Effective matter superpotentials from Wishart random matrices

Abstract

We show how within the Dijkgraaf-Vafa prescription one can derive superpotentials for matter fields. The ingredients forming the non-perturbative Affleck-Dine-Seiberg superpotentials arise from constrained matrix integrals, which are equivalent to classical complex Wishart random matrices. The mechanism is similar to the way the Veneziano-Yankielowicz superpotential arises from the matrix model measure.Comment: 9 pages; v2: published versio

Massless flavor in geometry and matrix models

“…Thus, its contribution to the superpotential is its derivative multiplied by the rank of the gauge group (the unbroken as well as the broken part!). The equation (58) recovers the suggestion [24] for the inclusion of massless quarks in the matrix model, and is also equivalent [39] with the suggestion of [30] that one first deforms the matrix model by mass terms and then takes the massless limit. This can be easily seen by using an integral representation for the δ-function in equation (58) and noticing that the new variable plays the role of mass parameter for the quarks Q M andQ M .…”

“…While the inclusion of massive quarks in this framework was easily achieved without reference to geometry, certain difficulties were encountered in dealing with massless ones. The two solutions to this problem, proposed in [24] and [30] on a field-theoretic basis only, were shown to be equivalent in [39]. From our analysis we see that this identification appears naturally from the geometrical picture and its relation to brane configurations.…”

“…We now turn to the discussion of fields in the fundamental representation and discuss the geometric justification of the various procedures of dealing with massless quarks [24,30,39].…”

“…Since there exists a holomorphic change of coordinates which casts equation (24) into that of the conifold, one may say that the two geometries describe the same physics. This is, however, not the case as various boundary conditions change under these transformations.…”

“…Indeed, in this case the low energy theory is not described only in terms of the glueball superfield as the naive matrix model predicts, but it must also contain quark bilinears. An attempt to handle this case was introduced in [24] and requires the introduction of delta functions relating the matrix model flavor fields with the corresponding gauge theory mesons. An alternate suggestion was presented in [30] and involves deforming the naive matrix model by mass terms for all massless fields and then computing the superpotential from the one-boundary free energy.…”