Anisotropic Weyl invariance

TL;DR

This paper introduces a non-relativistic scalar field theory with anisotropic scale invariance, extending it to curved backgrounds with a local Weyl symmetry by using a novel geometric construction involving Cartesian products of curved spaces.

Contribution

It proposes a new geometric framework for extending anisotropic scale-invariant theories to curved backgrounds with local Weyl symmetry.

Findings

Identifies a family of curved geometries compatible with the theory.

Demonstrates promotion of anisotropic scale invariance to local Weyl symmetry.

Provides a geometric construction involving Cartesian products of curved spaces.

Abstract

We consider a non-relativistic free scalar field theory with a type of anisotropic scale invariance in which the number of coordinates "scaling like time" is generically greater than one. We propose the Cartesian product of two curved spaces, the metric of each space being parameterized by the other space, as a notion of curved background to which the theory can be extended. We study this type of geometries, and find a family of extensions of the theory to curved backgrounds in which the anisotropic scale invariance is promoted to a local, Weyl-type symmetry.