  We consider N = 1 supersymmetric U(N) field theories in four dimensions with
adjoint chiral matter and a multi-trace tree-level superpotential. We show that
the computation of the effective action as a function of the glueball
superfield localizes to computing matrix integrals. Unlike the single-trace
case, holomorphy and symmetries do not forbid non-planar contributions.
Nevertheless, only a special subset of the planar diagrams contributes to the
exact result. Some of the data of this subset can be computed from the large-N
limit of an associated multi-trace Matrix model. However, the prescription
differs in important respects from that of Dijkgraaf and Vafa for single-trace
superpotentials in that the field theory effective action is not the derivative
of a multi-trace matrix model free energy. The basic subtlety involves the
correct identification of the field theory glueball as a variable in the Matrix
model, as we show via an auxiliary construction involving a single-trace matrix
model with additional singlet fields which are integrated out to compute the
multi-trace results. Along the way we also describe a general technique for
computing the large-N limits of multi-trace Matrix models and raise the
challenge of finding the field theories whose effective actions they may
compute. Since our models can be treated as N = 1 deformations of pure N =2
gauge theory, we show that the effective superpotential that we compute also
follows from the N = 2 Seiberg-Witten solution. Finally, we observe an
interesting connection between multi-trace local theories and non-local field
theory.
