Torsional string Newton-Cartan geometry for non-relativistic strings

TL;DR

This paper introduces a novel formulation of non-relativistic string theory with a unique target space geometry containing a two-form field, derived from limits and reductions of relativistic strings, and characterized by a gauged algebra without extraneous fields.

Contribution

It presents a new geometric formulation of non-relativistic strings involving a two-form field and a gauged algebra, derived from null reduction and infinite speed of light limits.

Findings

Target space geometry includes a two-form coupling to tension current.

Formulation derived from null reduction and infinite speed of light limit.

No superfluous fields or foliation constraints in the geometry.

Abstract

We revisit the formulation of non-relativistic (NR) string theory and its target space geometry. We obtain a new formulation in which the geometry contains a two-form field that couples to the tension current and that transforms under string Galilei boosts. This parallels the Newton-Cartan one-form that couples to the mass current of a non-relativistic point particle. We show how this formulation of the NR string arises both from an infinite speed of light limit and a null reduction of the relativistic closed bosonic string. In both cases, the two-form originates from a combination of metric quantities and the Kalb-Ramond field. The target space geometry of the NR string is seen to arise from the gauging of a new algebra that is obtained by an Inonu-Wigner contraction of the Poincar\'e algebra extended by the symmetries of the Kalb-Ramond field. In this new formulation, there are no superfluous target space fields that can be removed by fixing a St\"uckelberg symmetry. Classically, there are no foliation/torsion constraints imposed on the target space geometry.