The Glueball Superpotential

TLDR: In this article, the glueball superpotential for four-dimensional super-ymmetric gauge theories with arbitrary gauge groups and massive matter representations is computed, by perturbatively integrating massive charged fields.

Abstract: We compute glueball superpotentials for four-dimensional, ${\cal N}=1$ supersymmetric gauge theories, with arbitrary gauge groups and massive matter representations. This is done by perturbatively integrating out massive charged fields. The Feynman diagram computations simplify, and are related to the corresponding matrix model. This leads to a natural notion of ``projection to planar diagrams'' for arbitrary gauge groups and representations. We discuss a general ambiguity in the glueball superpotential $W(S)$ for terms, $S^n$, whose order, $n$ is greater than the dual Coxeter number. This ambiguity can be resolved for all classical gauge groups $(A,B,C,D)$, via a natural embedding in an infinite rank supergroup. We use this to resolve some recently raised puzzles. For exceptional groups, we compute the superpotential terms for low powers of the glueball field and propose an all-order completion for some examples including ${\cal N}=1^*$ for all simply-laced groups. We also comment on compactification of these theories to lower dimensions.