Some Explorations in Holomorphy

TL;DR

This paper explores the role of holomorphy in supersymmetric theories, emphasizing the importance of Wilsonian effective action and field-dependent cutoffs in deriving holomorphic properties of the superpotential and gauge couplings.

Contribution

It elaborates on the Shifman-Vainshtein approach, applying it to grand unification, supersymmetric QCD, and string theory, highlighting the significance of field-dependent cutoffs.

Findings

Holomorphy constrains superpotential and gauge couplings.

Wilsonian effective action is crucial for understanding holomorphy.

Field-dependent cutoffs are necessary for holomorphic results.

Abstract

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.