Schr\"odinger Holography with $z=2$

TL;DR

This paper develops a holographic dictionary for asymptotically Schr"odinger spacetimes with dynamical exponent z=2, highlighting differences from previous models and demonstrating the existence of solutions with arbitrary sources.

Contribution

It proposes a new holographic dictionary for z=2 Schr"odinger spacetimes, emphasizing the unique role of the null direction and extending previous work for z<2.

Findings

Established a holographic dictionary for z=2 Schr"odinger spacetimes

Demonstrated solutions exist with arbitrary sources in the asymptotic expansion

Identified key differences from previous AdS deformation approaches

Abstract

We investigate holography for asymptotically Schr\"odinger spacetimes, using a frame formalism based on the anisotropic scaling symmetry. We build on our previous work on $z<2$ to propose a dictionary for $z=2$. For $z=2$, the scaling symmetry does not act on the additional null direction, which implies that in our dictionary it does not correspond to one of the field theory directions. This is significantly different from previous analyses based on viewing Schr\"odinger as a deformation of AdS. We study this dictionary in the linearised theory and in an asymptotic expansion. We show that a solution exists in an asymptotic expansion for arbitrary sources for the relevant operators in the stress energy complex.