  In supersymmetric theories, one can obtain striking results and insights by
exploiting the fact that the superpotential and the gauge coupling function are
holomorphic functions of the model parameters. The precise meaning of this
holomorphy is subtle, and has been explained most clearly by Shifman and
Vainshtein, who have stressed the role of the Wilsonian effective action. In
this note, we elaborate on the Shifman-Vainshtein program, applying it to
examples in grand unification, supersymmetric QCD and string theory. We stress
that among the ``model parameters" are the cutoffs used to define the Wilsonian
action itself, and that generically these must be defined in a field-dependent
manner to obtain holomorphic results.
