Abstract
JHEP 1512 (2015) 144 Double Field Theory provides a geometric framework capable of describing
string theory backgrounds that cannot be understood purely in terms of
Riemannian geometry -- not only globally (`non-geometry'), but even locally
(`non-Riemannian'). In this work, we show that the non-relativistic closed
string theory of Gomis and Ooguri [1] arises precisely as such a non-Riemannian
string background, and that the Gomis-Ooguri sigma model is equivalent to the
Double Field Theory sigma model of [2] on this background. We further show that
the target-space formulation of Double Field Theory on this non-Riemannian
background correctly reproduces the appropriate sector of the Gomis-Ooguri
string spectrum. To do this, we develop a general semi-covariant formalism
describing perturbations in Double Field Theory. We derive compact expressions
for the linearized equations of motion around a generic on-shell background,
and construct the corresponding fluctuation Lagrangian in terms of novel
completely covariant second order differential operators. We also present a new
non-Riemannian solution featuring Schr\"odinger conformal symmetry.