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Penrose Limits and RG Flows

Eric G. Gimon, Leopoldo A. Pando Zayas, Jacob Sonnenschein

TL;DR

The paper develops a bridge between RG flows in non-conformal gauge theories and the time evolution of strings on time-dependent pp-wave backgrounds generated by Penrose limits. It shows that the worldsheet dynamics reduces to an exactly solvable time-dependent harmonic oscillator, enabling a precise mapping between operator mixing under RG flow and transition amplitudes in the quantum-mechanical system. Two main testing grounds are analyzed: the Pilch-Warner flow from ${ m N}=4$ to ${ m N}=1$ and the near-horizon limits of large ${N}$ D$_p$ branes, revealing how different geodesics yield distinct pp-wave backgrounds and operator bases, and highlighting the role of RR/NS-NS flux as well as the dilaton. The study also uncovers subtleties in the IR for D$_p$ backgrounds, where effective negative worldsheet masses appear, necessitating a change of description (e.g., D1→F1 or D2→M2) to maintain a consistent Hilbert space, thus offering insights into non-conformal holography and the limits of the Penrose limit approach.

Abstract

The Penrose-Gueven limit simplifies a given supergravity solution into a pp-wave background. Aiming at clarifying its relation to renormalization group flow we study the Penrose-Guven limit of supergravity backgrounds that are dual to non-conformal gauge theories. The resulting backgrounds fall in a class simple enough that the quantum particle is exactly solvable. We propose a map between the effective time-dependent quantum mechanical problem and the RG flow in the gauge theory. As a testing ground we consider explicitly two Penrose limits of the infrared fixed point of the Pilch-Warner solution. We analyze the corresponding gauge theory picture and write down the operators which are the duals of the low lying string states. We also address RG flows of a different nature by considering the Penrose-Gueven limit of a stack of N D_p branes. We note that in the far IR (for p<3)the limit generically has negative mass-squared. This phenomenon signals, in the world sheet picture, the necessity to transform to another description. In this regard, we consider explicitly the cases of M2 from D2 and F1 from D1 .

Penrose Limits and RG Flows

TL;DR

The paper develops a bridge between RG flows in non-conformal gauge theories and the time evolution of strings on time-dependent pp-wave backgrounds generated by Penrose limits. It shows that the worldsheet dynamics reduces to an exactly solvable time-dependent harmonic oscillator, enabling a precise mapping between operator mixing under RG flow and transition amplitudes in the quantum-mechanical system. Two main testing grounds are analyzed: the Pilch-Warner flow from to and the near-horizon limits of large D branes, revealing how different geodesics yield distinct pp-wave backgrounds and operator bases, and highlighting the role of RR/NS-NS flux as well as the dilaton. The study also uncovers subtleties in the IR for D backgrounds, where effective negative worldsheet masses appear, necessitating a change of description (e.g., D1→F1 or D2→M2) to maintain a consistent Hilbert space, thus offering insights into non-conformal holography and the limits of the Penrose limit approach.

Abstract

The Penrose-Gueven limit simplifies a given supergravity solution into a pp-wave background. Aiming at clarifying its relation to renormalization group flow we study the Penrose-Guven limit of supergravity backgrounds that are dual to non-conformal gauge theories. The resulting backgrounds fall in a class simple enough that the quantum particle is exactly solvable. We propose a map between the effective time-dependent quantum mechanical problem and the RG flow in the gauge theory. As a testing ground we consider explicitly two Penrose limits of the infrared fixed point of the Pilch-Warner solution. We analyze the corresponding gauge theory picture and write down the operators which are the duals of the low lying string states. We also address RG flows of a different nature by considering the Penrose-Gueven limit of a stack of N D_p branes. We note that in the far IR (for p<3)the limit generically has negative mass-squared. This phenomenon signals, in the world sheet picture, the necessity to transform to another description. In this regard, we consider explicitly the cases of M2 from D2 and F1 from D1 .

Paper Structure

This paper contains 21 sections, 118 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Strings on a time-dependent pp--wave background: General structure
  3. The geometry of the pp-wave limit of non-conformal backgrounds
  4. The particle on the tdppw background
  5. The particle on the tdppw background
  6. The bosonic tdppw string
  7. The RG flow in the dual field theories
  8. The Penrose-Güven limit of the Pilch-Warner solution
  9. The Pilch-Warner solution-a brief review
  10. Penrose Limit of the Pilch-Warner solution in the IR
  11. Flows in the corresponding field theory
  12. The $\theta = {\pi \over 2}$ Flow
  13. The $\theta = 0$ Flow
  14. General Comments on the limits
  15. The pp--wave limit of large N $D_p$ branes
  16. ...and 6 more sections

Figures (6)

  • Figure 1: $\mu(x^+)$ for various values of $E$; $E_0 < E_1 < E_2$
  • Figure 2: Motion of a stiff string in a background of negative mass squared.
  • Figure 3: Motion and splitting of a string in a background of large negative mass squared.
  • Figure 4: The values of the masses for D1
  • Figure 5: The values of the masses for D2
  • ...and 1 more figures