# String Theory and String Newton-Cartan Geometry

## TL;DR

This paper derives nonrelativistic string theory from relativistic string theory via a limit of the target space geometry, establishing a geometric framework called string Newton-Cartan geometry and analyzing its equations of motion and T-duality.

## Contribution

It introduces string Newton-Cartan geometry as a limit of Riemannian geometry and applies this to nonrelativistic string theory, connecting it with beta-functions and T-duality.

## Key findings

- Derived string Newton-Cartan geometry as a limit of Riemannian geometry.
- Showed that nonrelativistic string theory can be obtained from relativistic string theory in curved backgrounds.
- Reproduced recent results on beta-functions and T-duality within this geometric framework.

## Abstract

Nonrelativistic string theory is described by a sigma model with a relativistic worldsheet and a nonrelativistic target spacetime geometry, that is called string Newton-Cartan geometry. In this paper we obtain string Newton-Cartan geometry as a limit of the Riemannian geometry of General Relativity with a fluxless two-form field. We then apply the same limit to relativistic string theory in curved background fields and show that it leads to nonrelativistic string theory in a string Newton-Cartan geometry coupled to a Kalb-Ramond and dilaton field background. Finally, we use our limiting procedure to study the spacetime equations of motion and the T-duality transformations of nonrelativistic string theory. Our results reproduce the recent studies of beta-functions and T-duality of nonrelativistic string theory obtained from the microscopic worldsheet definition of nonrelativistic string theory.