Non-relativistic holography

Abstract: We consider holography for d-dimensional scale invariant but non-Lorentz invariant field theories, which do not admit the full Schrodinger symmetry group. We find new realizations of the corresponding (d+1)-dimensional gravity duals, engineered with a variety of matter Lagrangians, and their finite temperature generalizations. The thermodynamic properties of the finite temperature backgrounds are precisely those expected for anisotropic, scale invariant field theories. The brane and string theory realizations of such backgrounds are briefly discussed, along with their holographic interpretation in terms of marginal but non Lorentz invariant deformations of conformal field theories. We initiate discussion of holographic renormalization in these backgrounds, and note that such systematic renormalization is necessary to obtain the correct behavior of correlation functions.

• The paper's main contribution is the construction of finite temperature anisotropic holographic duals for non-relativistic field theories.
• It employs two methodological frameworks—massive vector fields and massless scalars coupled to gauge fields—to realize consistent gravity backgrounds.
• The study offers potential applications in condensed matter physics, paving the way for deeper insights into quantum critical behavior and non-Fermi liquid phenomena.

Analysis of Non-relativistic Holography

The paper "Non-relativistic Holography" by Marika Taylor investigates the duality between certain non-Lorentz invariant field theories and their gravitational duals, expanding the current paradigms of the AdS/CFT correspondence to encompass scale invariant, anisotropic field theories. This exploration, rooted in theoretical physics, navigates the complexities of achieving holographic duals for condensed matter applications, specifically addressing theories that lack traditional Galilean boosts and mass conservation.

Theoretical Context and Objectives

Taylor's work is situated within a broader initiative to apply holographic principles to condensed matter physics, specifically exploring the applicability of gravity duals to Galilean conformal field theories. These theories, characterized by their scale invariance with anisotropic scaling, present novel opportunities and challenges for holographic research. The symmetry group relevant to these theories includes time and spatial translations, spatial rotations, and potentially the special conformal transformation. This setup departs from the full Schrödinger symmetry, presenting a nuanced backdrop for holographic duality investigations.

The paper's primary aim involves constructing gravity duals for such field theories within a $d+1$-dimensional framework, utilizing carefully devised matter Lagrangians. The work seeks to establish finite temperature generalizations of these dualities, enabling thermodynamic analysis and potential experimental correlations within condensed matter physics.

Methodological Approaches

Taylor's methodology features two primary frameworks for constructing the desired gravity duals:

1. Massive Vector Field Approach: This approach leverages massive vector fields to produce backgrounds exhibiting both spatial and temporal anisotropy. Some significant advancements include achieving finite temperature generalizations for these backgrounds and establishing their thermodynamic compatibility with anisotropic, scale-invariant theories.
2. Massless Scalar Coupled to Gauge Field: Alternatively, Taylor explores a framework involving massless scalars coupled to gauge fields, which yields complimentary advantages in crafting anisotropic backgrounds. Notably, this method successfully realizes finite temperature generalizations that align thermodynamically with the theoretical expectations for the targeted field theories.

Key Findings and Implications

Among the paper's critical contributions is the identification of finite temperature anisotropic scale-invariant backgrounds and their subsequent use in exploring transport coefficients and phase structures. The introduction of a holographic renormalization procedure in these non-relativistic backgrounds, a departure from traditional Lorentz-invariant contexts, underscores the necessary adaptation of renormalization techniques to accurately capture correlation behaviors in these regimes.

Taylor's insights not only advance the theoretical constructs of holographic duals but also aim to integrate these developments into the broader context of string theory and its associated compactifications. This integration remains subject to ongoing exploration, as the paper suggests potential consistent truncations and underlying string theory realizations.

Conclusion and Future Directions

The exploration of non-relativistic holography, as presented in this paper, opens pathways for future research dedicated to resolving the theoretical challenges of embedding these models into supersymmetric frameworks and brane configurations. The work calls for robust examination of string theory compactifications to fully realize these gravity duals and speculates on potential intersection with known supersymmetric solutions.

The paper's discourse suggests promising implications for condensed matter physics, especially in its capacity to provide an analytic scaffold for anisotropic, large $N$ theories. As speculative as these avenues might be, Taylor's methodical advancements offer a foundational basis for future studies, encouraging further refinement of non-relativistic holographic renormalization and the application of these theories to realistic physical systems. By doing so, the work aims to deepen our understanding of quantum critical behaviors and non-Fermi liquid metals, contributing to the ongoing dialogue between theoretical predictions and experimental discoveries.