We obtain the Lifshitz UV completion in a specific model for z ¼ 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.

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 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.

We consider two distinct limits of General Relativity that in contrast to the standard non-relativistic limit can be taken at the level of the Einstein-Hilbert action instead of the equations of motion. One is a non-relativistic limit and leads to a so-called Galilei gravity theory, the other is an ultra-relativistic limit yielding a so-called Carroll gravity theory. We present both gravity theories in a first-order formalism and show that in both cases the equations of motion (i) lead to constraints on the geometry and (ii) are not sufficient to solve for all of the components of the connection fields in terms of the other fields. Using a second-order formalism we show that these independent components serve as Lagrange multipliers for the geometric constraints we found earlier. We point out a few noteworthy differences between Carroll and Galilei gravity and give some examples of matter couplings.

Two dimensional Warped Conformal Field Theories (WCFTs) may represent the simplest examples of field theories without Lorentz invariance that can be described holographically. As such they constitute a natural window into holography in non AdS space-times, including the near horizon geometry of generic extremal black holes. It is shown in this paper that WCFTs posses a type of boost symmetry. Using this insight, we discuss how to couple these theories to background geometry. This geometry is not Riemannian. We call it Warped Geometry and it turns out to be a variant of a Newton-Cartan structure with additional scaling symmetries. With this formalism the equivalent of Weyl invariance in these theories is presented and we write two explicit examples of WCFTs. These are free fermionic theories. Lastly we present a systematic description of the holographic duals of WCFTs. It is argued that the minimal setup is not Einstein gravity but an SL(2, R) × U (1) Chern-Simons Theory, which we call Lower Spin Gravity. This point of view makes manifest the definition of boundary for these non AdS geometries. This case represents the first step towards understanding a fully invariant formalism for W N field theories and their holographic duals.