Some explorations in holomorphy.

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 eeective action. In this note, we elaborate on the Shifman-Vainshtein program, applying it to examples in grand uniication, supersymmetric QCD and string theory. We stress that among the \model parameters" are the cutoos used to deene the Wilsonian action itself, and that generically these must be deened in a eld-dependent manner to obtain holomorphic results.