Schr\"odinger Manifolds

TL;DR

This paper defines Schr"odinger manifolds as Poincaré extensions of non-relativistic conformal structures, explores their examples and topology, and discusses their symmetries and relation to non-relativistic AdS/CFT correspondence.

Contribution

It introduces a formal definition of Schr"odinger manifolds, constructs explicit examples, analyzes their topology, and clarifies their connection to non-relativistic holography.

Findings

Schr"odinger manifolds admit the Schr"odinger group as their maximal isometry group.

Examples are constructed as homogeneous spaces of the Schr"odinger group.

The topology of Schr"odinger manifolds is systematically analyzed.

Abstract

This article propounds, in the wake of influential work of Fefferman and Graham about Poincar\'e extensions of conformal structures, a definition of a (Poincar\'e-)Schr\"odinger manifold whose boundary is endowed with a conformal Bargmann structure above a non-relativistic Newton-Cartan spacetime. Examples of such manifolds are worked out in terms of homogeneous spaces of the Schr\"odinger group in any spatial dimension, and their global topology is carefully analyzed. These archetypes of Schr\"odinger manifolds carry a Lorentz structure together with a preferred null Killing vector field; they are shown to admit the Schr\"odinger group as their maximal group of isometries. The relationship to similar objects arising in the non-relativisitc AdS/CFT correspondence is discussed and clarified.