Classification of non-Riemannian doubled-yet-gauged spacetime

TL;DR

This paper classifies non-Riemannian geometries in Double Field Theory using two integers, revealing novel spacetime structures where traditional metrics do not exist, with implications for string theory and gravity.

Contribution

It provides a complete classification of non-Riemannian doubled-yet-gauged spacetimes in terms of two integers, expanding the understanding of possible geometries in Double Field Theory.

Findings

Identifies (0,0) as Riemannian manifolds.

Classifies non-Riemannian backgrounds with specific (n, ar;n) values.

Suggests new dimensional reduction schemes using non-Riemannian spacetimes.

Abstract

Assuming $\mathbf{O}(D,D)$ 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, $(n,\bar{n})$, $0\leq n+\bar{n}\leq D$. Upon these backgrounds, strings become chiral and anti-chiral over $n$ and $\bar{n}$ directions respectively, while particles and strings are frozen over the $n+\bar{n}$ directions. In particular, we identify $(0,0)$ as Riemannian manifolds, $(1,0)$ as non-relativistic spacetime, $(1,1)$ as Gomis-Ooguri non-relativistic string, $(D{-1},0)$ 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 $D=10$, $(3,3)$ may open a new scheme of the dimensional reduction from ten to four.