Classification of non-Riemannian doubled-yet-gauged spacetime

Abstract

Abstract Assuming $$\mathbf {O}(D,D)$$ 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, $$(n,\bar{n})$$ (n,n¯) , $$0\le n+\bar{n}\le D$$ 0≤n+n¯≤D . Upon these backgrounds, strings become chiral and anti-chiral over n and $$\bar{n}$$ n¯ directions, respectively, while particles and strings are frozen over the $$n+\bar{n}$$ n+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, $$(D{-1},0)$$ (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 $$D=10$$ D=10 , (3, 3) may open a new scheme for the dimensional reduction from ten to four.