Remarks on the non-Riemannian sector in Double Field Theory

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 $${n\\bar{n}\\ne 0}$$ n n ¯ ≠ 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.