# A worldsheet supersymmetric Newton-Cartan string

## TL;DR

This paper develops a supersymmetric string action in a non-relativistic Newton-Cartan background using a doubled string formalism, unifying relativistic and non-relativistic geometries and providing new tools for non-relativistic string theory.

## Contribution

It introduces a doubled string approach to construct a supersymmetric Newton-Cartan string, enabling a unified description of relativistic and non-relativistic geometries.

## Key findings

- Derived the Hamiltonian and Lagrangian forms of the supersymmetric Newton-Cartan string.
- Extracted geometric quantities and supersymmetry transformations from the worldsheet couplings.
- Showed the doubled formalism's potential for studying various non-relativistic backgrounds.

## Abstract

We construct a (locally) supersymmetric worldsheet action for a string in a non-relativistic Newton-Cartan background. We do this using a doubled string action, which describes the target space geometry in an $O(D,D)$ covariant manner using a doubled metric and doubled vielbeins. By adopting different parametrisations of these doubled background fields, we can describe both relativistic and non-relativistic geometries. We focus on the torsional Newton-Cartan geometry which can be obtained by null duality/reduction (such null duality is particularly simple for us to implement). The doubled action we use gives the Hamiltonian form of the supersymmetric Newton-Cartan string action automatically, from which we then obtain the equivalent Lagrangian. We extract geometric quantities of interest from the worldsheet couplings and write down the supersymmetry transformations. Our general results should apply to other non-relativistic backgrounds. We comment on the usefulness of the doubled approach as a tool for studying non-relativistic string theory.