`Stringy' Newton-Cartan Gravity

TL;DR

This work develops a stringy generalization of Newton-Cartan gravity that treats Galilean observers as fundamental, extending non-relativistic gravity from particles to strings via an $M_{1,1}$-foliation. The authors derive both the string geodesic equations and the bulk equations of motion in terms of a symmetric longitudinal gravitational potential $\Phi_{\alpha\beta}(x)$, and show two equivalent gauging approaches—transverse translations or the full stringy Galilei algebra—yielding the same results. They further extend the construction to include a negative cosmological constant, obtaining AdS$_2$-foliation and a stringy Newton-Hooke symmetry, with a modified Poisson equation for $\Phi_{\alpha\beta}$ that couples to the cosmological constant. This provides a robust framework for exploring non-relativistic AdS/CFT scenarios based on stringy Galilei symmetries and sets the stage for potential null reductions and supersymmetric extensions.

Abstract

We construct a "stringy" version of Newton-Cartan gravity in which the concept of a Galilean observer plays a central role. We present both the geodesic equations of motion for a fundamental string and the bulk equations of motion in terms of a gravitational potential which is a symmetric tensor with respect to the longitudinal directions of the string. The extension to include a non-zero cosmological constant is given. We stress the symmetries and (partial) gaugings underlying our construction. Our results provide a convenient starting point to investigate applications of the AdS/CFT correspondence based on the non-relativistic "stringy" Galilei algebra.