Phases of Script N = 1 supersymmetric SO/Sp gauge theories via matrix model

Abstract

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...

Phases of SO(Nc) gauge theories with flavors

“…According to relation (A.6), we have to take care of the second term of (A.6) on the left hand side when the number of flavors N f is greater than (2N c − 4) for SO(2N c ) gauge theories and (2N c − 3) for SO(2N c + 1) gauge theories. This is a kind of new phenomenon that did not appear in the pure case [19]. For example, on the non-baryonic branch of U(N 1 ) factor for SO (4) with N f = 1, SO(5) with N f = 2, SO(6) with N f = 3 and SO (7) with N f = 4 cases, we must be attentive to these modifications due to the presence of flavors.…”

“…In [19], the case with N f = 0 was discussed. Since the factorization problem was trivial, the characteristic function P 4 (x) could be represented as…”

“…In [19] the theory with N f = 0 was discussed intensively. For the breaking pattern SO(6) → SO(2) × U(2) where SO(2) is at the Special branch, there are two vacua which are confining vacua.…”

“…Then we will take the degree of superpotential to be arbitrary. Until now we have reviewed the discussion given in [19]. Next we will study some restriction that is an important idea in the discussion below.…”

On Nonperturbative Exactness of Konishi Anomaly and the Dijkgraaf-Vafa Conjecture

“…The generalized Konishi anomaly for the SO(N ) and Sp(N ) gauge theories has been derived in [17], [15] and [14] …”