Nonrelativistic String Theory and T-Duality

TL;DR

The paper formulates nonrelativistic string theory as a unitary, UV-complete worldsheet QFT with a string Galilei symmetry, propagating on a string Newton-Cartan target geometry and coupled to a Kalb-Ramond field and dilaton. It constructs the NR string sigma model with worldsheet fields $(x^^,,)$ and boost-invariant target-space tensors $H_{}$ and $ au_{ u}$, and analyzes the background constraints via Weyl invariance. Three distinct T-duality transformations are derived and implemented in the path integral: longitudinal spatial, longitudinal lightlike, and transverse, each yielding precise Buscher-like relations among dual backgrounds; in particular longitudinal spatial T-duality maps NR theory to relativistic string theory on a Lorentzian geometry with a compact lightlike isometry, providing a first-principles DLCQ of string theory. The results establish a robust link between NR string theory and relativistic DLCQ, while preserving NR structure under transverse duality, and point to future avenues in spacetime NC dynamics and holographic applications.

Abstract

Nonrelativistic string theory in flat spacetime is described by a two-dimensional quantum field theory with a nonrelativistic global symmetry acting on the worldsheet fields. Nonrelativistic string theory is unitary, ultraviolet complete and has a string spectrum and spacetime S-matrix enjoying nonrelativistic symmetry. The worldsheet theory of nonrelativistic string theory is coupled to a curved spacetime background and to a Kalb-Ramond two-form and dilaton field. The appropriate spacetime geometry for nonrelativistic string theory is dubbed string Newton-Cartan geometry, which is distinct from Riemannian geometry. This defines the sigma model of nonrelativistic string theory describing strings propagating and interacting in curved background fields. We also implement T-duality transformations in the path integral of this sigma model and uncover the spacetime interpretation of T-duality. We show that T-duality along the longitudinal direction of the string Newton-Cartan geometry describes relativistic string theory on a Lorentzian geometry with a compact lightlike isometry, which is otherwise only defined by a subtle infinite boost limit. This relation provides a first principles definition of string theory in the discrete light cone quantization (DLCQ) in an arbitrary background, a quantization that appears in nonperturbative approaches to quantum field theory and string/M-theory, such as in Matrix theory. T-duality along a transverse direction of the string Newton-Cartan geometry equates nonrelativistic string theory in two distinct, T-dual backgrounds.