We identify configurations of intersecting branes that correspond to the meta-stable supersymmetry breaking vacua in the four-dimensional N = 1 supersymmetric Yang-Mills theory coupled to massive flavors. We show how their energies, the stability properties, and the decay processes are described geometrically in terms of the brane configurations.

The supersymmetric SUN C Yang-Mills theory coupled to N F matter fields in the fundamental representation has metastable vacua with broken supersymmetry when N C < N F < 3 2 N C . By gauging the flavor symmetry, this model can be coupled directly to the standard model. We show that it is possible to make a slight deformation to the model so that gaugino masses are generated and the Landau pole problem can be avoided. The deformed model has simple realizations on intersecting branes in string theory, where various features of the metastable vacua are encoded geometrically as brane configurations.

The supersymmetric SUN C Yang-Mills theory coupled to N F matter fields in the fundamental representation has metastable vacua with broken supersymmetry when N C < N F < 3 2 N C . By gauging the flavor symmetry, this model can be coupled directly to the standard model. We show that it is possible to make a slight deformation to the model so that gaugino masses are generated and the Landau pole problem can be avoided. The deformed model has simple realizations on intersecting branes in string theory, where various features of the metastable vacua are encoded geometrically as brane configurations.

We study vacuum structure of N = 1 supersymmetric quiver gauge theories which can be realized geometrically by D brane probes wrapping cycles of local Calabi-Yau threefolds. In particular, we show that the A 2 quiver theory with gauge group U (N 1 ) × U (N 2 ) with 1 2 N 1 < N 2 < 2 3 N 1 has a regime with an infrared free description that is partially magnetic and partially electric. Using this dual description, we show that the model has a landscape of inequivalent meta-stable vacua where supersymmetry is dynamically broken and all the moduli are stabilized. Each vacuum has distinct unbroken gauge symmetry. The gaugino masses are generated by radiative corrections, and we are left with the bosonic pure Yang-Mills theory in the infrared. We also identify the supersymmetric vacua in this model using their infrared free descriptions and show that the decay rates of the supersymmetry breaking vacua can be made parametrically small.

We studied the phase structures of N = 1 supersymmetric SO(N c ) gauge theory with N f flavors in the vector representation as we deformed the N = 2 supersymmetric QCD by adding the superpotential of arbitrary polynomial for the adjoint chiral scalar field. Using weak and strong coupling analyses, we determined the most general factorization forms for various breaking patterns. We observed all kinds of smooth transitions for quartic superpotential.

We show that, for a generic choice of a point on the Coulomb branch of any N = 2 supersymmetric gauge theory, it is possible to find a superpotential perturbation which generates a metastable vacuum at the point. For theories with SU (N ) gauge group, such a superpotential can be expressed as a sum of single-trace terms for N = 2 and 3. If the metastable point is chosen at the origin of the moduli space, we can show that the superpotential can be a single-trace operator for any N . In both cases, the superpotential is a polynomial of degree 3N of the vector multiplet scalar field.

We extend the results of Cachazo, Seiberg and Witten to N = 1 supersymmetric gauge theories with gauge groups SO(2N), SO(2N + 1) and Sp(2N). By taking the superpotential which is an arbitrary polynomial of adjoint matter Φ as a small perturbation of N = 2 gauge theories, we examine the singular points preserving N = 1 supersymmetry in the moduli space where mutually local monopoles become massless. We derive the matrix model complex curve for the whole range of the degree of perturbed superpotential. Then we determine a generalized Konishi anomaly equation implying the orientifold contribution. We turn to the multiplication map and the confinement index K and describe both Coulomb branch and confining branch. In particular, we construct a multiplication map from SO(2N + 1) to SO(2KN − K + 2) where K is an even integer as well as a multiplication map from SO(2N) to SO(2KN − 2K + 2) (K is a positive integer), a map from SO(2N + 1) to SO(2KN − K + 2) (K is an odd integer) and a map from Sp(2N) to Sp(2KN + 2K − 2). Finally we analyze some examples which show some duality: the same moduli space has two different semiclassical limits corresponding to distinct gauge groups. the matrix model. If one wants n-th order instanton effect, one has only to compute Feynman diagram up to (n − 1) loop. In this context, they also claimed that the loop equation of matrix model, that plays an important role in the matrix model, is equivalent to the Riemann surface that comes from the dual geometry discussed above. After all, this Riemann surface leads to a fruitful system for studying the holomorphic information of four dimensional N = 1 gauge theories. After their works, the correspondence between the several matrix models and four dimensional gauge theories attracted wide attention, and many papers, which include the extension to other gauge groups and an adding flavors and so on, appeared in [21]- [74]. In particular, in [54,69], Ferrari has discussed the quantum parameter space of the N = 1 U(N) gauge theory with one adjoint matter Φ and a cubic tree level superpotential. These works are 1 one of the motivations of our paper.Recently in [25] they showed that these matrix model analysis could be interpreted within purely field theoretic point of view. In particular for the U(N) gauge theory with adjoint matter Φ and a polynomial superpotential W (Φ), a generalized Konishi anomaly equation, providing both a connection between the quark condensation and the gluino condensation in the supersymmetric gauge theories and possible ways to explore the nonperturbative aspects of the supersymmetric gauge theories, gives rise to the loop equation of matrix model. In other words, within gauge theories there is some aspect that can be interpreted as matrix models. Later, Cachazo, Seiberg and Witten [26] have discussed a new kind of duality. By changing the parameters of W (Φ), one can transit several vacua with different broken gauge groups continuously and holomorphically. There was no restrictions to the degree of the superpotential W (Φ) wh...

Dijkgraaf and Vafa have conjectured that the effective superpotentials for N = 1 four-dimensional supersymmetric gauge theories can be given by the planar diagrams of matrix models. We examine some special models with cubic and quartic tree level superpotentials for adjoint chiral superfield Φ. We consider the effective superpotentials for the classical vacuum Φ = 0 for U (N ) and SO(N )/Sp(N ) gauge theories. We evaluate the effective superpotentials exactly in terms of the matrix model and in terms of closed string theory on Calabi-Yau geometry with fluxes. As a result we find their perfect agreements.