Non-relativistic Holography Research Articles

Nonrelativistic Holography from AdS_{5}/CFT_{4}.

We show that a novel holographic correspondence appears after taking a suitable stringy nonrelativistic limit of the AdS_{5}/CFT_{4} duality. This correspondence relates string theory in string Newton-Cartan AdS_{5}×S^{5} with Galilean electrodynamics with five scalars defined on the Penrose conformal boundary. As a first test, we match the Killing vectors on the string theory side with the on-shell symmetries of the dual field theory.

The mass of a Lifshitz black hole

It is well known that massive 3D gravity admits solutions that describe Lifshitz black holes as those considered in non-relativistic holography. However, the determination of the mass of such black holes remained unclear as many different results were reported in the literature presenting discrepancies. Here, by using a robust method that permits to tackle the problem in the strong field regime, we determine the correct mass of the Lifshitz black hole of the higher-derivative massive gravity and compare it with other results obtained by different methods. Positivity of the mass spectrum demands an odd normalization of the gravity action. In spite of this fact, the result turns out to be consistent with computations inspired in holography.

General theory of Galilean gravity

We obtain the complete theory of Newton-Cartan gravity in a curved spacetime by considering the large $c$ limit of the vielbein formulation of General Relativity. Milne boosts originate from local Lorentzian transformations, and the special cases of torsionless and twistless torsional geometries are explained in the context of the larger locally Lorentzian theory. We write the action for Newton-Cartan fields in the first order Palatini formalism, and the large $c$ limit of the Einstein equations. Finally, we obtain the generalised Eisenhart-Duval lift of the metric that plays an important role in non-relativistic holography.

Disformal transformation in Newton–Cartan geometry

Newton-Cartan geometry has played a central role in recent discussions of non-relativistic holography and condensed matter systems. Although the conformal transformation in non-relativistic holography can be easily rephrased in Newton-Cartan geometry, we show that it requires a nontrivial procedure to get the consistent form of anisotropic disformal transformation in this geometry. Furthermore, as an application of the newly obtained disformal transformation, we use it to induce a new geometry.

Hyperscaling violating Lifshitz holography

We present an overview of the construction of the general holographic dictionary for asymptotically locally Lifshitz and hyperscaling violating Lifshitz backgrounds with arbitrary dynamical exponents z and θ, compatible with the null energy condition, which was recently developed in [W. Chemissany, I. Papadimitriou, Generalized dilatation operator method for non-relativistic holography, arXiv:1405.3965, W. Chemissany, I. Papadimitriou, Lifshitz holography: The whole shebang, arXiv:1408.0795]. A concrete definition of asymptotically locally Lifshitz and hyperscaling violating Lifshitz backgrounds is provided in the context of a generic bottom-up Einstein-Proca-Dilaton theory, and a systematic procedure for solving the radial Hamilton-Jacobi equation via a covariant expansion in eigenfunctions of two commuting operators is presented. The resulting asymptotic solution of the Hamilton-Jacobi equation is subsequently used to derive the full holographic dictionary, including the Fefferman-Graham asymptotic expansions and the non-relativistic holographic Ward identities.

Lightlike limit of entanglement entropy

We study certain classes of $g_{++}$ deformations of theories arising in gauge/string realizations of nonrelativistic holography, some of which pertain to $z=2$ Lifshitz theories while others (pertaining to hyperscaling violation) comprise certain classes of excited states. Building on previous work, we consider holographic entanglement entropy for spacelike strip subsystems in a highly boosted (lightlike) limit, where the strip is stretched along the null $x^+$-plane. The leading divergence in entanglement in this null limit for these excited states is milder than the usual area law for spacelike subsystems in ground states. For ground states, the entanglement vanishes, perhaps consistent with ultralocality. We discuss this briefly from a field theory perspective. We also present some simple free lightfront field theory examples in excited states where correlators are nonvanishing.

Holographic charge diffusion in non-relativistic branes

In this paper, based on the principles of gauge/gravity duality and considering the so called hydrodynamic limit we compute various charge transport properties for a class of strongly coupled non-relativistic CFTs corresponding to z=2 fixed point whose dual gravitational counter part could be realized as the consistent truncation of certain non-relativistic Dp branes in the non-extremal limit. From our analysis we note that unlike the case for the AdS black branes, the charge diffusion constant in the non-relativistic background scales differently with the temperature. This shows a possible violation of the universal bound on the charge conductivity to susceptibility ratio in the context of non-relativistic holography.

Effects of high-order operators in nonrelativistic Lifshitz holography

In this paper, we study the effects of high-order operators on the non-relativistic Lifshitz holography in the framework of the Ho\v{r}ava-Lifshitz (HL) theory of gravity, which naturally contains high-order operators in order for the theory to be power-counting renormalizble, and provides an ideal place for such studies. In particular, we show that the Lifshitz space-time is still a solution of the full theory of the HL gravity. The effects of the high-oder operators on the space-time itself is simply to shift the Lifshitz dynamical exponent. However, while in the infrared the asymptotic behavior of a (probe) scalar field near the boundary is similar to that studied in the literature, it gets dramatically modified in the UV limit, because of the presence of the high-order operators in this regime. Then, according to the gauge/gravity duality, this in turn affects the two-point correlation functions.

The Eisenhart lift: a didactical introduction of modern geometrical concepts from Hamiltonian dynamics

This work originates from part of a final year undergraduate research project on the Eisenhart lift for Hamiltonian systems. The Eisenhart lift is a procedure to describe trajectories of a classical natural Hamiltonian system as geodesics in an enlarged space. We point out that it can be easily obtained from basic principles of Hamiltonian dynamics, and as such it represents a useful didactical way to introduce graduate students to several modern concepts of geometry applied to physics: curved spaces, both Riemannian and Lorentzian, conformal transformations, geometrization of interactions and extra dimensions, and geometrization of dynamical symmetries. For all these concepts the Eisenhart lift can be used as a theoretical tool that provides easily achievable examples, with the added benefit of also being a topic of current research with several applications, among which are included the study of dynamical systems and non-relativistic holography.

Generalized dilatation operator method for non-relativistic holography

We present a general algorithm for constructing the holographic dictionary for Lifshitz and hyperscaling violating Lifshitz backgrounds for any value of the dynamical exponent z and any value of the hyperscaling violation parameter θ compatible with the null energy condition. The objective of the algorithm is the construction of the general asymptotic solution of the radial Hamilton–Jacobi equation subject to the desired boundary conditions, from which the full dictionary can be subsequently derived. Contrary to the relativistic case, we find that a fully covariant construction of the asymptotic solution for running non-relativistic theories necessitates an expansion in the eigenfunctions of two commuting operators instead of one. This provides a covariant but non-relativistic grading of the expansion, according to the number of time derivatives.

NONRELATIVISTIC HOLOGRAPHY — A GROUP-THEORETICAL PERSPECTIVE

We give a review of some group-theoretical results related to nonrelativistic holography. Our main playgrounds are the Schrödinger equation and the Schrödinger algebra. We first recall the interpretation of nonrelativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. One important result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory, and that these operators and the bulk-to-boundary operators are intertwining operators. Further, we recall the fact that there is a hierarchy of equations on the boundary, invariant with respect to Schrödinger algebra. We also review the explicit construction of an analogous hierarchy of invariant equations in the bulk, and that the two hierarchies are equivalent via the bulk-to-boundary intertwining operators. The derivation of these hierarchies uses a mechanism introduced first for semisimple Lie groups and adapted to the nonsemisimple Schrödinger algebra. These require development of the representation theory of the Schrödinger algebra which is reviewed in some detail. We also recall the q-deformation of the Schrödinger algebra. Finally, the realization of the Schrödinger algebra via difference operators is reviewed.

Non-relativistic holography from Hořava gravity

Many non-relativistic Quantum Field Theories with conserved particle number share a common set of symmetries: time dependent spatial diffeomorphisms acting on the background metric and U(1) invariance acting on the background fields which couple to particle number. Here we use these symmetries to deduce a gravity dual for any such theory in terms of a non-relativistic theory of gravity, a variant of Horava gravity. This duality allows the extension of holography to generic non-relativistic field theories. As Horava gravity is presumed to be a consistent quantum theory, this duality also allows holography to move beyond the limit of a large number of colors, in principle. In the case when the field theory is conformally invariant, we prove that our proposal reproduces the form of the two point function demanded by this symmetry.

Schrödinger holography with and without hyperscaling violation

We study the properties of the Schr\"odinger-type non-relativistic holography for general dynamical exponent z with and without hyperscaling violation exponent \theta. The scalar correlation function has a more general form due to general z as well as the presence of \theta, whose effects also modify the scaling dimension of the scalar operator. We propose a prescription for minimal surfaces of this "codimension 2 holography," and demonstrate the (d-1) dimensional area law for the entanglement entropy from (d+3) dimensional Schr\"odinger backgrounds. Surprisingly, the area law is violated for d+1 < z < d+2, even without hyperscaling violation, which interpolates between the logarithmic violation and extensive volume dependence of entanglement entropy. Similar violations are also found in the presence of the hyperscaling violation. Their dual field theories are expected to have novel phases for the parameter range, including Fermi surface. We also analyze string theory embeddings using non-relativistic branes.

Schrödinger algebra and non-relativistic holography

We give a group-theoretic interpretation of non-relativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. The main result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory (without specifying any action). Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the relativistic case, we show that each bulk field has two boundary fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary. Analogously to the relativistic result of Klebanov-Witten we give the conditions when both boundary fields are physical.

Real-time correlators and non-relativistic holography

We consider Lorentzian correlation functions in theories with non-relativistic Schrodinger symmetry. We employ the method developed by Skenderis and van Rees in which the contour in complex time defining a given correlation function is associated holographically with the gluing together of Euclidean and Lorentzian patches of spacetimes. This formalism extends appropriately to geometries with Schrodinger isometry.

Intertwining operator realization of non-relativistic holography

We give a group-theoretic interpretation of non-relativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. Our main result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory (without specifying any action). Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the relativistic case, we show that each bulk field has two boundary fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary. Analogously to the relativistic result of Klebanov–Witten we give the conditions when both boundary fields are physical. Finally, we recover in our formalism earlier non-relativistic results for scalar fields by Son and others.

Non-relativistic holography and singular black hole

We provide a framework for non-relativistic holography so that a covariant action principle ensuring the Galilean symmetry for dual conformal field theory is given. This framework is based on the Bargmann lift of the Newton–Cartan gravity to the one-dimensional higher Einstein gravity, or reversely, the null-like Kaluza–Klein reduction. We reproduce the previous zero temperature results, and our framework provides a natural explanation about why the holography is co-dimension 2. We then construct the black hole solution dual to the thermal CFT, and find the horizon is curvature singular. However, we are able to derive the sensible thermodynamics for the dual non-relativistic CFT with correct thermodynamical relations. Besides, our construction admits a null Killing vector in the bulk such that the Galilean symmetry is preserved under the holographic RG flow. Finally, we evaluate the viscosity and find it zero if we neglect the back reaction of the singular horizon, otherwise, it could be non-zero.

A holographic perspective on non-relativistic conformal defects

We study defects in non-relativistic conformal field theories. As in the well-studied case of relativistic conformal defects, we find that a useful tool to organize correlation functions is the defect operator expansion (dOPE). We analyze how the dOPE is implemented in theories with a holographic dual, highlighting some interesting aspects of the operator/state mapping in non-relativistic holography.

Correlation functions in non-relativistic holography

Recently constructed gravity solutions with Schrodinger symmetry provide a new example of AdS/CFT-type dualities for the type of non-relativistic field theories relevant to certain cold atom systems. In this paper we use the gravity side to calculate n-point correlation functions of scalar fields by reducing the computation to that in ordinary AdS space via a particular Fourier transform. We evaluate the relevant integrals for 3- and 4-point functions and show that the results are consistent with the requirements of Schrodinger invariance, the implications of which we also work out for general n-point functions.

Supersymmetric solutions for non-relativistic holography

We construct families of supersymmetric solutions of type IIB and D=11 supergravity that are invariant under the non-relativistic conformal algebra for various values of dynamical exponent z greater than or equal to 4 and 3, respectively. The solutions are based on five- and seven-dimensional Sasaki-Einstein manifolds and generalise the known solutions with dynamical exponent z=4 for the type IIB case and z=3 for the D=11 case, respectively.