# Holographic geometry for non-relativistic systems emerging from generalized flow equations

## TL;DR

This paper explores how non-relativistic holographic geometries, like Lifshitz and Schr"odinger spacetimes, can emerge from generalized flow equations applied to boundary theories with anisotropic scaling symmetries.

## Contribution

It introduces a new hybrid holographic geometry combining Lifshitz and Schr"odinger spacetimes derived from flow equations for non-relativistic systems.

## Key findings

- Derived a hybrid Lifshitz-Schr"odinger geometry from flow equations.
- Confirmed the geometry reduces to known Lifshitz and Schr"odinger spacetimes in special cases.
- Extended the holographic framework beyond conformal field theories.

## Abstract

An intriguing result presented by two of the present authors is that an anti de Sitter space can be derived from a conformal field theory by considering a flow equation. A natural expectation is that given a certain data on the boundary system, the associated geometry would be able to emerge from a flow, even beyond the conformal case. As a step along this line, we examine this scenario for non-relativistic systems with anisotropic scaling symmetries, such as Lifshitz field theories and Schr\"odinger invariant theories. In consequence we obtain a new hybrid geometry of Lifshitz and Schr\"odinger spacetimes as a general holographic geometry in this framework. We confirm that this geometry reduces to each of them by considering special non-relativistic models.