Supersymmetric Gauge Theories with Flavors and Matrix Models

We present two results concerning the relation between poles and cuts by using the example of N=1 U(N_c) gauge theories with matter fields in the adjoint, fundamental and anti-fundamental representations. The first result is the on-shell possibility of poles, which are associated with flavors and on the second sheet of the Riemann surface, passing through the branch cut and getting to the first sheet. The second result is the generalization of hep-th/0311181 (Intriligator, Kraus, Ryzhov, Shigemori, and Vafa) to include flavors. We clarify when there are closed cuts and how to reproduce the results of the strong coupling analysis by matrix model, by setting the glueball field to zero from the beginning. We also make remarks on the possible stringy explanations of the results and on generalization to SO(N_c) and USp(2N_c) gauge groups.Comment: 52 pages, 6 figure

Supersymmetric SO(N_c) Gauge Theory and Matrix Model

By applying the method of Dijkgraaf-Vafa, we study matrix model related to supersymmetric SO(N_c) gauge theory with N_f flavors of quarks in the vector representation found by Intriligator-Seiberg. By performing the matrix integral over tree level superpotential characterized by light meson fields (mass deformation) in electric theory, we reproduce the exact effective superpotential in the gauge theory side. Moreover, we do similar analysis in magnetic theory. It turns out the matrix descriptions of both electric and magnetic theories are the same: Seiberg duality in the gauge theory side.Comment: 9 pp:v2 Kept N_c for gauge theory and N for matrix model and modified the measure of matrix integral with the footnote and to appear in PL

Effective matter superpotentials from Wishart random matrices

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

Classical Spin Chains and Exact Three-dimensional Superpotentials

We study exact effective superpotentials of four-dimensional {\cal N} = 2 supersymmetric gauge theories with gauge group U(N) and various amounts of fundamental matter on R^3 x S^1, broken to {\cal N} = 1 by turning on a classical superpotential for the adjoint scalar. On general grounds these superpotentials can easily be constructed once we identify a suitable set of coordinates on the moduli space of the gauge theory. These coordinates have been conjectured to be the phase space variables of the classical integrable system which underlies the {\cal N} = 2 gauge theory. For the gauge theory under study these integrable systems are degenerations of the classical, inhomogeneous, periodic SL(2,C) spin chain. Ambiguities in the degeneration provide multiple coordinate patches on the gauge theory moduli space. By studying the vacua of the superpotentials in several examples we find that the spin chain provides coordinate patches that parametrize holomorphically the part of the gauge theory moduli space which is connected to the electric (as opposed to magnetic or baryonic) Higgs and Coulomb branch vacua. The baryonic branch root is on the edge of some coordinate patches. As a product of our analysis all maximally confining (non-baryonic) Seiberg-Witten curve factorizations for N_f \leq N_c are obtained, explicit up to one constraint for equal mass flavors and up to two constraints for unequal mass flavors. Gauge theory addition and multiplication maps are shown to have a natural counterpart in this construction. Furthermore it is shown how to integrate in the meson fields in this formulation in order to obtain three and four dimensional Affleck-Dine-Seiberg-like superpotentials.Comment: 47 pages, LaTeX, v2: several extensions of results, typos corrected, reference adde

Baryonic Corrections to Superpotentials from Perturbation Theory

We study the corrections induced by a baryon vertex to the superpotential of SQCD with gauge group SU(N) and N quark flavors. We first compute the corrections order by order using a standard field theory technique and derive the corresponding glueball superpotential by "integrating in" the glueball field. The structure of the corrections matches with the expectations from the recently introduced perturbative techniques. We then compute the first non-trivial contribution using this new technique and find exact quantitative agreement. This involves cancellations between diagrams that go beyond the planar approximation.Comment: 8 page

Massless Flavor in Geometry and Matrix Models

The proper inclusion of flavor in the Dijkgraaf-Vafa proposal for the solution of N=1 gauge theories through matrix models has been subject of debate in the recent literature. We here reexamine this issue by geometrically engineering fundamental matter with type IIB branes wrapped on non-compact cycles in the resolved geometry, and following them through the geometric transition. Our approach treats massive and massless flavor fields on equal footing, including the mesons. We also study the geometric transitions and superpotentials for finite mass of the adjoint field. All superpotentials we compute reproduce the field theory results. Crucial insights come from T-dual brane constructions in type IIA.Comment: 33 pages, 1 figur

Seiberg Duality in Matrix Models II

In this paper we continue the investigation, within the context of the Dijkgraaf-Vafa Programme, of Seiberg duality in matrix models as initiated in hep-th/0211202, by allowing degenerate mass deformations. In this case, there are some massless fields which remain and the theory has a moduli space. With this illustrative example, we propose a general methodology for performing the relevant matrix model integrations and addressing the corresponding field theories which have non-trivial IR behaviour, and which may or may not have tree-level superpotentials.Comment: 11 pages, comments adde

Seiberg--Witten Duality in Dijkgraaf--Vafa Theory

We show that a suitable rescaling of the matrix model coupling constant makes manifest the duality group of the N=2 SYM theory with gauge group SU(2). This is done by first identifying the possible modifications of the SYM moduli preserving the monodromy group. Then we show that in matrix models there is a simple rescaling of the pair $(S_D,S)$ which makes them dual variables with $\Gamma(2)$ monodromy. We then show that, thanks to a crucial scaling property of the free energy derived perturbatively by Dijkgraaf, Gukov, Kazakov and Vafa, this redefinition corresponds to a rescaling of the free energy which in turn fixes the rescaling of the coupling constant. Next, we show that in terms of the rescaled free energy one obtains a nonperturbative relation which is the matrix model counterpart of the relation between the $u$--modulus and the prepotential of N=2 SYM. This suggests considering a dual formulation of the matrix model in which the expansion of the prepotential in the strong coupling region, whose QFT derivation is still unknown, should follow from perturbation theory. The investigation concerns the SU(2) gauge group and can be generalized to higher rank groups.Comment: 1+14 pages, LaTeX. v2: typos fixed, references added v3: some numerical factor corrected, typos fixed, version to appear in NP