What do non-relativistic CFTs tell us about Lifshitz spacetimes?

Abstract

Abstract:We study the reconstructability of (d + 2)-dimensional bulk spacetime from (d+1)-dimensional boundary data, particularly concentrating on backgrounds which break (d+1)-dimensional Lorentz invariance. For a large class of such spacetimes, there exist null geodesics which do not reach the boundary. Therefore classically one might guess some information is trapped in the bulk and thus invisible at the boundary. We show that this classical intuition correctly predicts the quantum situation: whenever there are null geodesics which do not reach the boundary, there are also "trapped scalar modes" whose boundary imprint is exponentially suppressed. We use these modes to show that no smearing function exists for pure Lifshitz spacetime, nor for any flow which includes a Lifshitz region. Indeed, for any (planar) spacetime which breaks (d + 1)-dimensional Lorentz invariance at any radius, we show that local boundary data cannot reconstruct complete local bulk data.

Boundary stress-energy tensor and Newton-Cartan geometry in Lifshitz holography

“…First of all, it would be interesting to study the probes in the Lifshitz space-time that we have briefly considered in section 2.2 and the associated two-point functions. Using the relation between AdS and Lifshitz probes one may get another perspective on the interesting results of [53]. An interesting generalization of our setup, which we leave for future work, will be to add charge to the five-dimensional theory and to compute the effects in the reduced theory.…”

“…[14,[44][45][46][47][48]. Probe fields and correlation functions have been studied in [13,14,[49][50][51][52][53][54].…”

Kasner-type singularities and solitons with Lifshitz asymptotics

“…The geodesic structure of different special cases of the solution (6) have been studied in various works. For instance, geodesics of asymptotically Lifshitz spacetimes have been studied by [32,33]. The D = 3 case in particular have been considered in [39,40].…”

“…It is well-known that only radial null geodesics are able to reach the boundary of Lifshitz spacetime [32], leading to various subtleties in the treatment of holographic quantities and must be dealt with care [33]. In any case, this is a consequence of the spacetime being dual to non-relativistic boundary by design, so that the speed of light is effectively infinite there.…”

Homogeneous nonrelativistic geometries as coset spaces

“…Instead of the Cartan-Killing metric, [44] uses the most general non-degenerate group invariant symmetric bilinear form, as [48] did for the Nappi-Witten WZW model on the non-semi simple group E c 2 , which is the centrally extended 2-dimensional Euclidean group. Thus [44] recovers the so-called Lifshitz and Schrödinger spacetimes, first proposed in [49, 50] and [51, 52], from a pseudo-Riemannian coset construction by performing cosets on the Lifshitz and Schrödinger groups respectively.However, aside from the somewhat counterintuitive concept of proposing locally relativistic spacetimes as duals to nonrelativistic field theories [34], difficulties with field profile reconstruction [53][54][55] as well as spacetime reconstruction [56] motivate the consideration of alternatives. As advocated in e.g.…”