An Ultralight Axion in Supersymmetry and Strings and Cosmology at Small
  Scales
  Dynamical mechanisms to generate an ultralight axion of mass $\sim
10^{-21}-10^{-22}$ eV in supergravity and strings are discussed. An ultralight
particle of this mass provides a candidate for dark matter that may play a role
for cosmology at scales $10\, {\rm kpc}$ or less. An effective operator
approach for the axion mass provides a general framework for models of
ultralight axions, and in one case recovers the scale $10^{-21}-10^{-22}$ eV as
the electroweak scale times the square of the hierarchy with an $O(1)$ Wilson
coefficient. We discuss several classes of models realizing this framework
where an ultralight axion of the necessary size can be generated. In one class
of supersymmetric models an ultralight axion is generated by instanton like
effects. In the second class higher dimensional operators involving couplings
of Higgs, standard model singlets, and axion fields naturally lead to an
ultralight axion. Further, for the class of models considered the hierarchy
between the ultralight scale and the weak scale is maintained. We also discuss
the generation of an ultralight scale within string based models. Here it is
shown that in the single modulus KKLT moduli stabilization scheme an ultralight
axion would require an ultra-low weak scale. However, within the Large Volume
Scenario, the desired hierarchy between the axion scale and the weak scale is
achieved. A general analysis of couplings of Higgs fields to instantons within
the string framework is discussed and it is shown that the condition necessary
for achieving such couplings is the existence of vector-like zero modes of the
instanton. Some of the phenomenological aspects of these models are also
discussed.
