Jelle Hartong

Torsional Newton-Cartan Geometry and Lifshitz Holography

We obtain the Lifshitz UV completion in a specific model for z=2 Lifshitz geometries. We use a vielbein formalism which enables identification of all the sources as leading components of well-chosen bulk fields. We show that the geometry induced from the bulk onto the boundary is a novel extension of Newton-Cartan geometry with a specific torsion tensor. We explicitly compute all the vacuum expectation values (VEVs) including the boundary stress-energy tensor and their Ward identities. After using local symmetries or Ward identities the system exhibits 6+6 sources and VEVs. The Fefferman-Graham expansion exhibits, however, an additional free function which is related to an irrelevant operator whose source has been turned off. We show that this is related to a second UV completion.

Holographic models for theories with hyperscaling violation

A bstractWe study in detail a variety of gravitational toy models for hyperscaling-violating Lifshitz (hvLif) space-times. These space-times have been recently explored as holographic dual models for condensed matter systems. We start by considering a model of gravity coupled to a massive vector field and a dilaton with a potential. This model supports the full class of hvLif space-times and special attention is given to the particular values of the scaling exponents appearing in certain non-Fermi liquids. We study linearized perturbations in this model, and consider probe fields whose interactions mimic those of the perturbations. The resulting equations of motion for the probe fields are invariant under the Lifshitz scaling. We derive Breitenlohner-Freedman-type bounds for these new probe fields. For the cases of interest the hvLif space-times have curvature invariants that blow up in the UV. We study the problem of constructing models in which the hvLif space-time can have an AdS or Lifshitz UV completion. We also analyze reductions of Schrödinger space-times and reductions of waves on extremal (intersecting) branes, accompanied by transverse space reductions, that are solutions to supergravity-like theories, exploring the allowed parameter range of the hvLif scaling exponents.

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

A bstractFor 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 Hořava-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.

Lifshitz space–times for Schrödinger holography

Abstract We show that asymptotically locally Lifshitz space–times are holographically dual to field theories that exhibit Schrodinger invariance. This involves a complete identification of the sources, which describe torsional Newton–Cartan geometry on the boundary and transform under the Schrodinger algebra. We furthermore identify the dual vevs from which we define and construct the boundary energy–momentum tensor and mass current and show that these obey Ward identities that are organized by the Schrodinger algebra. We also point out that even though the energy flux has scaling dimension larger than z + 2 , it can be expressed in terms of computable vev/source pairs.

Schrödinger Invariance from Lifshitz Isometries in Holography and Field Theory

We study non-relativistic field theory coupled to a torsional Newton-Cartan geometry both directly as well as holographically. The latter involves gravity on asymptotically locally Lifshitz space-times. We define an energy-momentum tensor and a mass current and study the relation between conserved currents and conformal Killing vectors for flat Newton-Cartan backgrounds. It is shown that flat NC space-time realizes two copies of the Lifshitz algebra that together form a Schroedinger algebra (without the central element). We show why the Schroedinger scalar model has both copies as symmetries and the Lifshitz scalar model only one. Finally we discuss the holographic dual of this phenomenon by showing that the bulk Lifshitz space-time realizes the same two copies of the Lifshitz algebra.

Seven-branes and supersymmetry

We re-investigate the construction of half-supersymmetric 7-brane solutions of IIB supergravity. Our method is based on the requirement of having globally well-defined Killing spinors and the inclusion of SL(2,)-invariant source terms. In addition to the well-known solutions going back to Greene, Shapere, Vafa and Yau we find new supersymmetric configurations, containing objects whose monodromies are not related to the monodromy of a D7-brane by an SL(2,) transformation.

From D3-branes to Lifshitz spacetimes

We present a simple embedding of a z = 2 Lifshitz spacetime into type IIB supergravity. This is obtained by considering a stack of D3-branes in type IIB supergravity and deforming the world-volume by a plane wave. The plane wave is sourced by the type IIB axion. The superposition of the plane wave and the D3-branes is 1/4 BPS. The near horizon geometry of this configuration is a five-dimensional z = 0 Schr?dinger spacetime times a 5-sphere. This geometry is also 1/4 BPS. Upon compactification along the direction in which the wave is traveling the five-dimensional z = 0 Schr?dinger spacetime reduces to a four-dimensional z = 2 Lifshitz spacetime. The compactification is such that the circle is small for weakly coupled type IIB string theory. This reduction breaks the supersymmetries. Further, we propose a general method to construct analytic z = 2 Lifshitz black brane solutions. The method is based on deforming AdS5 black strings by an axion wave and reducing to four dimensions. We illustrate this method with an example.

Gauge Theories, Duality Relations and the Tensor Hierarchy

We compute the complete 3- and 4-dimensional tensor hierarchies, i.e. sets of p-form fields, with 1 ≤ p ≤ D, which realize an off-shell algebra of bosonic gauge transformations. We show how these tensor hierarchies can be put on-shell by introducing a set of duality relations, thereby introducing additional scalars and a metric tensor. These so-called duality hierarchies encode the equations of motion of the bosonic part of the most general gauged supergravity theories in those dimensions, including the (projected) scalar equations of motion. We construct gauge-invariant actions that include all the fields in the tensor hierarchies. We elucidate the relation between the gauge transformations of the p-form fields in the action and those of the same fields in the tensor hierarchy.

Torsional Newton–Cartan geometry and the Schrödinger algebra

We show that by gauging the Schrodinger algebra with critical exponent z and imposing suitable curvature constraints, that make diffeomorphisms equivalent to time and space translations, one obtains a geometric structure known as (twistless) torsional Newton-Cartan geometry (TTNC). This is a version of torsional Newton-Cartan geometry (TNC) in which the timelike vielbein t mu must be hypersurface orthogonal. For z = 2 this version of TTNC geometry is very closely related to the one appearing in holographic duals of z = 2 Lifshitz space-times based on Einstein gravity coupled to massive vector fields in the bulk. For z not equal 2 there is however an extra degree of freedom b(0) that does not appear in the holographic setup. We show that the result of the gauging procedure can be extended to include a Stuckelberg scalar chi that shifts under the particle number generator of the Schrodinger algebra, as well as an extra special conformal symmetry that allows one to gauge away b(0). The resulting version of TTNC geometry is the one that appears in the holographic setup. This shows that Schrodinger symmetries play a crucial role in holography for Lifshitz space-times and that in fact the entire boundary geometry is dictated by local Schrodinger invariance. Finally we show how to extend the formalism to generic TNC geometries by relaxing the hypersurface orthogonality condition for the timelike vielbein tau(mu).

Geometry of Schrödinger space-times, global coordinates, and harmonic trapping

We study various geometrical aspects of Schrodinger space-times with dynamical exponent z > 1 and compare them with the properties of AdS (z = 1). The Schrodinger metrics are singular for 1 2, we show that the Schrodinger space-times admit no global timelike Killing vectors.