Remarks on the non-Riemannian sector in Double Field Theory

TL;DR

This paper explores the non-Riemannian sectors within Double Field Theory, highlighting their classification, variational principles, and implications for string theory landscapes beyond traditional Riemannian geometries.

Contribution

It provides a detailed analysis of the variational principle in non-Riemannian sectors of Double Field Theory, revealing subtleties in their classification and suggesting they are solution sectors rather than independent theories.

Findings

Non-Riemannian backgrounds classified by (n, n̄) integers.

Variational principle subtlety for n n̄ ≠ 0 sectors.

Potential to extend string landscape beyond Riemannian geometries.

Abstract

Taking $\mathbf{O}(D,D)$ 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, $(n,\bar{n})$. Such non-Riemannian backgrounds render a propagating string chiral and anti-chiral over $n$ and $\bar{n}$ 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 $(n,\bar{n})$ non-Riemannian sector. We recognize a nontrivial subtlety for ${n\bar{n}\neq 0}$ that infinitesimal variations generically include those which change $(n,\bar{n})$. 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.