Generalized Newton-Cartan Geometries for Particles and Strings

TL;DR

This paper develops a framework for generalized Newton-Cartan geometries with torsion, providing invariant affine connections for particles and strings, and explores constraints on these geometries that align with existing literature.

Contribution

It introduces torsionful Newton-Cartan geometries with intrinsic torsion tensors and analyzes constraints for particles and strings, extending previous results in non-Lorentzian geometry.

Findings

Defined invariant affine connections in torsionful geometries

Identified intrinsic torsion tensors that cannot be absorbed into spin connections

Derived consistent geometric constraints for particles and strings

Abstract

We discuss the generalized Newton-Cartan geometries that can serve as gravitational background fields for particles and strings. In order to enable us to define affine connections that are invariant under all the symmetries of the structure group, we describe torsionful geometries with independent torsion tensors. A characteristic feature of the non-Lorentzian geometries we consider is that some of the torsion tensors are so-called `intrinsic torsion' tensors that cannot be absorbed in any of the spin connections. Setting some components of these intrinsic torsion tensors to zero leads to constraints on the geometry. For both particles and strings, we discuss various such constraints that can be imposed consistently with the structure group symmetries. In this way, we reproduce several results in the literature.