# Non-Relativistic Strings and Limits of the AdS/CFT Correspondence

## TL;DR

This paper introduces a new non-relativistic string theory framework via null reduction, connecting it to Spin Matrix theory limits of AdS/CFT and integrability, and providing a covariant Landau-Lifshitz model.

## Contribution

It develops a covariant action for non-relativistic strings in $U(1)$-Galilean geometry and links zero tension limits to Spin Matrix theory within AdS/CFT.

## Key findings

- Derived a covariant action for non-relativistic strings in torsional Newton-Cartan geometry.
- Connected zero tension string limits to Spin Matrix theory in AdS/CFT.
- Presented a covariant Landau-Lifshitz sigma-model as an example.

## Abstract

Using target space null reduction of the Polyakov action we find a novel covariant action for strings moving in a torsional Newton-Cartan geometry. Sending the string tension to zero while rescaling the Newton-Cartan clock 1-form, so as to keep the string action finite, we obtain a non-relativistic string moving in a new type of non-Lorentzian geometry that we call $U(1)$-Galilean geometry. We apply this to strings on $AdS_5 \times S^5$ for which we show that the zero tension limit is realized by the Spin Matrix theory limits of the AdS/CFT correspondence. This is closely related to limits of spin chains studied in connection to integrability in AdS/CFT. The simplest example gives a covariant version of the Landau-Lifshitz sigma-model.