Particle Number and 3D Schroedinger Holography

TL;DR

This paper explores the structure of asymptotically locally Schrödinger space-times in three dimensions, analyzing their solutions, boundary conditions, and implications for holographic renormalization, with a focus on the role of particle number.

Contribution

It introduces a new class of space-times called asymptotically locally Schrödinger, examines their solutions, and discusses conditions for Fefferman-Graham expansions related to particle number.

Findings

No universal Fefferman-Graham expansion for general solutions.

Exact solutions when the massive vector field is null.

Solutions where particle number is broken, either on the boundary or in the bulk.

Abstract

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.