  In this paper we discuss quantization of the Fayet-Iliopoulos parameter in
supergravity theories with altered nonperturbative sectors, which were recently
used to argue a fractional quantization condition. Nonlinear sigma models with
altered nonperturbative sectors are the same as nonlinear sigma models on
special stacks known as gerbes. After reviewing the existing results on such
theories in two dimensions, we discuss examples of gerby moduli `spaces'
appearing in four-dimensional field theory and string compactifications, and
the effect of various dualities. We discuss global topological defects arising
when a field or string theory moduli space has a gerbe structure. We also
outline how to generalize results of Bagger-Witten and more recent authors on
quantization issues in supergravities from smooth manifolds to smooth moduli
stacks, focusing particular attention on stacks that have gerbe structures.
