Remarks on the non-Riemannian sector in Double Field Theory

Abstract

Abstract Taking $$\mathbf {O}(D,D)$$ O(D,D) covariant field variables as its truly fundamental constituents, Double Field Theory can accommodate not only conventional supergravity but also non-Riemannian gravities that may be classified by two non-negative integers, $$(n,\bar{n})$$ (n,n¯) . Such non-Riemannian backgrounds render a propagating string chiral and anti-chiral over n and $$\bar{n}$$ n¯ dimensions respectively. Examples include, but are not limited to, Newton–Cartan, Carroll, or Gomis–Ooguri. Here we analyze the variational principle with care for a generic $$(n,\bar{n})$$ (n,n¯) non-Riemannian sector. We recognize a nontrivial subtlety for $${n\bar{n}\ne 0}$$ nn¯≠0 that infinitesimal variations generically include those which change $$(n,\bar{n})$$ (n,n¯) . This seems to suggest that the various non-Riemannian gravities should better be identified as different solution sectors of Double Field Theory rather than viewed as independent theories. Separate verification of our results as string worldsheet beta-functions may enlarge the scope of the string landscape far beyond Riemann.