Holographic Reconstruction of 3D Flat Space-Time

We study asymptotically flat space-times in 3 dimensions for Einstein gravity near future null infinity and show that the boundary is described by Carrollian geometry. This is used to add sources to the BMS gauge corresponding to a non-trivial boundary metric in the sense of Carrollian geometry. We then solve the Einstein equations in a derivative expansion and derive a general set of equations that take the form of Ward identities. Next, it is shown that there is a well-posed variational problem at future null infinity without the need to add any boundary term. By varying the on-shell action with respect to the metric data of the boundary Carrollian geometry we are able to define a boundary energy-momentum tensor at future null infinity. We show that its diffeomorphism Ward identity is compatible with Einstein's equations. There is another Ward identity that states that the energy flux vanishes. It is this fact that is responsible for the enhancement of global symmetries to the full BMS$_3$ algebra when we are dealing with constant boundary sources. Using a notion of generalized conformal boundary Killing vector we can construct all conserved BMS$_3$ currents from the boundary energy-momentum tensor.Comment: v3: clarifications added, matches published versio

Gauging the Carroll Algebra and Ultra-Relativistic Gravity

It is well known that the geometrical framework of Riemannian geometry that underlies general relativity and its torsionful extension to Riemann-Cartan geometry can be obtained from a procedure known as gauging the Poincare algebra. Recently it has been shown that gauging the centrally extended Galilei algebra, known as the Bargmann algebra, leads to a geometrical framework that when made dynamical gives rise to Horava-Lifshitz gravity. Here we consider the case where we contract the Poincare algebra by sending the speed of light to zero leading to the Carroll algebra. We show how this algebra can be gauged and we construct the most general affine connection leading to the geometry of so-called Carrollian space-times. Carrollian space-times appear for example as the geometry on null hypersurfaces in a Lorentzian space-time of one dimension higher. We also construct theories of ultra-relativistic (Carrollian) gravity in 2+1 dimensions with dynamical exponent z<1 including cases that have anisotropic Weyl invariance for z=0.Comment: 27 page

Asymptotically Schroedinger Space-Times: TsT Transformations and Thermodynamics

We study the complete class of 5-dimensional asymptotically Schroedinger space-times that can be obtained as the TsT transform of a 5-dimensional asymptotically AdS space-time. Based on this we identify a conformal class of Schroedinger boundaries. We use a Fefferman-Graham type expansion to study the on-shell action for this class of asymptotically Schroedinger space-times and we show that its value is TsT invariant. In the second part we focus on black hole space-times and prove that black hole thermodynamics is also TsT invariant. We use this knowledge to argue that thermal global Schroedinger space-time at finite chemical potential undergoes a Hawking-Page type phase transition.Comment: References adde

Particle Number and 3D Schroedinger Holography

We define a class of space-times that we call asymptotically locally Schroedinger space-times. We consider these space-times in 3 dimensions, in which case they are also known as null warped AdS. The boundary conditions are formulated in terms of a specific frame field decomposition of the metric which contains two parts: an asymptotically locally AdS metric and a product of a lightlike frame field with itself. Asymptotically we say that the lightlike frame field is proportional to the particle number generator N regardless of whether N is an asymptotic Killing vector or not. We consider 3-dimensional AlSch space-times that are solutions of the massive vector model. We show that there is no universal Fefferman-Graham (FG) type expansion for the most general solution to the equations of motion. We show that this is intimately connected with the special role played by particle number. Fefferman-Graham type expansions are recovered if we supplement the equations of motion with suitably chosen constraints. We consider three examples. 1). The massive vector field is null everywhere. The solution in this case is exact as the FG series terminates and has N as a null Killing vector. 2). N is a Killing vector (but not necessarily null). 3). N is null everywhere (but not necessarily Killing). The latter case contains the first examples of solutions that break particle number, either on the boundary directly or only in the bulk. Finally, we comment on the implications for the problem of holographic renormalization for asymptotically locally Schroedinger space-times.Comment: 56 pages, v3: matches version published in JHE

BPS Open Strings and A-D-E-singularities in F-theory on K3

We improve on a recently constructed graphical representation of the supergravity 7-brane solution and apply this refined representation to re-study the open string description of the A-D-E-singularities in F-theory on K3. A noteworthy feature of the graphical representation is that it provides the complete global branch cut structure of the 7-brane solution which plays an important role in our analysis. We first identify those groups of branes which when made to coincide lead to the A-D-E-gauge groups. We next show that there is always a sufficient number of open BPS strings to account for all the generators of the gauge group. However, as we will show, there is in general no one-to-one relation between BPS strings and gauge group generators. For the D_{n+4}- and E-singularities, in order to relate BPS strings with gauge group generators, we make an SU(n+4), respectively SU(5) subgroup of the D_{n+4}- and E-gauge groups manifest. We find that only for the D-series (and for the standard A-series) this is sufficient to identify, in a one-to-one manner, which BPS strings correspond to which gauge group generators.Comment: 37 pages, 15 figure

Horava-Lifshitz Gravity From Dynamical Newton-Cartan Geometry

Recently it has been established that torsional Newton-Cartan (TNC) geometry is the appropriate geometrical framework to which non-relativistic field theories couple. We show that when these geometries are made dynamical they give rise to Horava-Lifshitz (HL) gravity. Projectable HL gravity corresponds to dynamical Newton-Cartan (NC) geometry without torsion and non-projectable HL gravity corresponds to dynamical NC geometry with twistless torsion (hypersurface orthogonal foliation). We build a precise dictionary relating all fields (including the scalar khronon), their transformations and other properties in both HL gravity and dynamical TNC geometry. We use TNC invariance to construct the effective action for dynamical twistless torsional Newton-Cartan geometries in 2+1 dimensions for dynamical exponent 1<z\le 2 and demonstrate that this exactly agrees with the most general forms of the HL actions constructed in the literature. Further, we identify the origin of the U(1) symmetry observed by Horava and Melby-Thompson as coming from the Bargmann extension of the local Galilean algebra that acts on the tangent space to TNC geometries. We argue that TNC geometry, which is manifestly diffeomorphism covariant, is a natural geometrical framework underlying HL gravity and discuss some of its implications.Comment: 48 page

Torsional Newton-Cartan Geometry and the Schr\"odinger Algebra

We show that by gauging the Schr\"odinger 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 $\tau_\mu$ 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 $z\neq 2$ 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 St\"uckelberg scalar $\chi$ that shifts under the particle number generator of the Schr\"odinger 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 Schr\"odinger symmetries play a crucial role in holography for Lifshitz space-times and that in fact the entire boundary geometry is dictated by local Schr\"odinger invariance. Finally we show how to extend the formalism to generic torsional Newton-Cartan geometries by relaxing the hypersurface orthogonality condition for the timelike vielbein $\tau_\mu$.Comment: v2: 38 pages, references adde

Schroedinger 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.Comment: 5 pages, modified abstract, clarifications added, typos fixed, refs update