Boundary Stress-Energy Tensor and Newton-Cartan Geometry in Lifshitz Holography (1311.6471v1)

Abstract: For a specific action supporting z=2 Lifshitz geometries we identify the Lifshitz UV completion by solving for the most general solution near the Lifshitz boundary. We identify all the sources as leading components of bulk fields which requires a vielbein formalism. This includes two linear combinations of the bulk gauge field and timelike vielbein where one asymptotes to the boundary timelike vielbein and the other to the boundary gauge field. The geometry induced from the bulk onto the boundary is a novel extension of Newton-Cartan geometry that we call torsional Newton-Cartan (TNC) geometry. There is a constraint on the sources but its pairing with a Ward identity allows one to reduce the variation of the on-shell action to unconstrained sources. We compute all the vevs along with their Ward identities and derive conditions for the boundary theory to admit conserved currents obtained by contracting the boundary stress-energy tensor with a TNC analogue of a conformal Killing vector. We also obtain the anisotropic Weyl anomaly that takes the form of a Horava-Lifshitz action defined on a TNC geometry. The Fefferman-Graham expansion contains a free function that does not appear in the variation of the on-shell action. We show that this is related to an irrelevant deformation that selects between two different UV completions.

• The paper introduces a novel frame formalism to connect Lifshitz holography with torsional Newton-Cartan geometry at z=2.
• It derives explicit boundary Ward identities and stress-energy tensor computations under gauge and Galilean boost symmetries.
• The study reveals an anisotropic Weyl anomaly resembling Hořava-Lifshitz models, paving the way for advances in non-relativistic holography.

Boundary Stress-Energy Tensor and Newton-Cartan Geometry in Lifshitz Holography

The paper entitled "Boundary Stress-Energy Tensor and Newton-Cartan Geometry in Lifshitz Holography" by Morten H. Christensen et al. explores the holographic framework within the specific context of z = 2 Lifshitz geometries, unveiling deep connections with Newton-Cartan geometry. The investigation is rooted in the possibility of defining a holographic dual for anisotropic scale-invariant field theories.

Core Contributions and Methodology

The authors tackle the challenge of identifying a viable ultraviolet (UV) completion for Lifshitz spacetimes. Their methodology is grounded in the application of a novel frame formalism to identify the sources as leading components of the bulk fields. Notably, this includes linear combinations of the bulk gauge field and timelike vielbein, which asymptote to the boundary fields such as the timelike vielbein and gauge field. This approach unveils a unique geometric structure on the boundary, termed as torsional Newton-Cartan (TNC) geometry.

The focus on TNC geometry marks a significant departure from the traditional AdS/CFT correspondence, extending the theoretical insights into non-relativistic holography. The usage of vielbein in the gravitational setup allows for a coherent description of boundary conditions and the identification of sources and corresponding vacuum expectation values (vevs).

Detailed Findings and Results

1. Torsional Newton-Cartan Geometry: The paper identifies that the boundary geometry, when subjected to specific reduction constraints, aligns with Newton-Cartan structures. TNC geometry extends Newton-Cartan by incorporating a specific torsion tensor which becomes pertinent when the timelike vielbein is not closed.
2. Frame Formalism and Ward Identities: The authors calculate all vevs, taking into account their transformations under various symmetry operations, including gauge transformations and local tangent space symmetries like Galilean boosts. The paper meticulously derives Ward identities associated with the boundary stress-energy tensor and provides conditions under which conserved boundary currents can be defined.
3. Anisotropic Weyl Anomaly: The paper presents the anisotropic Weyl anomaly associated with Lifshitz theories, revealing it in a form that resembles Hořava-Lifshitz Lagrangians. However, the crucial distinction lies in the underlying geometry, which is non-relativistic compared to the typical Lorentzian geometries in HL theories.
4. Numerical and Analytical Insights: The computations of the boundary stress-energy tensor in the presence of torsion provide outcomes aligned with the proposed theoretical framework, demonstrating that the analyses hold under the constraints imposed by the holographic renormalization group flows.
5. Second UV Completion: Beyond the Lifshitz UV completion, the authors consider a scenario involving a hyperscaling violation, which introduces distinctive UV features that affect the holographic correspondence between the boundary field theory and the bulk gravitational solution.

Implications and Future Developments

The paper opens avenues for further exploration into complex field theories described by non-relativistic holography, potentially enhancing our understanding of quantum critical phenomena within condensed matter physics. The utilization of TNC geometry can lead to advanced formulations that address how holographic principles manifest in non-traditional settings, greatly impacting the paper of quantum gravity and anisotropic field theories.

Future developments may include examining different values of the dynamical exponent beyond z = 2 and investigating the implications of additional holographic dualities involving more intricate geometric setups. Furthermore, leveraging this work in the context of real-world condensed matter systems or exploring its ramifications in the quest for a deeper understanding of quantum gravity through the lens of holography could yield substantial advancements.

Overall, the depth of the investigation by Christensen et al. showcases a crucial intersection of geometry, holography, and theoretical physics, providing a robust platform for theoretical exploration and practical insights into the field of non-relativistic holography.