Extended Galilean symmetries of non-relativistic strings

TL;DR

The paper studies two non-relativistic limits of a relativistic string—the NR particle limit (non-vibrating) and the NR stringy limit (vibrating)—and derives the most general point transformations that leave the NR actions invariant via Noether symmetry. It solves the resulting NR Killing equations to uncover two distinct infinite-dimensional symmetry algebras that extend Galilean symmetry: an z = −1 extension for the NR particle/string and a stringy Galilean conformal-type structure for the vibrating case, with explicit conserved charges. In the vibrating case, non-central extensions appear in the charge algebra, while the non-vibrating case yields a clean infinite-dimensional Galilean extension without central terms. These results deepen the understanding of non-relativistic holography and provide soluble sectors of AdS/CFT-like correspondences in NR string theories.

Abstract

We consider two non-relativistic strings and their Galilean symmetries. These strings are obtained as the two possible non-relativistic (NR) limits of a relativistic string. One of them is non-vibrating and represents a continuum of non-relativistic massless particles, and the other one is a non-relativistic vibrating string. For both cases we write the generator of the most general point transformation and impose the condition of Noether symmetry. As a result we obtain two sets of non-relativistic Killing equations for the vector fields that generate the symmetry transformations. Solving these equations shows that NR strings exhibit two extended, infinite dimensional space-time symmetries which contain, as a subset, the Galilean symmetries. For each case, we compute the associated conserved charges and discuss the existence of non-central extensions.