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Meta-Stable Brane Configurations of Triple Product Gauge Groups

From an N=1 supersymmetric electric gauge theory with the gauge group SU(N_c) x SU(N_c') x SU(N_c'') with fundamentals for each gauge group and the bifundamentals, we apply Seiberg dual to each gauge group and obtain the N=1 supersymmetric dual magnetic gauge theories with dual matters including the additional gauge singlets. By analyzing the F-term equations of the dual magnetic superpotentials, we describe the intersecting brane configurations of type IIA string theory corresponding to the meta-stable nonsupersymmetric vacua of this gauge theory. We apply also to the case for N=1 supersymmetric electric gauge theory with the gauge group Sp(N_c) x SO(2N_c') x Sp(N_c'') with flavors for each gauge group and the bifundamentals. Finally, we describe the meta-stable brane configurations of multiple product gauge groups.


Introduction
The nonsupersymmetric meta-stable vacua exist in N = 1 SQCD with massive fundamental quarks where the masses are much smaller than the dynamical scale of the gauge sector [1]. The supersymmetry is broken by the rank condition and the classical flat directions can be lifted by quantum corrections which generate positive mass terms for the pseudomoduli leading to long-lived meta-stable vacua. When the direction which was not properly lifted by the one-loop potential occurs, by adding a set of singlets, one gets meta-stable vacua in the quiver gauge theory on fractional branes [2]. Other possible embedding by brane probes wrapping cycles of local Calabi-Yau and corresponding quiver gauge theories have inequivalent meta-stable vacua [3]. See the review paper [4] for the recent developments of dynamical supersymmetry breaking.
The geometrical approach for these meta-stable vacua, using the brane configuration in type IIA string theory, has many interesting features [5,6,7]. Turning on the quark masses in the electric theory corresponds to deform the superpotential by adding a term in linear in a singlet field in the dual magnetic theory. This is equivalent to move D6-branes in particular directions in type IIA brane configuration. The misalignment of D4-branes connecting NS5'-brane can be analyzed as a nontrivial F-term conditions providing nonzero vacuum expectation values of dual quarks. The fact that some of D4-branes connecting NS5'-brane can move in other two directions freely is exactly the classical moduli space of nonsupersymmetric vacua. See the review paper [8] for the gauge theory and the brane dynamics. When we add an adjoint matter field into above N = 1 SQCD with massive fundamental quarks, a meta-stable nonsupersymmetric long-lived vacuum was found by considering an addition of gauge singlet terms [9]. The corresponding brane configuration for these nonsupersymmetric meta-stable vacua was studied by realizing these deformation terms in the magnetic dual theory geometrically [10]. The lifting of the tree-level supersymmetry breaking brane configuration to M-theory was described in [7] by computing the equations of motion for the minimal area nonholomorphic curves. The behavior at infinity of this nonsupersymmetric brane configuration was different from that of the standard supersymmetric ground state of MQCD. The M-theory lift for symplectic and orthogonal gauge groups by introducing an orientifold 4-plane to the brane configuration for unitary gauge group was presented in [11].
When we add a symmetric flavor and a conjugate symmetric flavor to an N = 1 SQCD with massive fundamental quarks, a meta-stable nonsupersymmetric long-lived vacuum was found by adding an orientifold 6-plane to the above brane configuration [12]. Moreover, theory are the same as those for the third(first) gauge group in the electric theory.
One can easily generalize the meta-stable brane configuration to the ones corresponding to a multiple product of gauge groups. Then one takes the Seiberg dual for the first gauge group factor, the last gauge group factor and for any gauge group factor except the first and last gauge group factors. One can write down the magnetic superpotentials in terms of the cubic interactions between the gauge singlets and dual matters.
We also add an orientifold 4-plane to the brane configurations consisting of four NS-branes, three kinds of D4-branes and three kinds of D6-branes in type IIA string theory. Then the gauge group will be a product of symplectic and orthogonal gauge groups alternatively. The meta-stable brane configuration for the multiple product gauge groups is also discussed.
In section 2, we describe the type IIA brane configuration corresponding to the electric theory based on the N = 1 SU(N c ) × SU(N ′ c ) × SU(N ′′ c ) gauge theory with fundamentals and bifundamentals, and deform this theory by adding the mass term for the quarks for each gauge group. Then we construct the Seiberg dual magnetic theories for each gauge group factor with corresponding dual matters as well as additional gauge singlets, by brane motion and linking number counting. Finally, the nonsupersymmetric brane configurations are found by recombination and splitting for the flavor D4-branes. One generalizes to the meta-stable brane configurations corresponding to a multiple product of gauge groups and describe them very briefly.
In section 3, we describe the type IIA brane configuration corresponding to the electric theory based on the N = 1 Sp(N c ) × SO(2N ′ c ) × Sp(N ′′ c ) gauge theory with fundamentals, vectors, and bifundamentals, and deform this theory by adding the mass term for the quarks for each gauge group. In the brane configuration, this can be obtained from the brane configuration given in section 2 by inserting the appropriate orientifold 4-planes. Then we construct the Seiberg dual magnetic theories for each gauge group factor with corresponding dual matters as well as extra gauge singlets, by brane motion and linking number counting. Finally, the nonsupersymmetric brane configurations are found. These can be seen from those brane configurations in section 2 by the action of orientifold 4-plane. The generalization to a multiple product of gauge groups is described. Compared with the ones in section 2, in this case, there exist more magnetic dual theories because the gauge group factors have both symplectic and orthogonal gauge groups.
Finally, in section 4, we summarize what we have found in this paper and make some comments for the future directions 1 .
2 Nonsupersymmetric meta-stable brane configurations of SU (N c ) × SU (N ′ c ) × SU (N ′′ c ) and its multiple product gauge theories For the meta-stable brane configurations, it is necessary to have nonzero masses for the quarks(corresponding to relative displacement between D6-branes and D4-branes in type IIA string theory) and to take the Seiberg dual theory(corresponding to the magnetic theory obtained from the electric theory via brane motion or field theory analysis) [1]. We need to understand the brane configurations both from the electric theory(where one has massdeformed superpotential) and from the magnetic theory(where the meta-stable states are long-lived parametrically). There exist four possible magnetic brane configurations for the triple product gauge group SU(N c ) × SU(N ′ c ) × SU(N ′′ c ), depending on whether the dual gauge group we take is the first gauge group, the second gauge group(in which there exist two magnetic brane configurations), or the third gauge group.
After reviewing the electric brane configuration, we describe the four magnetic brane configurations, and then the nonsupersymmetric brane configurations are found by recombination of some flavor D4-branes and color D4-branes and splitting procedure between those flavor D4-branes and the remnant of flavor D4-branes.

Electric theory
The gauge group is given by SU(N c ) × SU(N ′ c ) × SU(N ′′ c ) and the matter contents [46,47]  whole gauge group • The flavor-singlet field F is in the bifundamental representation (N c , N ′ c , 1) under the whole gauge group and its complex conjugate field F is in the bifundamental representation (N c , N ′ c , 1) under the whole gauge group • The flavor-singlet field G is in the bifundamental representation (1, N ′ c , N ′′ c ) under the whole gauge group and its complex conjugate field G is in the bifundamental representation (1, N ′ c , N ′′ c ) under the whole gauge group This is a simple generalization to triple product gauge groups from the two product gauge groups [48,49,46,47]. Then the gauge group and matter contents we consider are summarized as follows: In the electric theory, since there exist N f quarks Q, N f quarks Q, one bifundamental field F (which will give rise to the contribution of N ′ c ), and its complex conjugate field F (which will give rise to the contribution of N ′ c ), the coefficient of the beta function of the first gauge group factor is b SU (Nc) = 3N c − N f − N ′ c . Similarly, since there exist N ′ f quarks Q ′ , N ′ f quarks Q ′ , one bifundamental field F (which gives the contribution of N c ), its complex conjugate field F (which will give the contribution of N c ), one bifundamental field G(which will give rise to the contribution of N ′′ c ), and its complex conjugate field G(which will give the contribution of N ′′ c ), the coefficient of the beta function for the second gauge group factor is given by Finally, since there exist N ′′ f quarks Q ′′ , N ′′ f quarks Q ′′ , one bifundamental field G(which will give rise to the contribution of N ′ c ), and its complex conjugate field G(which will give rise to the contribution of N ′ c ), the coefficient of the beta function of the third gauge group factor can be read off b SU (N ′′ There is a symmetry between b SU (Nc) and b SU (N ′′ c ) such that the former becomes the latter by replacing N c and N f with N ′′ c and N ′′ f respectively. The anomaly free global symmetry contains [SU(N f ) × SU(N ′ f ) × SU(N ′′ f )] 2 × U(1) R [46] and let us denote the strong coupling scales for SU(N c ) as Λ 1 , for SU(N ′ c ) as Λ 2 and for SU(N ′′ c ) as Λ 3 respectively. The electric theory is asymptotically free when b SU (Nc) > 0 for the SU(N c ) gauge theory, when b SU (N ′ c ) > 0 for the SU(N ′ c ) gauge theory, and when b SU (N ′′ c ) > 0 for the SU(N ′′ c ) gauge theory. The classical electric superpotential is where the coefficient functions µ, µ ′ , µ ′′ , λ, λ ′ and λ ′′ are given by six rotation angles for the branes in type IIA string theory [46,50]. We do not write down the dependences on these angles explicitly. Each gauge group factor has two rotation angles on NS-brane and D6-branes.
Here the adjoint field for SU(N c ) gauge group is denoted by A, the adjoint field for SU(N ′ c ) gauge group is denoted by A ′ , and the adjoint field for SU(N ′′ c ) gauge group is denoted by A ′′ . The mass terms of these adjoint fields are related to the rotation angles of NS-branes.
The couplings of fundamentals with these adjoint fields are related also to the rotation angles of NS-branes as well as the rotation angles of D6-branes. We add the mass terms for each fundamental flavor. Setting the fields Q ′′ , Q ′′ , G, G and A ′′ to zero, the superpotential becomes the one described in [46,50,14].
After integrating out the adjoint fields A, A ′ and A ′′ , this superpotential (2.1) at the particular orientations for branes, i.e., the case where any two neighboring NS-branes are perpendicular to each other, will reduce to the last three mass-deformed terms since the coefficient functions 1 µ , 1 µ ′ and 1 µ ′′ vanish at these particular rotation angles for branes. It does not matter whether λ, λ ′ or λ ′′ vanishes because even if these coefficient functions are not zero, λ, λ ′ -or λ ′′ -dependent terms all vanish because they contain 1 µ -, 1 µ ′ -or 1 µ ′′ -prefactors. Then the classical superpotential by deforming the massless case by adding the mass terms for the quarks Q(Q ′ )[Q ′′ ] and Q( Q ′ )[ Q ′′ ] is given by When we discuss the meta-stable brane configurations for the particular dual gauge group with corresponding massive flavors, the other flavors corresponding to other two gauge groups will be massless. The type IIA brane configuration for this mass-deformed theory can be described by as follows 2 . The N c -color D4-branes (01236) are suspended between the NS5 L -brane (012345) and the NS5 ′ L -brane (012389) along x 6 direction, together with N f D6-branes (0123789) which are parallel to NS5 ′ L -brane and have nonzero v direction. The NS5 R -brane is located at the right hand side of the NS5 ′ L -brane along the x 6 direction and there exist N ′ c -color D4-branes suspended between them, with N ′ f D6-branes which have nonzero v direction. Moreover, the NS5 ′ R -brane is located at the right hand side of the NS5 R -brane along the x 6 direction and there exist N ′′ c -color D4-branes suspended between them, with N ′′ f D6-branes which have nonzero v direction 3 .
We summarize the N = 1 supersymmetric electric brane configuration in type IIA string theory as follows: (0123789) directions. Now we draw this electric brane configuration in Figure 1 and we put the coincident for each gauge group and bifundamentals F (G) and F ( G). The two NS5-branes can be written as NS5 L,R -branes while the two NS5'-branes can be denoted by NS5 ′ L,R -branes. 3 Basically, if there are no N S5 L -brane, N c D4-branes and N f D6-branes, this brane configuration is exactly the same as the one in [14]. One can consider the other brane configuration by rotations of NS-branes in above brane configuration by π 2 angles. That is, N S5 L,R -branes go to N S5 ′ L,R -branes and vice versa. In other words, this other brane configuration can be obtained from the one in [14], by adding N S5 R -brane, N ′′ c D4-branes and N ′′ f D6-branes to the right. Then we will see that all the meta-stable brane configurations from this other brane configuration can be obtained by viewing the Figures 2B, 3B, 4B and 5B from the negative w direction.

Magnetic theory with dual for third gauge group
One considers dualizing one of the gauge groups regarding as the other gauge groups as a spectator. In this subsection, one takes the Seiberg dual for the third gauge group factor SU(N ′′ c ) while remaining the first and second gauge group factors SU(N c ) and SU(N ′ c ) unchanged. One ignores the dynamics of the other gauge groups. We consider the case where Λ 3 >> Λ 1 , Λ 2 , in other words, the dualized group's dynamical scale is far above that of the other spectator groups.
Let us move the NS5 R -brane to the right all the way past the NS5 ′ R -brane. After this brane motion, one arrives at the Figure 2A. Note that there exists a creation of N ′′ f D4-branes connecting N ′′ f D6-branes and NS5 ′ R -brane because the N ′′ f D6-branes are perpendicular(or are not parallel) to the NS5 R -brane in Figure 1 and we consider massless quarks for Q(Q ′ ) and Q( Q ′ ). The linking number [51] of NS5 R -brane from Figure 2A can be read off and is given by On the other hand, the linking number of NS5 R -brane from Figure  1 Figure  1, the presence of N ′ c in the linking number arises. This N ′ c dependence will appear in many places later and is common feature when we discuss about the product gauge group. From these two relations, one obtains the number of colors of dual magnetic theory Let us draw this magnetic brane configuration in Figure 2A and recall that we put the coincident N ′′ f D6-branes in the nonzero v-direction in the electric theory and consider massless flavors for Q and Q ′ by putting N f and N ′ f D6-branes at v = 0. The N ′′ f created D4-branes connecting between D6-branes and NS5 ′ R -brane can move freely in the w-direction. Moreover, since N ′ c D4-branes are suspending between two equal NS5 ′ L,R -branes located at different x 6 coordinate, these D4-branes can slide along the w-direction also. If we ignore the NS5 L -brane, N c D4-branes, N f D6-branes, the NS5 ′ L -brane, N ′ c D4-branes and N ′ f D6-branes(detaching these branes from Figure 2A), then this brane configuration leads to the standard N = 1 SQCD with the magnetic gauge group SU( N ′′ [3,6,7]. The IR dynamics of the electric theory is described by Seiberg dual and one uses the variables of dual theory instead of the original variables for the electric theory. Then the dual magnetic gauge group is given by SU(N c ) × SU(N ′ c ) × SU( N ′′ c ) and the matter contents are given by • for each gauge group, bifundamentals F (g) and F ( g), and gauge singlets in Figure 2A. In Figure 2B, the nonsupersymmetric minimal energy brane configuration with the same gauge group and matter contents above for massless Q(Q ′ ) and Q( Q ′ ) is given. • The flavor-singlet field F is in the bifundamental representation (N c , N ′ c , 1) under the whole gauge group and its complex conjugate field F is in the bifundamental representation (N c , N ′ c , 1) under the whole gauge group • The flavor-singlet field g is in the bifundamental representation (1, N ′ c , N ′′ c ) under the whole gauge group and its complex conjugate field g is in the bifundamental representation (1, N ′ c , N ′′ c ) under the whole gauge group There are (N ′′ f + N ′ c ) 2 gauge singlets in the third gauge group factor as follows: • N ′′ f -fields X ′ are in the fundamental representation under the SU(N ′ c ), its complex conjugate N ′′ f -fields X ′ are in the antifundamental representation under the SU(N ′ c ) and then X ′ are in the representation (1, N ′ c , 1) under the whole gauge group while X ′ are in the representation (1, N ′ c , 1) under the whole gauge group These additional N ′′ f -SU(N ′ c ) fundamentals and N ′′ f -SU(N ′ c ) antifundamentals are originating from the SU(N ′′ c ) chiral mesons GQ ′′ and G Q ′′ respectively where the color indices are contracted each other. Therefore, there are free indices for a single color and a single flavor.
Then the strings stretching between the N ′′ f D6-branes and N ′ c D4-branes will give rise to these additional N ′′ f -fields M ′′ are in the representation (1, 1, 1) under the whole gauge group This corresponds to the SU(N ′′ c ) chiral meson Q ′′ Q ′′ where the color indices are contracted. It is clear to see that since the N ′′ f D6-branes are parallel to the NS5 ′ R -brane from Figure  2A Figure 2A. As we will see later, for the nonsupersymmetric brane configuration, a misalignment for the N ′′ f -flavor D4-branes arises and some of the vacuum expectation value of M ′′ is fixed and the remaining components are arbitrary.
• The N ′2 c -fields Φ ′ is in the representation (1, N ′2 c − 1, 1)⊕(1, 1, 1) under the whole gauge group This corresponds to the SU(N ′′ c ) chiral meson G G where the color indices for the third gauge group are contracted each other and note that G has a representation (1, N ′ c , N ′′ c ) of an electric theory while G has a representation (1, N ′ c , N ′′ c ) of an electric theory. The fluctuations of the singlet Φ ′ correspond to the motion of N ′ c D4-branes suspended two NS5 ′ L,R -branes along the (789) directions in Figure 2A. The fact that these D4-branes can slide along the w-direction implies that the vacuum expectation value of Φ is arbitrary.
In the dual theory, since there exist N ′′ f quarks q ′′ , N ′′ f quarks q ′′ , one bifundamental field g(which will give rise to the contribution of N ′ c ), and its complex conjugate field g(which will give rise to the contribution of N ′ c ), the coefficient of the beta function for the third gauge group factor is b mag where we inserted the number of colors given in (2.3) in the second equality. Since there exist N ′ f quarks Q ′ , N ′ f quarks Q ′ , one bifundamental field g(which will give rise to the contribution of N ′′ c ), its complex conjugate field g(which will give rise to the contribution of N ′′ c ), N ′′ f -fields X ′ , its complex conjugate N ′′ f -fields X ′ , the singlet Φ ′ (which will give rise to N ′ c ), one bifundamental field F (which will give rise to the contribution of N c ), and its complex conjugate field F (which will give rise to the contribution of N c ), the coefficient of the beta function of second gauge group factor is b mag Similarly, since there exist N f quarks Q, N f quarks Q, one bifundamental field F (which will give rise to the contribution of N ′ c ), and its complex conjugate field F (which will give rise to the contribution of N ′ c ), the coefficient of the beta function of the first gauge group factor is b mag Nc) . Note that the SU(N c ) fields in the magnetic theory are the same as those of the electric theory. Therefore, SU(N c ), SU(N ′ c ) and SU( N ′′ c ) gauge couplings are IR free by requiring the negativeness of the coefficients of beta function. One relies on the perturbative calculations at low energy for this magnetic IR free region b mag SU (Nc Note that the SU(N ′ c ) fields in the magnetic theory are different from those of the electric theory.
gauge theory, by performing the dualizing the third gauge group, we have an SU(N c ) × SU(N ′ c ) × SU( N ′′ c ) gauge theory with additional fields X ′ , X ′ , M ′′ and Φ ′ and the gauge group and matter contents are summarized as follows: The dual magnetic superpotential, by taking the mass term (2.2) for Q ′′ and Q ′′ and massless flavors for m = m ′ = 0 in the electric theory, which is equal to put a linear term in M ′′ in the dual magnetic theory, in addition to the cubic interaction terms, takes the form Here q ′′ and q ′′ are fundamental and antifundamental for the third gauge group index respectively. Then, q ′′ q ′′ has rank N ′′ c while m ′′ has a rank N ′′ f . Therefore, the F-term condition, the derivative the superpotential W dual with respect to M ′′ , cannot be satisfied if the rank N ′′ f exceeds N ′′ c . This is so-called rank condition and the supersymmetry is broken [14]. Other F-term equations are satisfied by taking the vacuum expectation values of g, g, X ′ and X ′ to vanish. More explicitly, the classical moduli space of vacua can be obtained from the following F-term equations Then, it is easy to see that there exist three reduced equations and other F-term equations are satisfied if one takes the zero vacuum expectation values for the fields g, g, X ′ and X ′ . Then the solutions for the equations can be written as follows: Let us expand around a point on these vacua, as done in [1]. Then the remaining relevant terms of superpotential can be written as W rel dual = M ′′ 0 (δϕ δ ϕ + m ′′ ) + δZ δϕ q ′′ 0 + δ Z q ′′ 0 δ ϕ by following the same fluctuations for the various fields as in [12]: as well as the fluctuations of g, g, X ′ and X ′ . Note that there exist also three kinds of terms, the vacuum < q ′′ > multiplied by δ gδ X ′ , the vacuum < q ′′ > multiplied by δX ′ δg, and the vacuum < Φ ′ > multiplied by δgδ g. However, by redefining these, they do not enter the contributions for the one loop result, up to quadratic order. As done in [52], it turns out that will contain (log 4 − 1) > 0 which implies that these vacua are stable. Now let us recombine N ′′ c flavor D4-branes among N ′′ f flavor D4-branes(connecting between D6-branes and NS5 ′ R -brane) with the same number of color D4-branes(connecting between NS5 ′ R -brane and NS5 R -brane) and push them in +v direction from Figure 2A. We assume that N ′′ c ≥ N ′ c . After this procedure, there are no color D4-branes between NS5 ′ R -brane and NS5 R -brane. For the flavor D4-branes, we are left with only (N ′′ f − N ′′ c ) = N ′′ c − N ′ c flavor D4-branes connecting between D6-branes and NS5 ′ R -brane. Then the minimal energy supersymmetry breaking brane configuration is shown in Figure   2B. If we ignore the NS5 ′ L -brane, N ′ c D4-branes, N ′ f D6-branes, NS5 L -brane, N c D4-branes and N f D6-branes(detaching these from Figure 2B), as observed already, then this brane configuration corresponds to the minimal energy supersymmetry breaking brane configuration for the N = 1 SQCD with the magnetic gauge group SU( N ′′ [3,6,7]. The nonsupersymmetric minimal energy brane configuration in Figure 2B leads to the Figure 3 of [14] if we ignore the NS5 L -brane, N f D6-branes and N c D4-branes. Moreover, the Figure 2B with a replacement N ′′ f D6-branes by the NS5'-brane(neglecting the NS5 L -brane, N f D6-branes and N c D4-branes) will become the Figure 2B of [19] together with a reflection with respect to the NS5 L -brane and a rotation of NS5 R -brane by π 2 angle. When one moves the NS5 ′ R -brane to the left all the way past the NS5 R -brane, as long as the N ′′ f D6-branes are not parallel to NS5 ′ R -brane in an electric theory, then one arrives at the other magnetic brane configuration similar to the Figure 2. The only difference is that the N ′′ f D6-branes are located at the right hand side of the NS5 R -brane. Starting with NS5 ′ L -NS5 L -NS5 ′ R -NS5 R branes configuration, as in the footnote 3, and moving the NS5 R -brane to the left, one gets the nonsupersymmetric minimal energy brane configuration which is exactly the Figure 5B(which will appear later) with a reflection with respect to the NS5-brane. Furthermore, by moving the NS5 ′ L -brane to the right (as long as the N f D6-branes are not parallel to NS5 ′ L -brane in an electric theory), one gets the nonsupersymmetric minimal energy brane configuration which is exactly the new figure in previous paragraph with a reflection with respect to the NS5 L -brane.
After lifting the type IIA description to M-theory, the corresponding magnetic M5-brane configuration, in a background space of where e k is the position of the N ′′ f D6-branes in the v direction and this four dimensional space replaces (45610) directions, is described by [53] where we have ignored the lower power terms in v in t 3 , t 2 and t and the scales for the gauge groups in front of the first term and the last term, for simplicity. For fixed x, the coordinate t corresponds to y.
From this curve of quartic equation for t above, the asymptotic regions for four NS5branes can be classified by looking at the first two terms providing NS5 L -brane asymptotic region, next two terms providing NS5 ′ L -brane asymptotic region, next two terms providing NS5 ′ R -brane asymptotic region, and the last two terms giving NS5 R -brane asymptotic region as follows: Here the two NS5 ′ L,R -branes are moving in the +v direction holding everything else fixed instead of moving N ′′ f D6-branes in the +v direction, as in [7]. The harmonic function sourced by the D6-branes can be written explicitly by summing over three contributions from the N f , N ′ f and N ′′ f D6-branes and similar analysis to both solve the differential equation and find out the nonholomorphic curve can be done. We expect that an instability from a new M5-brane mode arises.

Magnetic theory with dual for second gauge group
Based on the procedure in previous subsection, one continues to consider dualizing one of the gauge groups regarding as the other gauge groups as a spectator. One takes the Seiberg dual for the second gauge group factor SU(N ′ c ) while remaining the first and third gauge group factor SU(N c ) and SU(N ′′ c ) unchanged. Also, since the dualized group's dynamical scale is far above that of the other spectator groups, we consider the case where Λ 2 >> Λ 1 , Λ 3 .
Let us move the NS5 R -brane in Figure 1 to the left all the way past the NS5 ′ L -brane. There is also a possibility to move NS5 ′ L -brane to the right instead. We'll come to this next subsection. After this brane motion, one arrives at the Figure 3A. The linking number of NS5 R -brane from Figure 3A is Figure 1, the presence of N ′′ c in the linking number arises. From these two relations, one obtains the number of colors of dual magnetic theory Compared with the previous case given by (2.3), the dependence on the ranks of both the first gauge group and the third gauge group occurs. Let us draw this magnetic brane configuration in Figure 3A and recall that we put the coincident N ′ f D6-branes in the nonzero v-direction in the electric theory as well as massless flavors for Q(Q ′′ ) and Q( Q ′′ ) for simplicity. If we ignore the NS5 L -brane, N c D4-branes, N f D6-branes, N ′′ f D6-branes, the NS5 ′ R -brane and N ′′ c D4-branes(detaching these branes from Figure 3A), then this brane configuration leads to the standard N = 1 SQCD with the magnetic gauge group SU( for each gauge group and bifundamentals F (g) and F ( g) and gauge singlets in Figure 3A. In Figure 3B, the nonsupersymmetric minimal energy brane configuration with the same gauge group and matter contents above for massless Q(Q ′′ ) and Q( Q ′′ ) is given.
In the dual theory, since there exist N ′ f quarks q ′ , N ′ f quarks q ′ , one bifundamental field g(which gives rise to the contribution of N ′′ c ) and its complex conjugate field g(which gives the contribution of N ′′ c ), one bifundamental field F (which will give rise to the contribution of N c ), and its complex conjugate field F (which will give rise to the contribution of N c ), the coefficient of the beta function for the second gauge group factor is b mag where we inserted the number of colors given in (2.4) in the second equality. Since there exist N ′′ f quarks Q ′′ , N ′′ f quarks Q ′′ , one bifundamental field g(which will give rise to the contribution of N ′ c ), its complex conjugate g(which will give rise to the contribution of N ′ c ), N ′ f -fields X ′′ , its complex conjugate N ′ f -fields X ′′ , and the singlet Φ ′′ (which will give rise to N ′′ c ), the coefficient of the beta function of second gauge group factor is b mag Similarly, since there exist N f quarks Q, N f quarks Q, one bifundamental field F (which will give rise to the contribution of N ′ c ), and its complex conjugate field F (which will give rise to the contribution of N ′ c ), the coefficient of the beta function of the first gauge group factor is b mag and SU(N ′′ c ) gauge couplings are IR free by requiring the negativeness of the coefficients of beta function. One relies on the perturbative calculations at low energy for this magnetic IR free region b mag The SU(N c ) fields in the magnetic theory are different from those of electric theory and also the SU(N ′′ c ) fields in the magnetic theory are different from those of electric theory. Neglecting the SU(N c ) and Here the N c -and N ′′ c -dependence appears. Then one can summarize the gauge group and matter contents where there are additional fields X ′′ , X ′′ , M ′ and Φ ′′ as follows: The dual magnetic superpotential, by adding the mass term (2.2) for Q ′ and Q ′ in the electric theory which is equal to put a linear term in M ′ in the dual magnetic theory, is given by One sees also the usual cubic interaction terms between the additional fields X ′′ , X ′′ , M ′ and Φ ′′ and their dual expressions, gq ′ , g q ′ , q ′ q ′ and g g respectively [14]. Here q ′ and q ′ are fundamental and antifundamental for the gauge group index respectively. Then, q ′ q ′ has rank N ′ c while m ′ has a rank N ′ f . Therefore, the F-term condition, the derivative the superpotential W dual with respect to M ′ , cannot be satisfied if the rank N ′ f exceeds N ′ c . This is so-called rank condition and the supersymmetry is broken. Other F-term equations are satisfied by taking the vacuum expectation values of g, g, X ′′ and X ′′ to vanish.
Then the solutions for these equations can be written as follows: As we did in previous case, one can analyze the one loop computation by expanding the fields around the vacua and it will lead to the fact that states are stable by realizing the mass of m 2 Then the minimal energy supersymmetry breaking brane configuration is shown in Figure  3B. Figure 3B), as observed already, then this brane configuration corresponds to the minimal energy supersymmetry breaking brane configuration The nonsupersymmetric minimal energy brane configuration Figure 3B with vanishing N ′′ f D6-branes leads to the Figure 3 of [14] with a reflection with respect to the NS5-brane if we ignore the NS5 L -brane, N f D6-branes and N c D4-branes. Moreover, this Figure 3B with a replacement N ′ f D6-branes by the NS5'-brane(and neglecting the NS5 L -brane, N f D6-branes and N c D4-branes) will become the Figure 2B of [19] with a rotation of NS5 R -brane by π 2 angle or the Figure 4B of [19] with a reflection with respect to the NS5 R -brane if we ignore the NS5 ′ R -brane, N ′′ f D6-branes and N ′′ c D4-branes from the Figure 3B. Starting with NS5 ′ L -NS5 L -NS5 ′ R -NS5 R branes configuration, as in the footnote 3, by moving the NS5 L -brane to the right, one gets the nonsupersymmetric minimal energy brane configuration which is exactly the Figure 3 with a reflection with respect to the NS5-brane.
By lifting the type IIA description to M-theory, the corresponding magnetic M5-brane configuration, in a background space where e k is the position of the N ′ f D6-branes in the v direction, is described by From this curve of quartic equation for t above, the asymptotic regions for four NS5-branes can be classified as follows:

Magnetic theory with dual for second gauge group
For the dualizing the middle gauge group, there exists another magnetic brane configuration.
One considers dualizing one of the gauge groups regarding as the other gauge groups as a spectator, as we did in previous subsections. Also we consider the case where in other words, the dualized group's dynamical scale is far above that of the other spectator groups.
Let us move the NS5 ′ L -brane in Figure 1 to the right all the way past the NS5 R -brane. After this brane motion, one arrives at the Figure 4A. That is, we rotate N ′ f D6-branes a little bit(this does not change the classical electric superpotential as we explained before) and after dualizing the second gauge group, we rotate those N ′ f D6-branes with opposite direction. The linking number of NS5 ′ L -brane from Figure 4A is On the other hand, the linking number of NS5 ′ L -brane from modified Figure 1 is Figure 1, the presence of N c in the linking number arises. From these two relations, one obtains the number of colors of dual magnetic theory which is the same as before given in (2.4). Let us draw this magnetic brane configuration in Figure 4A and recall that we put the coincident N ′ f D6-branes in the nonzero v-direction and the coincident N f and N ′′ f D6-branes at v = 0 in the electric theory.
In the dual theory, since there exist N ′ f quarks q ′ , N ′ f quarks q ′ , one bifundamental field G(which will give rise to the contribution of N ′′ c ), its complex conjugate field G(which will give rise to the contribution of N ′′ c ), one bifundamental field f which(will give rise to the for each gauge group and bifundamentals f (G) and f ( G) and gauge singlets in Figure 4A. In Figure 4B, the nonsupersymmetric minimal energy brane configuration with the same gauge group and matter contents above for massless Q(Q ′′ ) and Q( Q ′′ ) is given.
contribution of N c ), and its complex conjugate field f (which will give rise to the contribution of N c ), the coefficient of the beta function for the second gauge group factor is b mag where we inserted the number of colors given in (2.5) in the second equality and this is equal to the b mag SU ( e N ′ c ) in previous subsection. Since there exist N ′′ f quarks Q ′′ , N ′′ f quarks Q ′′ , one bifundamental field G(which will give rise to the contribution of N ′ c ), and its complex conjugate field G(which will give rise to the contribution of N ′ c ), the coefficient of the beta function of second gauge group factor is b mag Similarly, since there exist N f quarks Q, N f quarks Q, one bifundamental field f (which will give rise to the contribution of N ′ c ), its complex conjugate field f (which will give rise to the contribution of N ′ c ), N ′ f -fields X, its complex conjugate N ′ f -fields X, and the singlet Φ(which will give rise to N c ), the coefficient of the beta function of the first gauge group factor is b mag and SU(N ′′ c ) gauge couplings are IR free by requiring the negativeness of the coefficients of beta function. One relies on the perturbative calculations at low energy for this magnetic IR free region b mag Then one can summarize the gauge group and matter contents where there are additional fields X, X, M ′ and Φ as follows: The dual magnetic superpotential, by adding the mass term (2.2) for Q ′ and Q ′ in the electric theory, which is equal to put a linear term in M ′ in the dual magnetic theory, is given by Here q ′ and q ′ are fundamental and antifundamental for the gauge group index respectively [14]. Then, q ′ q ′ has rank N ′ c while m ′ has a rank N ′ f . Therefore, the F-term condition, the derivative the superpotential W dual with respect to M ′ , cannot be satisfied if the rank N ′ f exceeds N ′ c . This is so-called rank condition and the supersymmetry is broken. Other F-term equations are satisfied by taking the vacuum expectation values of f, f , X and X to vanish.
Then the solutions can be written as follows: As we did in previous case, one can analyze the one loop computation by expanding the fields around the vacua and it will lead to the fact that states are stable by realizing the mass of positive. Then the minimal energy supersymmetry breaking brane configuration is shown in Figure   4B. If we ignore the NS5 L -brane, N c D4-branes, N f D6-branes, NS5 ′ R -brane, N ′′ c D4-branes and N ′′ f D6-branes(detaching these from Figure 4B), then this brane configuration looks similar to the minimal energy supersymmetry breaking brane configuration for the N = 1 SQCD with the magnetic gauge group SU( The difference occurs in the position of D6-branes. The nonsupersymmetric minimal energy brane configuration Figure 4B with a replacement N ′ f D6-branes by the NS5'-brane(neglecting the NS5 ′ R -brane, N ′′ f D6-branes and N ′′ c D4branes with vanishing N f D6-branes) leads to the Figure 5B of [19] with a rotation of NS5 ′ Lbrane by π 2 angle. Starting with NS5 ′ L -NS5 L -NS5 ′ R -NS5 R branes configuration, as in the footnote 3, by moving the NS5 ′ R -brane to the left, one gets the nonsupersymmetric minimal energy brane configuration which is exactly the Figure 4 with a reflection with respect to the NS5-brane.
After lifting the type IIA description to M-theory, the magnetic M5-brane configuration, From this curve of quartic equation for t above, the asymptotic regions for four NS5-branes can be classified as follows

Magnetic theory with dual for first gauge group
Now we turn to the last magnetic brane configuration. One considers dualizing one of the gauge groups regarding as the other gauge groups as a spectator, as done in previous subsections. Also we consider the case where Λ 1 >> Λ 2 , Λ 3 , in other words, the dualized group's dynamical scale is far above that of the other spectator groups.
Let us move the NS5 L -brane in Figure 1 to the right all the way past the NS5 ′ L -brane. After this brane motion, one arrives at the Figure 5A. Recall that the N f D6-branes are perpendicular to the NS5 L -brane in Figure 1. The linking number of NS5 L -brane from Figure 5A is Figure 5A, the presence of N ′ c in the linking number arises. On the other hand, the linking number of NS5 L -brane from Figure 1 From these two relations, one obtains the number of colors of dual magnetic theory Let us draw this magnetic brane configuration in Figure 5A and recall that we put the coincident N f D6-branes in the nonzero v-direction as well as f D6-branes and N ′′ c D4-branes(detaching these branes from Figure 5A), then this brane configuration leads to the standard N = 1 SQCD with the magnetic gauge group [3,6,7].
for each gauge group and bifundamentals F (g) and F ( g) and gauge singlets in Figure 5A. In Figure 5B, the nonsupersymmetric minimal energy brane configuration with the same gauge group and matter contents above for massless Q ′ (Q ′′ ) and Q ′ ( Q ′′ ) is given.
In the dual theory, since there exist N f quarks q, N f quarks q, one bifundamental field f (which will give rise to the contribution of N ′ c ), and its complex conjugate f (which will give rise to the contribution of N ′ c ), the coefficient of the beta function for the third gauge group factor is b mag where we inserted the number of colors given in (2.6) in the second equality. Since there exist N ′ f quarks Q ′ , N ′ f quarks Q ′ , one bifundamental field f (which will give rise to the contribution of N c ), its complex conjugate field f(which will give rise to the contribution of N c ), one bifundamental field G(which will give rise to the contribution of N ′′ c ), its complex conjugate G(which will give rise to the contribution of N ′′ c ), N f fields X ′ , its complex conjugate N f fields X ′ , and the singlet Φ ′ (which will give rise to N ′ c ), the coefficient of the beta function of second gauge group factor is b mag one bifundamental field G(which will give rise to the contribution of N ′ c ), and its complex conjugate field G(which will give rise to the contribution of N ′ c ), the coefficient of the beta function of the third gauge group factor is b mag and SU(N ′′ c ) gauge couplings are IR free by requiring the negativeness of the coefficients of beta function. One relies on the perturbative calculations at low energy for this magnetic IR free region b mag fields in the magnetic theory are different from those of the electric theory.
Then one can summarize the gauge group and matter contents where there are additional fields X ′ , X ′ , M and Φ ′ as follows: The dual magnetic superpotential, by adding the mass term (2.2) for Q and Q in the electric theory, which is equal to put a linear term in M in the dual magnetic theory, is given by [14] W dual = Mq q + f X ′ q + f qX ′ + Φ ′ f f + mM.
Here q and q are fundamental and antifundamental for the gauge group index respectively. Then, q q has rank N c while m has a rank N f . Therefore, the F-term condition, the derivative the superpotential W dual with respect to M, cannot be satisfied if the rank N f exceeds N c . This is so-called rank condition and the supersymmetry is broken. Other F-term equations are satisfied by taking the vacuum expectation values of f, f , X ′ and X ′ to vanish.
Then the solutions can be written as follows: As we did in previous case, one can analyze the one loop computation by expanding the fields around the vacua and it will lead to the fact that states are stable by realizing the mass of Then the minimal energy supersymmetry breaking brane configuration is shown in Figure   5B. If we ignore the NS5 R -brane, N ′ c D4-branes, N ′ f D6-branes, NS5 ′ R -brane, N ′′ c D4-branes and N ′′ f D6-branes(detaching these from Figure 5B), as observed already, then this brane configuration is the minimal energy supersymmetry breaking brane configuration for the N = 1 SQCD with the magnetic gauge group SU( N c = N f − N c ) with N f massive flavors [3,6,7].
The nonsupersymmetric minimal energy brane configuration Figure 5B with a replacement N f D6-branes by the NS5'-brane(neglecting the NS5 ′ R -brane, N ′′ f D6-branes and N ′′ c D4branes and N ′ f D6-branes) leads to the Figure 4B of [19]. When we move the NS5 ′ L -brane in Figure 1 to the left all the way past the NS5 L -brane, then one arrives at the magnetic brane configuration similar to the Figure 5. The only difference is that the N f D6-branes are located at the right hand side of the NS5 L -brane. Then this nonsupersymmetric minimal energy brane configuration with a replacement N f D6branes by the NS5'-brane(by neglecting the NS5 ′ R -brane, N ′′ f D6-branes and N ′′ c D4-branes) leads to the Figure 5B of [19] with a reflection with respect to the NS5-brane and an extra rotation of NS5 ′ L -brane by π 2 angle. Starting with NS5 ′ L -NS5 L -NS5 ′ R -NS5 R branes configuration, as in the footnote 3, by moving the NS5 L -brane to the left, one gets the nonsupersymmetric minimal energy brane configuration which is exactly the Figure 2 with a reflection with respect to the NS5-brane. Furthermore, by moving the NS5 ′ R -brane to the right, one gets the nonsupersymmetric minimal energy brane configuration which is exactly the new figure in previous paragraph with a reflection with respect to the NS5 L -brane.
The corresponding magnetic M5-brane configuration, in a background space where this four dimensional space replaces (45610) directions, is de-scribed by From this curve of quartic equation for t above, the asymptotic regions for four NS5-branes can be classified as follows: Nc+N f + · · · NS L asymptotic region,
There exist (2n − 2) magnetic theories and they can be classified as four cases as follows.
• Case 1 When the Seiberg dual is taken for the first gauge group factor by assuming that Λ 1 >> Λ i where i = 2, · · · , n, one follows the procedure given in the subsection 2.5. The gauge group is and the matter contents are given by the dual quarks q 1 ( 1 , 1, · · · , 1) and q 1 in the representation ( 1 , 1, · · · , 1) as well as (n − 1) quarks Q i and Q i where i = 2, · · · , n, the bifundamentals f 1 in the representation ( 1 , 2 , 1, · · · , 1) under the dual gauge group, · · · , and f 1 in the representation ( 1 , 2 , 1, · · · , 1) under the dual gauge group in addition to (n − 2) bifundamentals G i and G i , and various gauge singlets X 2 , X 2 , M 1 and Φ 2 . The corresponding brane configuration can be obtained similarly and the extra (n − 3) NSbranes, (n − 3) sets of D6-branes and (n − 3) sets of D4-branes are present at the right hand side of the NS5 ′ R -brane of Figure 5. The magnetic superpotential can be written as By computing the contribution for the one loop as in the subsection 2.5, the vacua are stable and the asymptotic behavior of (n + 1) NS-branes can be obtained also.
• Case 2 When the Seiberg dual is taken for the last gauge group factor by assuming that Λ n >> Λ i where i = 1, 2, · · · , (n − 1), one follows the procedure given in the subsection 2.2. The gauge group is given by The corresponding brane configuration can be obtained similarly and the extra (n − 3) NSbranes, (n − 3) sets of D6-branes and (n − 3) sets of D4-branes are present at the left hand side of the NS5 L -brane of Figure 2. The magnetic superpotential can be written as W dual = M n q n q n + g n−1 X n−1 q n + g n−1 q n X n−1 + Φ n−1 g n−1 g n−1 + m n M n .
• Case 3 When the Seiberg dual is taken for the middle gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 2.3. The gauge group is given by where 2 ≤ i ≤ n−1 implying that the number of possible magnetic gauge group is (n−2). The corresponding brane configuration can be obtained similarly and the extra (i − 2) NS-branes, • Case 4 When the Seiberg dual is taken for the middle gauge group factor with different brane motion by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 2.4. The gauge group is given by where 2 ≤ i ≤ n − 1 implying that the number of possible magnetic gauge group is given by (n − 2). The corresponding brane configuration can be obtained similarly and the extra (i − 2) NS-branes, (i − 2) sets of D6-branes and (i − 2) sets of D4-branes are present at the left hand side of the NS5 L -brane and the extra (n − i − 1) NS-branes, (n − i − 1) sets of D6-branes and (n − i − 1) sets of D4-branes are present at the right hand side of the NS5 ′ R -brane of Figure 4. The magnetic superpotential can be written as and its multiple product gauge theories

Electric theory
The gauge group and matter contents are summarized as follows: In the electric theory 4 , since there exist 2N f quarks Q, and one bifundamental field F (which will give rise to the contribution of 2N ′ c ), the coefficient of the beta function of the first gauge group factor is b quarks Q ′ , one bifundamental field F (which will give rise to the contribution of 2N c ), and one bifundamental field G(which will give rise to the contribution of 2N ′′ c ), the coefficient of the beta function of the second gauge group factor is Finally, since there exist 2N ′′ f quarks Q ′′ , and one bifundamental field G(which will give rise to the contribution of 2N ′ c ), the coefficient of the beta function of the third gauge group factor The classical electric superpotential can be obtained from (2.1) by orientifolding where the coefficient functions µ, µ ′ , µ ′′ , λ, λ ′ and λ ′′ are given by six rotation angles. Here the adjoint field for Sp(N c ) gauge group is denoted by A, the adjoint field for SO(2N ′ c ) gauge group is denoted by A ′ , and the adjoint field for Sp(N ′′ c ) gauge group is denoted by A ′′ . The mass terms of these adjoint fields are related to the rotation angles of NS-branes in type IIA brane configuration. The couplings of flavors with these adjoint fields are related also to the rotation angles of NS-branes as well as the rotation angles of D6-branes. We add the mass terms for each flavor. Setting the fields Q ′′ , G, and A ′′ to zero, the superpotential becomes the one described in [15].
After integrating out the adjoint fields A, A ′ and A ′′ , this superpotential (3.1) at the particular orientations for branes, i.e., the case where any two neighboring NS-branes are perpendicular to each other, will reduce to the last three mass-deformed terms since the coefficient functions 1 µ , 1 µ ′ and 1 µ ′′ vanish at this particular rotation angles for branes. Then the classical superpotential by deforming this theory by adding the mass terms for the quarks Q(Q ′ )[Q ′′ ] is given by The type IIA brane configuration for this mass-deformed theory can be described by as follows. The 2N c -color D4-branes (01236) are suspended between the NS5 L -brane (012345) and the NS5 ′ L -brane (012389) along x 6 direction, together with N f D6-branes (0123789) which are parallel to NS5 ′ L -brane and have nonzero +v direction(and their mirrors). The NS5 Rbrane is located at the right hand side of the NS5 ′ L -brane along the x 6 direction and there exist 2N ′ c -color D4-branes suspended between them, with N ′ f D6-branes which have nonzero +v direction(and their mirrors). Moreover, the NS5 ′ R -brane is located at the right hand side of the NS5 R -brane along the x 6 direction and there exist 2N ′′ c -color D4-branes suspended between them, with N ′′ f D6-branes which have nonzero +v direction(and their mirrors). One summarizes the brane configuration as follows: Now we draw this electric brane configuration in Figure 6 and we put the coincident

Magnetic theory with dual for third gauge group
Let us take the Seiberg dual for the third gauge group factor Sp(N ′′ c ) while keeping the first and the second gauge group factors Sp(N c ) and SO(2N ′ c ) untouched. Suppose that Λ 3 >> Λ 1 , Λ 2 . This can be done by type IIA string theory side via brane motion. After we move the NS5 R -brane in Figure 6 to the right all the way past the NS5 ′ R -brane, we arrives at the Figure 7A. Then the linking number of NS5 R -brane from Figure 7A is given by Note that O4 + -plane with RR charge +1 realizes a symplectic gauge group while O4 − -plane with RR charge −1 does an orthogonal gauge group. Originally, it was Figure 6 before the brane motion. Therefore, by the linking number conservation and equating these two L 5 's each other, we are left with the number of colors in the magnetic theory N ′′ Note the dependence on N ′ c here and the constant piece −2 comes from the presence of O4-plane. Let us draw this magnetic brane configuration in Figure 7A and we put the coincident N ′′ f D6-branes in the nonzero +v direction(and its mirrors) as well as the massless Q(Q ′ ). If we ignore the NS5 L -brane, 2N c D4-branes, 2N f D6-branes, the NS5 ′ L -brane, 2N ′ c D4-branes and 2N ′ f D6-branes(detaching these branes from Figure 7A), then this brane configuration leads to the standard N = 1 SQCD with the magnetic gauge group Sp( N ′′ f massive flavors [6,11].  Figure 7A. In Figure 7B, the nonsupersymmetric minimal energy brane configuration with the same gauge group and matter contents above for massless Q(Q ′ ) is given.
In the dual theory, since there exist 2N ′′ f fundamental fields q ′′ , and one bifundamental field g(which will give rise to the contribution of 2N ′ c ), the coefficient of the beta function is b mag fundamental fields g for the second factor, 2N c fundamental fields F for the first factor, an antisymmetric tensor Φ ′ (which will contribute to (2N ′ c −2)), and 2N ′′ f fields X ′ , the coefficient of the beta function is b mag Since there exist 2N f quarks Q, and one bifundamental field F (which will give rise to the contribution of 2N ′ c ), the coefficient of the beta function of the first gauge group factor is b mag Nc) . Note that the Sp(N c ) fields in the magnetic theory are the same as those of the electric theory.
Then one can summarize the gauge group and matter contents where there are additional fields X ′ , M ′′ and Φ ′ as follows: The dual magnetic tree level superpotential, by adding the mass term for the Q ′′ in electric theory corresponding to add a linear term in M ′′ in dual magnetic theory, is given by Here q ′′ is fundamental for the gauge group index [15]. Then, q ′′ q ′′ has rank 2 N ′′ c while m ′′ has a rank 2N ′′ f . Therefore, the F-term condition, the derivative the superpotential W dual with respect to M ′′ , cannot be satisfied if the 2N ′′ f exceeds 2 N ′′ c . This is so-called rank condition and the supersymmetry is broken.
More explicitly, the classical moduli space of vacua can be obtained from F-term equations.
From the F-terms F q ′′ and F M ′′ , one gets M ′′ q ′′ + gX ′ = 0 = q ′′ q ′′ + m. Similarly, one obtains Φ ′ g + X ′ q ′′ = 0 = gg from the F-terms F g and F Φ ′ . Moreover, there is a relation q ′′ g = 0 from the F-term F X ′ . Then, one obtains the following solutions c ) ⊗ iσ 2 as well as the fluctuations for g and X ′ . There are two kinds of terms, the vacuum of < q ′′ > multiplied by δX ′ δg and the vacuum of < Φ ′ > multiplied by δgδg. By redefining these as δX ′ δĝ and δĝδĝ respectively, they do not enter the contributions for the one loop result. At one loop, the effective potential V  Figure 7B), then this brane configuration corresponds to the minimal energy supersymmetry breaking brane configuration for the N = 1 SQCD with the magnetic gauge group Sp( N ′′ c ) with N ′′ f massive flavors [6,11]. The nonsupersymmetric minimal energy brane configuration Figure 7B with vanishing 2N ′ f D6-branes leads to the Figure 3 of [15] if we ignore the NS5 L -brane, 2N f D6-branes and 2N c D4-branes. Moreover, this Figure 7B with a replacement N ′′ f D6-branes by the NS5'brane(neglecting the NS5 L -brane, 2N f D6-branes and 2N c D4-branes) will become the Figure  7B of [19] with a reflection with respect to the NS5 L -brane and a rotation of NS5 R -brane by π 2 angle.
Starting with NS5 ′ L -NS5 L -NS5 ′ R -NS5 R branes configuration, as in the footnote 3, by moving the NS5 R -brane to the left, one gets the nonsupersymmetric minimal energy brane configuration which is exactly the Figure 10B(which will appear later) with a reflection with respect to the NS5-brane. Furthermore, by moving the NS5 ′ L -brane to the right, one gets the nonsupersymmetric minimal energy brane configuration which is exactly the new figure in previous paragraph with a reflection with respect to the NS5 L -brane.
After lifting the type IIA description we explained so far to M-theory, the corresponding magnetic M5-brane configuration, in a background space of where this four dimensional space replaces (45610) directions, is characterized by [55] where we ignored both the lower power terms in v and the scales for the gauge groups for simplicity.
From this curve of quartic equation for t above, the asymptotic regions can be classified as follows:

Magnetic theory with dual for second gauge group
Let us take the Seiberg dual for the second gauge group factor SO(2N ′ c ) while keeping the first and the third gauge group factors Sp(N c ) and Sp(N ′′ c ) untouched. Suppose that Λ 2 >> Λ 1 , Λ 3 . After we move the NS5 R -brane in Figure 6 to the left all the way past the NS5 ′ L -brane, the linking number of NS5 R -brane from Figure 8A is given by Figure 6 in electric theory before the brane motion. Therefore, by the linking number conservation and equating these two L 5 's each other, we are left with the number of colors in the magnetic theory N ′ Here the dependence on N c and N ′′ c arises. Let us draw this magnetic brane configuration in Figure 8A and we put the coincident N ′ f D6-branes in the nonzero v direction(and its mirrors). For the N f and N ′′ f D6-branes we consider the massless flavors and then they are located at v = 0. If we ignore the NS5 L -brane, 2N c D4-branes, 2N f D6-branes, the NS5 ′ R -brane, 2N ′′ c D4-branes and 2N ′′ f D6-branes(detaching these branes from Figure 8A), then this brane configuration leads to the standard N = 1 SQCD with the magnetic gauge group [6,11].
In the dual theory, since there exist 2N ′ f fundamental fields q ′ , one bifundamental field g(which will give rise to the contribution of 2N ′′ c ), and one bifundamental field F (which will give rise to the contribution of 2N c ), the coefficient of the beta function is b mag and since there are 2 N ′ c fundamental fields g for the first factor, 2N ′′ f fundamental fields Q ′′ , an antisymmetric tensor Φ ′′ (which will contribute to (2N ′′ c + 2)), and a field X ′′ (which will contribute to 2N ′ f ), the coefficient of the beta function is b mag Since there exist 2N f quarks Q, and one  Figure 8A. In Figure 8B, the nonsupersymmetric minimal energy brane configuration with the same gauge group and matter contents above for massless Q(Q ′′ ) is given.
bifundamental field F (which will give rise to the contribution of 2 N ′ c ), the coefficient of the beta function of the first gauge group factor is b mag Then the dual gauge group and matter contents with the additional fields X ′′ , M ′ and Φ ′′ are gauge group : The dual magnetic tree level superpotential, by adding the mass term for the Q ′ in electric theory corresponding to add a linear term in M ′ in dual magnetic theory, is given by Here q ′ is fundamental for the gauge group index. Then, q ′ q ′ has rank 2 N ′ c while m ′ has a rank 2N ′ f . Therefore, the F-term condition, the derivative the superpotential W dual with respect to M ′ , cannot be satisfied if the rank 2N ′ f exceeds 2 N ′ c . This is so-called rank condition and the supersymmetry is broken [15].
The classical moduli space of vacua can be obtained from F-term equations. From the F-terms F q ′ and F M ′ , one gets M ′ q ′ +gX ′′ = 0 = q ′ q ′ +m ′ . Similarly, one obtains Φ ′′ g +X ′′ q ′ = 0 = gg from the F-terms F g and F Φ ′′ . Moreover, there is a relation q ′ g = 0 from the F-term F X ′′ . Then, one obtains the following solutions  Figure 8B), then this brane configuration corresponds to the minimal energy supersymmetry breaking brane configuration for the N = 1 SQCD with the magnetic gauge group SO(2 N ′ c ) with N ′ f massive flavors [6,11]. The nonsupersymmetric minimal energy brane configuration Figure 8B with vanishing 2N ′′ f D6-branes leads to the Figure 3 of [15] with a reflection with respect to the NS5-brane if we ignore the NS5 L -brane, 2N f D6-branes and 2N c D4-branes. Moreover, this Figure 8B with a replacement N ′ f D6-branes by the NS5'-brane(neglecting the NS5 L -brane, N f D6-branes and N c D4-branes) will become the Figure 7B of [19] with a rotation of NS5 R -brane by π 2 angle or the Figure 9B of [19] with a reflection with respect to the NS5 R -brane if we ignore the NS5 ′ R -brane, N ′′ f D6-branes and N ′′ c D4-branes from the Figure 8B. Starting with NS5 ′ L -NS5 L -NS5 ′ R -NS5 R branes configuration, as in the footnote 3, by moving the NS5 L -brane to the right, one gets the nonsupersymmetric minimal energy brane configuration which is exactly the Figure 8 with a reflection with respect to the NS5-brane.
After lifting the type IIA description to M-theory, the corresponding magnetic M5-brane configuration with equal mass for the quarks, in a background space of where this four dimensional space replaces (45610) directions, is characterized by [55] where we ignored both the lower power terms in v and the scales for the gauge groups for simplicity.
From this curve of quartic equation for t above, the asymptotic regions can be classified by as follows:

Magnetic theory with dual for second gauge group
Let us take the Seiberg dual for the second gauge group factor SO(2N ′ c ) while keeping the first and the third gauge group factors Sp(N c ) and Sp(N ′′ c ) untouched. Suppose that Λ 2 >> Λ 1 , Λ 3 . Let us move the NS5 ′ L -brane in Figure 6 to the right all the way past the NS5 R -brane. After this brane motion, one arrives at the Figure 9A. We need to change the Figure 6 a little bit such that the N ′ f D6-branes are not parallel to the NS5 ′ L -brane. That is, we rotate N ′ f D6-branes a little bit(this does not change the classical electric superpotential as we explained before) and after dualizing the second gauge group, we rotate those N ′ f D6-branes with opposite direction. The linking number of NS5 ′ L -brane from Figure 9A is given by Figure 6 before the brane motion. Therefore, by the linking number conservation and equating these two L 5 's each other, we are left with the number of colors in the magnetic Let us draw this magnetic brane configuration in Figure 9A and we put the coincident N ′ f D6-branes in the nonzero v direction(and its mirrors).  Figure 9A. In Figure 9B, the nonsupersymmetric minimal energy brane configuration with the same gauge group and matter contents above for massless Q(Q ′′ ) is given.
Then the dual gauge group and matter contents with the additional fields X, M ′ and Φ are gauge group : In the dual theory, since there exist 2N ′ f flavor fields q ′ and one bifundamental field G(which will give rise to the contribution of 2N ′′ c ), and one bifundamental field f (which will give rise to the contribution of 2N c ), the coefficient of the beta function is b mag c ) in previous subsection and since there are 2 N ′ c fields G for the first factor, and 2N ′′ f fundamental fields Q ′′ , the coefficient of the beta function is b mag Since there exist 2N f quarks Q, one bifundamental field f (which will give rise to the contribution of 2 N ′ c ), an antisymmetric tensor Φ(which will contribute to (2N c + 2)), and a field X(which will contribute to 2N f ), the coefficient of the beta function of the first gauge group factor is b mag The dual magnetic tree level superpotential, by adding the mass term for the Q ′ in electric theory corresponding to add a linear term in M ′ in dual magnetic theory, is given by Here q ′ is fundamental for the gauge group index. Then, q ′ q ′ has rank 2 N ′ c while m ′ has a rank 2N ′ f . Therefore, the F-term condition, the derivative the superpotential W dual with respect to M ′ , cannot be satisfied if the rank 2N ′ f exceeds 2 N ′ c . This is so-called rank condition [15] and the supersymmetry is broken.
The classical moduli space of vacua can be obtained from F-term equations. From the F-terms F q ′ and F M ′ , one gets M ′ q ′ + f X = 0 = q ′ q ′ + m ′ . Similarly, one obtains Φf + Xq ′ = 0 = f f from the F-terms F f and F Φ . Moreover, there is a relation q ′ f = 0 from the F-term F X . Then, one obtains the following solutions The nonsupersymmetric minimal energy brane configuration Figure 9B with a replacement N ′ f D6-branes by the NS5'-brane(neglecting the NS5 ′ R -brane, 2N ′′ f D6-branes and 2N ′′ c D4branes with vanishing 2N f D6-branes) leads to the Figure 10B of [19] with a rotation of NS5 ′ L -brane by π 2 angle. Starting with NS5 ′ L -NS5 L -NS5 ′ R -NS5 R branes configuration, as in the footnote 3, by moving the NS5 ′ R -brane to the left, one gets the nonsupersymmetric minimal energy brane configuration which is exactly the Figure 9 with a reflection with respect to the NS5-brane.
After lifting the type IIA description we explained so far to M-theory, the corresponding magnetic M5-brane configuration, in a background space of where this four dimensional space replaces (45610) directions, is characterized by [55] From this curve of quartic equation for t above, the asymptotic regions can be classified as follows:

Magnetic theory with dual for first gauge group
Let us take the Seiberg dual for the first gauge group factor Sp(N c ) while keeping the second and the third gauge group factors SO(2N ′ c ) and Sp(N ′′ c ) untouched. Suppose that Λ 1 >> Λ 2 , Λ 3 . After we move the NS5 L -brane in Figure 6 to the right all the way past the NS5 ′ Lbrane, one arrives at Figure 10A and the linking number of NS5 L -brane from Figure 10A is given by Figure 6 before the brane motion. Therefore, by the linking number conservation and equating these two L 5 's each other, we are left with the number of colors in the magnetic Let us draw this magnetic brane configuration in Figure 10A and we put the coincident N f D6-branes in the nonzero v direction(and its mirrors). If we ignore the NS5 R -brane, N ′ c D4-branes, N ′ f D6-branes, the NS5 ′ R -brane, N ′′ f D6-branes and N ′′ c D4-branes(detaching these branes from Figure 10A), then this brane configuration leads to the standard N = 1 SQCD with the magnetic gauge group Sp( N c = N f − N c − 2) with N f massive flavors [6,11].
for each gauge group, the bifundamentals F (g), and gauge singlets in Figure 10A. In Figure 10B, the nonsupersymmetric minimal energy brane configuration with the same gauge group and matter contents above for massless Q ′ (Q ′′ ) is given.
In the dual theory, since there exist 2N f fundamental fields q, and one bifundamental field f (which will give rise to the contribution of 2N ′ c ), the coefficient of the beta function is b mag c and since there are 2N ′ f fundamental fields Q ′ , 2 N c fundamental fields f for the second factor, 2N ′′ c fundamental fields G for the flavor index of the first factor, an antisymmetric tensor Φ ′ (which will contribute to (2N ′ c −2)), and 2N f fields X ′ , the coefficient of the beta function is b mag f quarks Q ′′ , and one bifundamental field G(which will give rise to the contribution of 2N ′ c ), the coefficient of the beta function of the third gauge group factor is b mag is more asymptotically free than SO(2N ′ c ) mag . Neglecting the SO(2N ′ c ) and Sp(N ′′ c ) dynamics, the magnetic Sp( N c ) is IR free when Then the dual gauge group and matter contents with the additional fields X ′ , M and Φ ′ are gauge group : The dual magnetic tree level superpotential, by adding the mass term for the Q in electric theory corresponding to add a linear term in M in dual magnetic theory, is given by [15] Here q is fundamental for the gauge group index. Then, qq has rank 2 N c while m has a rank 2N f . Therefore, the F-term condition, the derivative the superpotential W dual with respect to M, cannot be satisfied if the rank 2N f exceeds 2 N c . This is so-called rank condition and the supersymmetry is broken.
The classical moduli space of vacua can be obtained from F-term equations. From the Fterms F q and F M , one gets Mq+f X ′ = 0 = qq+m. Similarly, one obtains Φ ′ f +X ′ q = 0 = f f from the F-terms F f and F Φ ′ . Moreover, there is a relation qf = 0 from the F-term F X ′ . Then, one obtains the following solutions antisymmetric matrix and the zeros of < q > are 2(N f − N c ) × 2 N c zero matrices. Similarly, the zeros of 2N f × 2N f matrix M are assumed also. As we did in previous case, one can analyze the one loop computation by expanding the fields around the vacua and it will lead to the fact that states are stable by realizing the mass of m 2 M 0 positive. Then the minimal energy supersymmetry breaking brane configuration is shown in Figure  10B. If we ignore the NS5 R -brane, 2N ′ c D4-branes, 2N ′ f D6-branes, NS5 ′ R -brane, 2N ′′ c D4branes and 2N ′′ f D6-branes(detaching these from Figure 10B), as observed already, then this brane configuration is the minimal energy supersymmetry breaking brane configuration for the N = 1 SQCD with the magnetic gauge group Sp( N c = N f − N c − 2) with N f massive flavors [6,11].
The nonsupersymmetric minimal energy brane configuration Figure 10B with a replacement N f D6-branes by the NS5'-brane(neglecting the NS5 ′ R -brane, 2N ′′ f D6-branes and 2N ′′ c D4-branes with vanishing 2N ′ f D6-branes) leads to the Figure 10B of [19]. When we move the NS5 ′ L -brane in Figure 6 to the left all the way past the NS5 L -brane, then one arrives at the magnetic brane configuration similar to the Figure 10. The only difference is that the N f D6-branes are located at the right hand side of the NS5 L -brane.
Then this nonsupersymmetric minimal energy brane configuration with a replacement N f D6branes by the NS5'-brane(neglecting the NS5 ′ R -brane, 2N ′′ f D6-branes and 2N ′′ c D4-branes) leads to the Figure 10B of [19] with a reflection with respect to the NS5-brane and an extra rotation of NS5 ′ L -brane by π 2 angle. Starting with NS5 ′ L -NS5 L -NS5 ′ R -NS5 R branes configuration, as in the footnote 3, by moving the NS5 L -brane to the left, one gets the nonsupersymmetric minimal energy brane configuration which is exactly the Figure 7 with a reflection with respect to the NS5-brane.
Furthermore, by moving the NS5 ′ R -brane to the right, one gets the nonsupersymmetric minimal energy brane configuration which is exactly the new Figure in previous paragraph with a reflection with respect to the NS5 L -brane.
After lifting the type IIA description we explained so far to M-theory, the corresponding magnetic M5-brane configuration, in a background space of where this four dimensional space replaces (45610) directions, is characterized by [55] From this curve of quartic equation for t above, the asymptotic regions can be classified as follows: Nc−2+2N f + · · · NS L asymptotic region,

Other magnetic theories with opposite O4-plane charges
By changing the charges of O4-plane in previous brane configuration given in Figure 6, the type IIA brane configuration is realized by an N = 1 supersymmetric gauge theory with [54] SO and corresponding matter contents. Then by deforming this electric theory by mass term for the quarks and taking the magnetic dual on each gauge group factor, one gets the possible meta-stable brane configurations. There exists an N = 1 magnetic supersymmetric gauge theory with with matters and the corresponding brane configuration is given by the Figure 7 with opposite O4-plane charges(O4 + goes to O4 − and vice versa). The constant term +2 in the dual color has opposite sign, compared with the discussion of subsection 3.2 where it has −2. This is due to the fact that the linking number counting in this case has opposite sign for the contribution of O4-plane since we have different charge in the electric and magnetic theories.
Since the dual gauge group is an orthogonal gauge group, the corresponding gauge singlet M ′′ is a symmetric two index tensor in the flavor indices(leading to different structure of vacuum expectation value, compared with the solution in the subsection 3.2) and the field Φ ′ is a symmetric under the second gauge group. Also there is an N = 1 magnetic supersymmetric gauge theory with with matters and the corresponding brane configuration is given by the Figure 8 with opposite O4-plane charges(O4 + goes to O4 − and vice versa). Since the dual gauge group is a symplectic gauge group, the corresponding gauge singlet M ′ is an antisymmetric two index tensor in the flavor indices(leading to different structure of vacuum expectation value, compared with the solution in the subsection 3.3) and the field Φ ′′ is an antisymmetric under the third gauge group.
Moreover, there is an N = 1 magnetic supersymmetric gauge theory with with matters and the corresponding brane configuration is given by the Figure 9 with opposite O4-plane charges(O4 + goes to O4 − and vice versa). Since the dual gauge group is a symplectic gauge group, the corresponding gauge singlet M ′ is an antisymmetric two index tensor in the flavor indices(leading to different structure of vacuum expectation value, compared with the solution in the subsection 3.4) and the field Φ is an antisymmetric under the first gauge group.
Finally, there exists an N = 1 magnetic supersymmetric gauge theory with with matters and the corresponding brane configuration is given by the Figure 10 with opposite O4-plane charges(O4 + goes to O4 − and vice versa). Since the dual gauge group is a symplectic gauge group, the corresponding gauge singlet M is a symmetric two index tensor in the flavor indices(leading to different structure of vacuum expectation value, compared with the solution in the subsection 3.5) and the field Φ ′ is a symmetric under the second gauge group.
The remaining analysis can be done easily without any difficulty.
• Case 1 When the Seiberg dual is taken for the first gauge group factor by assuming that Λ 1 >> Λ i where i = 2, · · · , n, one follows the procedure given in the subsection 3.5. The gauge group is and the matter contents are given by the dual quark q 1 in the representation ( 1 , 1, · · · , 1) as well as (n − 1) quarks Q i where i = 2, · · · , n, the bifundamentals f 1 in the representation ( 1 , 2 , 1, · · · , 1) under the gauge group, · · · in addition to (n − 2) bifundamentals G i , and various gauge singlets X 2 , M 1 and Φ 2 . The corresponding brane configuration can be obtained similarly and the extra (n − 3) NS-branes, (n − 3) sets of D6-branes and (n − 3) sets of D4-branes are present at the right hand side of the NS5 ′ R -brane of Figure 10. The magnetic superpotential is W dual = (M 1 q 1 q 1 + f 1 X 2 q 1 + Φ 2 f 1 f 1 ) + m 1 M 1 . By computing the contribution for the one loop as in the subsection 3.5, the vacua are stable and the asymptotic behavior of (n + 1) NS-branes can be obtained also.
• Case 2 When the Seiberg dual is taken for the last gauge group factor by assuming that Λ n >> Λ i where i = 1, 2, · · · , (n − 1), one follows the procedure given in the subsection 3.2. The gauge group is given by  Figure 7. The magnetic superpotential is W dual = (M n q n q n + g n−1 X n−1 q n + Φ n−1 g n−1 g n−1 ) + m n M n .
• Case 3 When the Seiberg dual is taken for the middle gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 3.3. The gauge group is given by where i = 2, 4, · · · , (n − 1) implying that the number of possible magnetic gauge group is When the Seiberg dual is taken for the middle gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 3.4. The gauge group is given by where i = 2, 4, · · · , (n − 1) implying that the number of possible magnetic gauge group is  Figure  9. The magnetic superpotential is • Case 5 When the Seiberg dual is taken for the symplectic gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 3.6. The gauge group is given by where i = 3, 5, · · · , (n − 2) implying that the number of possible magnetic gauge group is When the Seiberg dual is taken for the symplectic gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 3.6. The gauge group is given by where i = 3, 5, · · · , (n − 2) implying that the number of possible magnetic gauge group is n−3 2 . The corresponding brane configuration can be obtained similarly and the extra (i − 2) NS-branes, (i − 2) sets of D6-branes and (i − 2) sets of D4-branes are present at the left hand side of the NS5 L -brane and the extra (n − i − 1) NS-branes, (n − i − 1) sets of D6-branes and (n − i − 1) sets of D4-branes are present at the right hand side of the NS5 ′ R -brane of Figure  9. The magnetic superpotential is
• Case 1 ′ When the Seiberg dual is taken for the first gauge group factor by assuming that Λ 1 >> Λ i where i = 2, · · · , n, one follows the procedure given in the subsection 3.5. The gauge group is SO(2 N c,1 ≡ 2(N f,1 + N c,2 − N c,1 + 2)) × Sp(N c,2 ) × · · · × SO(2N c,n ) and the matter contents are given by the dual quark q 1 in the representation ( 1 , 1, · · · , 1) as well as (n − 1) quarks Q i where i = 2, · · · , n, the bifundamentals f 1 in the representation ( 1 , 2 , 1, · · · , 1), · · · in addition to (n − 2) bifundamentals G i , and various gauge singlets X 2 , M 1 and Φ 2 . The corresponding brane configuration can be obtained similarly and the extra (n − 3) NS-branes, (n − 3) sets of D6-branes and (n − 3) sets of D4-branes are present at the right hand side of the NS5 ′ R -brane of Figure 10. The magnetic superpotential is When the Seiberg dual is taken for the last gauge group factor by assuming that Λ n >> Λ i where i = 1, 2, · · · , (n − 1), one follows the procedure given in the subsection 3.2. The gauge group is given by The corresponding brane configuration can be obtained similarly and the extra (n − 3) NS-branes, (n − 3) sets of D6-branes and (n − 3) sets of D4-branes are present at the left hand side of the NS5 L -brane of Figure 7. The magnetic superpotential is W dual = (M n q n q n + g n−1 X n−1 q n + Φ n−1 g n−1 g n−1 ) + m n M n .
• Case 3 ′ When the Seiberg dual is taken for the middle gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 3.3. The gauge group is given by where i = 2, 4, · · · , (n − 1) implying that the number of possible magnetic gauge group is • Case 4 ′ When the Seiberg dual is taken for the middle gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 3.4. The gauge group is given by where i = 2, 4, · · · , (n − 1) implying that the number of possible magnetic gauge group is When the Seiberg dual is taken for the symplectic gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 3.6. The gauge group is given by where i = 3, 5, · · · , (n − 2) implying that the number of possible magnetic gauge group is • Case 6 ′ When the Seiberg dual is taken for the symplectic gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 3.6. The gauge group is given by where i = 3, 5, · · · , (n − 2) implying that the number of possible magnetic gauge group is  Figure  9. The magnetic superpotential is

The symplectic gauge group at the start and the orthogonal gauge group at end of the chain
When the electric theory is described by with the (n−1) bifundametals ( 1 , 2 , 1, · · · , 1), · · · , and (1, · · · , 1, n−1 , n ), and n-quarks in the representation ( 1 , 1, · · · , 1), · · · , and (1, · · · , 1, n ), there exist (2n − 2) magnetic theories and they can be classified as six cases as follows. Note that there are the symplectic gauge group at the start and the orthogonal gauge group at the end of the chain.
• Case 2 ′′ When the Seiberg dual is taken for the last gauge group factor by assuming that Λ n >> Λ i where i = 1, 2, · · · , (n − 1), one follows the procedure given in the subsection 3.2. The gauge group is given by The corresponding brane configuration can be obtained similarly and the extra (n − 3) NS-branes, (n − 3) sets of D6-branes and (n − 3) sets of D4-branes are present at the left hand side of the NS5 L -brane of Figure 7. The magnetic superpotential is W dual = (M n q n q n + g n−1 X n−1 q n + Φ n−1 g n−1 g n−1 ) + m n M n .
• Case 5 ′′ When the Seiberg dual is taken for the symplectic gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 3.6. The gauge group is given by where i = 3, 5, · · · , (n − 2) implying that the number of possible magnetic gauge group is n−3 2 . The corresponding brane configuration can be obtained similarly and the extra (i − 2) NS-branes, (i − 2) sets of D6-branes and (i − 2) sets of D4-branes are present at the left hand side of the NS5 L -brane and the extra (n − i − 1) NS-branes, (n − i − 1) sets of D6-branes and (n − i − 1) sets of D4-branes are present at the right hand side of the NS5 ′ R -brane of Figure  8. The magnetic superpotential is W dual = (M i q i q i + g i X i+1 q i + Φ i+1 g i g i ) + m i M i .
The magnetic superpotential is W dual = (M n q n q n + g n−1 X n−1 q n + Φ n−1 g n−1 g n−1 ) + m n M n . The corresponding brane configuration can be obtained similarly and the extra (n − 3) NSbranes, (n − 3) sets of D6-branes and (n − 3) sets of D4-branes are present at the left hand side of the NS5 L -brane of Figure 7.
• Case 3 ′′′ When the Seiberg dual is taken for the middle gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 3.3. The gauge group is given by · · · × Sp(N c,i−1 ) × SO(2 N c,i ≡ 2(N f,i + N c,i+1 + N c,i−1 − N c,i + 2)) × Sp(N c,i+1 ) × · · · where i = 2, 4, · · · , (n − 1) implying that the number of possible magnetic gauge group is When the Seiberg dual is taken for the middle gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 3.4. The gauge group is given by · · · × Sp(N c,i−1 ) × SO(2 N c,i ≡ 2(N f,i + N c,i+1 + N c,i−1 − N c,i + 2)) × Sp(N c,i+1 ) × · · · where i = 2, 4, · · · , (n − 1) implying that the number of possible magnetic gauge group is n−1 2 . The magnetic superpotential is W dual = (M i q i q i + f i−1 X i−1 q i + Φ i−1 f i−1 f i−1 ) + m i M i . The corresponding brane configuration can be obtained similarly and the extra (i − 2) NS-branes, (i − 2) sets of D6-branes and (i − 2) sets of D4-branes are present at the left hand side of the NS5 L -brane and the extra (n − i − 1) NS-branes, (n − i − 1) sets of D6-branes and (n − i − 1) sets of D4-branes are present at the right hand side of the NS5 ′ R -brane of Figure 9. • Case 5 ′′′ When the Seiberg dual is taken for the symplectic gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 3.6. The gauge group is given by · · · × SO(2N c,i−1 ) × Sp( N c,i ≡ N f,i + N c,i+1 + N c,i−1 − N c,i − 2) × SO(2N c,i+1 ) × · · · where i = 3, 5, · · · , (n − 2) implying that the number of possible magnetic gauge group is n−3 2 . The magnetic superpotential is W dual = (M i q i q i + g i X i+1 q i + Φ i+1 g i g i ) + m i M i . The corresponding brane configuration can be obtained similarly and the extra (i − 2) NS-branes, When the Seiberg dual is taken for the symplectic gauge group factor by assuming that Λ i >> Λ j where j = 1, 2, · · · , i − 1, i + 1, · · · , n, one follows the procedure given in the subsection 3.6. The gauge group is given by where i = 3, 5, · · · , (n − 2) implying that the number of possible magnetic gauge group is n−3 2 . The magnetic superpotential is W dual = (M i q i q i + f i−1 X i−1 q i + Φ

Conclusions and outlook
The meta-stable brane configurations we have found are summarized by Figures 2B, 3B, 4B, 5B(and 7B, 8B, 9B, and 10B). The nonsupersymmetric minimal energy brane configurations in Figure 2B and 3B lead to the Figure 3 of [14] if we ignore the NS5 L -brane, N f D6-branes and N c D4-branes. Similarly the nonsupersymmetric minimal energy brane configurations in Figure 7B and 8B lead to the Figure 3 of [15] if we ignore the NS5 L -brane, 2N f D6-branes and 2N c D4-branes.
The Figures 2B and 7B with a replacement N ′′ f D6-branes by the NS5'-brane by neglecting the NS5 L -brane, N f D6-branes and N c D4-branes become the Figures 2B and 7B of [19] together with a reflection with respect to the NS5 L -brane and a rotation of NS5 R -brane by π 2 angle respectively. The Figures 3B and 8B with a replacement N ′ f D6-branes by the NS5'brane by neglecting the NS5 L -brane, N f D6-branes and N c D4-branes become the Figures  2B and 7B of [19] with a rotation of NS5 R -brane by π 2 angle respectively or the Figures 4B  and 9B of [19] with a reflection with respect to the NS5 R -brane if we ignore the NS5 ′ R -brane, N ′′ f D6-branes and N ′′ c D4-branes from the Figure 3B(8B) respectively. The Figures 4B and  9B with a replacement N ′ f D6-branes by the NS5'-brane by neglecting the NS5 ′ R -brane, N ′′ f D6-branes and N ′′ c D4-branes with vanishing N f D6-branes lead to the Figures 5B and 10B of [19] with a rotation of NS5 ′ L -brane by π 2 angle respectively. Finally, the Figures 5B and 10B with a replacement N f D6-branes by the NS5'-brane, when we neglect the NS5 ′ R -brane, N ′′ f D6-branes and N ′′ c D4-branes and N ′ f D6-branes, lead to the Figures 4B and 10B of [19] respectively.
For the same gauge groups in this paper, one can add an orientifold 6-planes together with three more NS-branes. Totally there exist seven NS-branes. The relevant previous works are given in [56,57,58,50,59]. Depending on the matter contents, there are two possibilities.
On the other hand, for the different triple product gauge groups, one can add an orientifold 6-planes together with two more NS-branes. Totally, there exist six NS-branes. It would be interesting to study these meta-stable brane configurations in type IIA string theory.