  The majority of models of inflation in string theory predict an absence of
measurable gravitational waves, r << 10^{-3}. The most promising proposals for
making string theoretic models that yield measurable tensor fluctuations
involve axion fields with slightly broken shift symmetry. We consider such
models in detail, with a particular focus on the N-flation scenario and on
axion valley/natural inflation models. We find that in Calabi-Yau threefold
compactifications with logarithmic Kahler potentials K it appears to be
difficult to meet the conditions required for axion inflation in the
supergravity regime. However, in supergravities with an (approximately)
quadratic shift-symmetric K, axion inflation may be viable. Such Kahler
potentials do arise in some string models, in specific limits of the moduli
space. We describe the most promising classes of models; more detailed study
will be required before one can conclude that working models exist.
