Nonrelativistic String Theory and T-Duality

TL;DR

This paper develops nonrelativistic string theory with nonrelativistic symmetry, explores its coupling to curved backgrounds via string Newton-Cartan geometry, and analyzes T-duality transformations revealing connections to relativistic string theory and DLCQ.

Contribution

It introduces the sigma model for nonrelativistic strings in curved backgrounds and demonstrates T-duality transformations, linking nonrelativistic and relativistic string theories in novel ways.

Findings

Nonrelativistic string theory is unitary and UV complete.

T-duality along the longitudinal direction relates to relativistic string theory with a lightlike isometry.

T-duality along transverse directions relates different nonrelativistic backgrounds.

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.