Can Non-Relativistic Strings Propagate Without Geometric Baggage?

TL;DR

This paper develops a minimal, consistent formulation of non-relativistic bosonic string theory in Newton-Cartan backgrounds, deriving geometric data dynamically and confirming the sufficiency of Newton-Cartan geometry for string dynamics.

Contribution

It introduces a Hamiltonian and Lagrangian framework for non-relativistic strings that derives geometric structures dynamically, contrasting with prior approaches requiring additional background fields.

Findings

The model exhibits first-class constraints generating worldsheet diffeomorphisms.

All geometric data are derived from the string evolution, not imposed externally.

Newton-Cartan geometry suffices for consistent non-relativistic string dynamics.

Abstract

We present a minimal and dynamically consistent formulation of non-relativistic bosonic string theory in a Newton-Cartan (NC) background. Starting from a reparametrization-invariant Nambu-Goto action, we develop the Hamiltonian framework and perform a complete Dirac constraint analysis. The resulting structure exhibits first-class constraints that generate worldsheet diffeomorphisms, confirming the internal gauge consistency of the model. Using an interpolating Lagrangian, we derive a Polyakov-type action that enables a direct comparison with symmetry-based constructions known as gauging the algebra (GTA) approaches, which promote non-relativistic symmetry algebras to local symmetries. In contrast to GTA formulations, which require additional background fields to achieve algebraic closure, our model derives all necessary geometric data dynamically from the string evolution itself. This establishes that standard Newton-Cartan geometry is sufficient to support consistent non-relativistic string dynamics. Our results provide a conceptually transparent and technically robust foundation for future studies of non-relativistic string theory in curved backgrounds.