Remark About Non-Relativistic String in Newton-Cartan Background and Null Reduction

TL;DR

The paper addresses the canonical structure of non-relativistic strings in Newton-Cartan backgrounds derived from null reduction. It carries out a Hamiltonian analysis showing the dynamics are governed by two first-class, Galilean-invariant constraints $\mathcal{H}_\tau$ and $\mathcal{H}_\sigma$, whose algebra closes on the constraint surface, and it explores a gauge-fixed reduction yielding a simple reduced Hamiltonian. An alternative derivation via a canonical T-duality along a null direction reproduces the same constraint structure and covariant Lagrangian, reinforcing the consistency of Harmark’s NC-string formulation. Additionally, a scaling limit $T\to 0$ is implemented, connecting to a distinct non-relativistic regime and matching the scaled action in Harmark:2017rpg. The results provide a robust foundation for future extensions, including Green-Schwarz-type strings and explicit Newton-Cartan backgrounds.

Abstract

We analyze non-relativistic string in Newton-Cartan background that was found recently in [arXiv:1705.03535 [hep-th]]. We find its Hamiltonian formulation and study structure of constraints. We also discuss a relation between string in Newton-Cartan Background and T-duality along null reduction.