  The (heterotic) double field theories and the exceptional field theories are
recently developed for manifestly duality covariant formulation of various
supergravity theories, describing low-energy limit of various (heterotic)
superstring and M-theory compactifications. These field theories are known to
be reduced to the standard descriptions by introducing appropriately
parameterized generalized metric and by applying suitably chosen section
conditions. We generalize this development to non-geometric backgrounds by
utilizing dual fields pertinent to non-geometric fluxes. We introduce different
parameterizations for the generalized metric, in terms of the conventional
supergravity fields or the dual fields. Under certain simplifying assumptions,
we construct new effective action for non-geometric backgrounds. We then obtain
the non-geometric backgrounds sourced by exotic branes. From them, we construct
their $U$-duality monodromy matrices. The charge of exotic branes obtained from
these monodromy matrices agrees perfectly with the charge obtained from the
non-geometric flux integral.
