Regular Article - Theoretical Physics

Classification of non-Riemannian doubled-yet-gauged spacetime

¹ Departamento de Ciencias Físicas, Universidad Andres Bello, Republica 220, Santiago de Chile, Chile
² Departamento de Física, Universidad Técnica Federico Santa María, Centro Científico-Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, Chile
³ Department of Physics, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul, 04107, Korea
⁴ Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), Seoul, 08826, Korea

^* e-mail: park@sogang.ac.kr

Received: 22 August 2017
Accepted: 25 September 2017
Published online: 16 October 2017

Abstract

Assuming covariant fields as the ‘fundamental’ variables, double field theory can accommodate novel geometries where a Riemannian metric cannot be defined, even locally. Here we present a complete classification of such non-Riemannian spacetimes in terms of two non-negative integers, , . Upon these backgrounds, strings become chiral and anti-chiral over n and directions, respectively, while particles and strings are frozen over the directions. In particular, we identify (0, 0) as Riemannian manifolds, (1, 0) as non-relativistic spacetime, (1, 1) as Gomis–Ooguri non-relativistic string, as ultra-relativistic Carroll geometry, and (D, 0) as Siegel’s chiral string. Combined with a covariant Kaluza–Klein ansatz which we further spell, (0, 1) leads to Newton–Cartan gravity. Alternative to the conventional string compactifications on small manifolds, non-Riemannian spacetime such as , (3, 3) may open a new scheme for the dimensional reduction from ten to four.