SOME EXPLORATIONS IN HOLOMORPHY

In supersymmetric theories, one can obtain striking results and insigh ts by exploiting the fact that the superpotential and the gauge coupli ng function are holomorphic functions of the model parameters. The pre cise 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 article, we elaborate on the Shifm an-Vainshtein program, applying it to examples in grand unification, s upersymmetric QCD, and string theory. We stress that among the ''model parameters'' are the cutoffs used to define the Wilsonian action itse lf, and that generically these must be defined in a field-dependent ma nner to obtain holomorphic results.