a conformal-to-conformal RG flow between 2D CFTs using holographic renormalization in a exact domain-wall background of a 3D gauged supergravity, driven by an operator of dimension $3/2$ and preserving $1/4$ of the supersymmetry. The authors derive the near-boundary expansions, construct the full renormalized action with covariant counterterms, and obtain correlators for the stress-energy tensor and scalars, including inert and active multiplets, while unveiling a SUSY QM prepotential structure that ensures stability of fluctuations. They analyze the two-point functions throughout the flow, establish a SUSY Ward identity relating inert and active sectors, and extract a monotonic holographic $C$ function which they compare with Zamolodchikov’s construction, finding consistent IR/UV behavior $c_{ m IR}/c_{ m UV}=1/2$ and a close but scheme-dependent agreement with $C_{ m hol}$. The work also resolves puzzles encountered in similar five-dimensional flows by exploiting exact solvability, including reduction of fluctuation equations to a biconfluent Heun form and the role of finite counterterms in preserving supersymmetry. Overall, the results provide a concrete, exact holographic framework for analyzing deformations of 2D CFTs, offering quantitative predictions for correlators and central-function behavior along RG flows between fixed points.

Abstract

Using holographic renormalization, we study correlation functions throughout a renormalization group flow between two-dimensional superconformal field theories. The ultraviolet theory is an N=(4,4) CFT which can be thought of as a symmetric product of U(2) super WZW models. It is perturbed by a relevant operator which preserves one-quarter supersymmetry and drives the theory to an infrared fixed point. We compute correlators of the stress-energy tensor and of the relevant operators dual to supergravity scalars. Using the former, we put together Zamolodchikov's C function, and contrast it with proposals for a holographic C function. In passing, we address and resolve two puzzles also found in the case of five-dimensional bulk supergravity.

Holographic Correlators in a Flow to a Fixed Point

TL;DR

This work computes two-point functions along a conformal-to-conformal RG flow between 2D CFTs using holographic renormalization in a exact domain-wall background of a 3D gauged supergravity, driven by an operator of dimension and preserving of the supersymmetry. The authors derive the near-boundary expansions, construct the full renormalized action with covariant counterterms, and obtain correlators for the stress-energy tensor and scalars, including inert and active multiplets, while unveiling a SUSY QM prepotential structure that ensures stability of fluctuations. They analyze the two-point functions throughout the flow, establish a SUSY Ward identity relating inert and active sectors, and extract a monotonic holographic function which they compare with Zamolodchikov’s construction, finding consistent IR/UV behavior and a close but scheme-dependent agreement with . The work also resolves puzzles encountered in similar five-dimensional flows by exploiting exact solvability, including reduction of fluctuation equations to a biconfluent Heun form and the role of finite counterterms in preserving supersymmetry. Overall, the results provide a concrete, exact holographic framework for analyzing deformations of 2D CFTs, offering quantitative predictions for correlators and central-function behavior along RG flows between fixed points.

Abstract

Using holographic renormalization, we study correlation functions throughout a renormalization group flow between two-dimensional superconformal field theories. The ultraviolet theory is an N=(4,4) CFT which can be thought of as a symmetric product of U(2) super WZW models. It is perturbed by a relevant operator which preserves one-quarter supersymmetry and drives the theory to an infrared fixed point. We compute correlators of the stress-energy tensor and of the relevant operators dual to supergravity scalars. Using the former, we put together Zamolodchikov's C function, and contrast it with proposals for a holographic C function. In passing, we address and resolve two puzzles also found in the case of five-dimensional bulk supergravity.

Paper Structure

This paper contains 24 sections, 146 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. An exact holographic conformal-to-conformal flow
  3. Field equations and near boundary analysis
  4. Perturbative solution
  5. Metric coefficients
  6. Scalar coefficients
  7. Counterterms and one-point functions
  8. Counterterms
  9. One-point functions: generalities
  10. Inert scalars
  11. Active scalar
  12. Stress-energy tensor
  13. Two-point functions of inert scalars
  14. Fluctuation equations and the existence of prepotentials
  15. Distinguishing $\Delta_+$ and $\Delta_-$
  16. ...and 9 more sections

Figures (2)

  • Figure 1: Contour plot of the scalar potential $V(Z_1,Z_2)$ (\ref{['Vxy']}) and the flow trajectory.
  • Figure 2: Holographic $C$ function (\ref{['Chol']}) (solid) and Zamolodchikov's $C$ function (\ref{['CZam']}) (dashed) from holographic correlators as functions of $\mu$ after normalizing and rescaling such as to match the first derivative in the UV.