Schroedinger Invariance from Lifshitz Isometries in Holography and Field Theory

TL;DR

This paper explores the connection between Lifshitz symmetries and Schrödinger invariance in non-relativistic field theories and their holographic duals, revealing how these symmetries manifest in Newton-Cartan geometry and Lifshitz space-times.

Contribution

It demonstrates the realization of Schrödinger algebra from Lifshitz isometries both in field theory and holographically, clarifying symmetry structures in non-relativistic holography.

Findings

Flat Newton-Cartan space-time exhibits two copies of Lifshitz algebra forming a Schrödinger algebra.

The Schrödinger scalar model possesses both copies of the algebra as symmetries.

Holographically, Lifshitz space-times realize the same two copies of the Lifshitz algebra.

Abstract

We study non-relativistic field theory coupled to a torsional Newton-Cartan geometry both directly as well as holographically. The latter involves gravity on asymptotically locally Lifshitz space-times. We define an energy-momentum tensor and a mass current and study the relation between conserved currents and conformal Killing vectors for flat Newton-Cartan backgrounds. It is shown that flat NC space-time realizes two copies of the Lifshitz algebra that together form a Schroedinger algebra (without the central element). We show why the Schroedinger scalar model has both copies as symmetries and the Lifshitz scalar model only one. Finally we discuss the holographic dual of this phenomenon by showing that the bulk Lifshitz space-time realizes the same two copies of the Lifshitz algebra.