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Comments on Holography with Broken Lorentz Invariance

Ivan Gordeli, Peter Koroteev

TL;DR

The paper shows that the Kachru–Liu–Mulligan gravitational solution with broken Lorentz invariance is a member of an earlier Koroteev bulk family, establishing a concrete map between LIV parameters and bulk stress-energy through explicit metric identifications $($e.g., $t=L\tau$, $\mathbf{x}=L\mathbf{X}$, $r^2=e^{-2 kz}$$)$ and flux data $(A,B)$. It derives the bulk–boundary dictionary at tree level by computing scalar Green functions in the LIV backgrounds, demonstrating correct boundary scaling for the KL and Dubovsky critical cases, where the operator dimensions become energy- or momentum-dependent. The work unifies two seemingly distinct holographic constructions, clarifies how anisotropy in the bulk (via one-form flux) translates into LIV on the boundary, and discusses the RG-flow interpretation with IR AdS behavior, while highlighting avenues for embedding in a full string-theory framework. This broadens holographic methods to LIV backgrounds and provides a robust footing for studying boundary dynamics in anisotropic spacetimes.

Abstract

Recently a family of solutions of Einstein equations in backgrounds with broken Lorentz invariance was found ArXiv:0712.1136. We show that the gravitational solution recently obtained by Kachru, Liu and Mulligan in ArXiv:0808.1725 is a part of the former solution which was derived earlier in the framework of extra dimensional theories. We show how the energy-momentum and Einstein tensors are related and establish a correspondence between parameters which govern Lorentz invariance violation. Then we demonstrate that scaling behavior of two point correlation functions of local operators in scalar field theory is reproduced correctly for two cases with critical values of scaling parameters. Therefore, we complete the dictionary of "tree-level" duality for all known solutions of the bulk theory. In the end we speculate on relations between RG flow of a boundary theory and asymptotic behavior of gravitational solutions in the bulk.

Comments on Holography with Broken Lorentz Invariance

TL;DR

The paper shows that the Kachru–Liu–Mulligan gravitational solution with broken Lorentz invariance is a member of an earlier Koroteev bulk family, establishing a concrete map between LIV parameters and bulk stress-energy through explicit metric identifications e.g., , , and flux data . It derives the bulk–boundary dictionary at tree level by computing scalar Green functions in the LIV backgrounds, demonstrating correct boundary scaling for the KL and Dubovsky critical cases, where the operator dimensions become energy- or momentum-dependent. The work unifies two seemingly distinct holographic constructions, clarifies how anisotropy in the bulk (via one-form flux) translates into LIV on the boundary, and discusses the RG-flow interpretation with IR AdS behavior, while highlighting avenues for embedding in a full string-theory framework. This broadens holographic methods to LIV backgrounds and provides a robust footing for studying boundary dynamics in anisotropic spacetimes.

Abstract

Recently a family of solutions of Einstein equations in backgrounds with broken Lorentz invariance was found ArXiv:0712.1136. We show that the gravitational solution recently obtained by Kachru, Liu and Mulligan in ArXiv:0808.1725 is a part of the former solution which was derived earlier in the framework of extra dimensional theories. We show how the energy-momentum and Einstein tensors are related and establish a correspondence between parameters which govern Lorentz invariance violation. Then we demonstrate that scaling behavior of two point correlation functions of local operators in scalar field theory is reproduced correctly for two cases with critical values of scaling parameters. Therefore, we complete the dictionary of "tree-level" duality for all known solutions of the bulk theory. In the end we speculate on relations between RG flow of a boundary theory and asymptotic behavior of gravitational solutions in the bulk.

Paper Structure

This paper contains 9 sections, 33 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Correspondence of the Solutions
  3. Solution with perfect fluid.
  4. KLM Solution.
  5. Correspondence between the solutions.
  6. Two Point Correlation Functions
  7. The KL model.
  8. The Dubovsky model.
  9. Discussion

Figures (1)

  • Figure 1: The parameter space of metrics. Backgrounds and their mirror duals with known bulk solutions are shown. Although mirror transformation does not make sence in dimensions different from two, on this figure we refer to it as formal interchanging of $\xi$ and $\zeta$ (See Koroteev:2009qr for the details). Models obtained by reflection of $\xi$ and $\zeta$ are called anti-models. In this note we shall KL and Dubovsky models as critical ones since one of the scaling parameters vanish.