Dark Matter with Stuckelberg Axions
  We review a class of models which generalize the traditional Peccei-Quinn
(PQ) axion solution by a St\"uckelberg pseudoscalar. Such axion models
represent a significant variant with respect to earlier scenarios where axion
fields were associated with global anomalies, because of the St\"uckelberg
field, which is essential for the cancellation of gauge anomalies in the
presence of extra $U(1)$ symmetries. The extra neutral currents associated to
these models have been investigated in the past in orientifold models with
intersecting branes, under the assumption that the St\"uckelberg scale was in
the multi-TeV region. Such constructions, at the field theory level, are quite
general and can be interpreted as the four-dimensional field theory realization
of the Green-Schwarz mechanism of anomaly cancellation of string theory. We
present an overview of models of this type in the TeV/multi TeV range in their
original formulation and their recent embeddings into an ordinary GUT theory,
presenting an $E_6\times U(1)_X$ model as an example. In this case the model
contains two axions, the first corresponding to a Peccei-Quinn axion, whose
misalignment takes place at the QCD phase transition, with a mass in the meV
region and which solves the strong CP problem. The second axion is ultralight,
in the $10^{-20}-10^{-22}$ eV region, due to a misalignment and a decoupling
taking place at the GUT scale. The two scales introduced by the PQ solution,
the PQ breaking scale and the misalignment scale at the QCD hadron transition,
become the Planck and the GUT scales respectively, with a global anomaly
replaced by a gauge anomaly. The periodic potential and the corresponding
oscillations are related to a particle whose De Broglie wavelength can reach 10
kpc. Such a sub-galactic scale has been deemed necessary in order to resolve
several dark matter issues at the astrophysical level.
