# Nonrelativistic pulsating strings

## TL;DR

This paper investigates nonrelativistic pulsating string configurations in torsion Newton-Cartan geometry, demonstrating their classical integrability through analytic criteria and conserved charges, with simpler structures in certain scaling limits.

## Contribution

It provides the first analysis of nonrelativistic pulsating strings in TNC geometry, establishing their Liouvillian integrability and exploring simpler integrable structures under specific scalings.

## Key findings

- Classical phase space is Liouvillian integrable.
- Conserved charges are computed for the 2D sigma model.
- Simpler integrable structures emerge under nonrelativistic scaling.

## Abstract

We explore nonrelativistic (NR) pulsating string configurations over torsion Newton-Cartan (TNC) geometry having topology $ R \times S^2 $ and check the corresponding analytic integrability criteria following Kovacic's algorithm. In the first part we consider pulsating strings propagating over TNC geometry whose world-sheet theory is described by relativistic CFTs. We compute conserved charges associated with the $ 2D $ sigma model and show that the classical phase space corresponding to these NR pulsating string configurations is Liouvillian integrable. Finally, we consider nonrelativisitc scaling associated with the world-sheet d.o.f. and show that the corresponding string configuration allows even simpler integrable structure.