aa r X i v : . [ h e p - t h ] S e p More Meta-Stable Brane Configurations without D6-Branes
Changhyun Ahn
Department of Physics, Kyungpook National University, Taegu 702-701, Korea [email protected]
Abstract
We describe the intersecting brane configurations, consisting of NS-branes, D4-branes(andanti-D4-branes), in type IIA string theory corresponding to the meta-stable nonsupersymmet-ric vacua of N = 1 SU ( N c ) × SU ( N ′ c ) × SU ( N ′′ c ) gauge theory with bifundamentals. By addingthe orientifold 4-plane to these brane configurations, we also discuss the meta-stable braneconfigurations for other gauge theory with bifundamentals. Furthermore, we study the inter-secting brane configurations corresponding to the nonsupersymmetric meta-stable vacua ofother gauge theory with bifundamentals, by adding the orientifold 6-plane. Introduction
In the standard type IIA brane configuration, the quark masses correspond to the relative dis-placement of the D6-branes(0123789) and D4-branes(01236) along the 45 directions geomet-rically. Then the eigenvalues of quark mass matrix correspond to the positions of D6-branesin 45 directions. See the review paper [1] for the gauge theory and the brane dynamics. TheSeiberg duality in the classical brane picture can be accomplished by exchanging the locationsof the NS5-brane(012345) and NS5’-brane(012389) along x direction each other.The geometric misalignment of D4-branes connecting both NS5’-brane and D6-branes inthe magnetic brane configuration can be interpreted as a nontrivial F-term condition in thegauge theory with massive flavors. Then the F-term equations can be partially cancelledby both recombination of flavor-D4-branes with the color-D4-branes and then movement ofthose D4-branes into the 45 directions. This phenomenon in magnetic brane configurationcorresponds to the fact that some entries in the magnetic dual quarks acquire nonzero vacuumexpectation values to minimize the F-term in the dual gauge theory side. Moreover, theremaining flavor-D4-branes that do not move to 45 directions, connecting to NS5’-brane, canmove along 89 directions freely since D6-branes and NS5’-brane are parallel and this geometricfreedom of meson field corresponds to the classical pseudomoduli space of nonsupersymmetricvacua of the gauge theory.On the other hand, it is known that the NS-brane configuration in type IIA string theory,where there exist only two types of NS5-brane and NS5’-brane, preserves N = 2 supersym-metry in four dimensions [1]. The geometry [2] of the coincident NS5-branes is characterizedby the metric, the dilaton, and the field strength and is useful to construct the DBI action forD4-branes. In order to break the supersymmetry, one adds D4-branes and anti-D4-branes [3].By adding D4-branes suspending between the NS5-brane and the NS5’-brane, and anti-D4-branes( D D . When one of the NS5’-branes goes to infinity along the x direction, then the corresponding gauge group becomes a global symmetry and the theoryleads to a standard N = 1 SQCD with fundamentals. In other regions, a generalization of [4]showing very similar qualitative phenomena in classical string theory occurs [3].The focus on the new meta-stable brane configurations by adding an orientifold 4-planeand an orientifold 6-plane to the above brane configuration studied by [3], along the line of[5, 6, 7, 8], was given in [9]. When the former was added, no extra NS-branes or D-braneswere needed. However, when the latter was added, the extra NS-branes or D-branes into theabove brane configuration were needed in order to have a product gauge group.In this paper, we continue to find out new meta-stable brane configurations which containfour NS-branes or six NS-branes, along the line of [3, 9], by starting from the known or newsupersymmetric brane configurations in type IIA string theory. Compared with the previousapproaches given by [5, 6, 7, 8], the superpotential in the magnetic theory has very simple formbecause there are no D6-branes in the brane configurations and this fact allows us to analyzethe meta-stable vacua easily using the F-term equations and one loop effective potential. But A replacement of D6-branes with NS5’-brane corresponds to the gauging of the flavor group(global sym-metry) of the gauge theory realized on the D4-branes and this replacement might be useful to construct thephenomenological model building. N = 1 SU ( N c ) × SU ( N ′ c ) × SU ( N ′′ c ) gauge theory with the bifundamentalsand deform this theory by adding the mass term for the bifundamental. We construct the threedifferent dual magnetic theories by taking the Seiberg dual for each gauge group factor. Thenwe consider the nonsupersymmetric meta-stable minimum and present the correspondingintersecting brane configurations of type IIA string theory.In section 3, we discuss the type IIA brane configuration, by adding the oreintifold 4-planeto the brane onfiguration in section 2, corresponding to the electric theory based on the N = 1 Sp ( N c ) × SO (2 N ′ c ) × Sp ( N ′′ c ) gauge theory with matters and deform this theory by adding themass term for the bifundamental. Then we construct the corresponding dual magnetic theoriesby taking the Seiberg dual for each gauge group factor. We consider the nonsupersymmetricmeta-stable minimum and present the corresponding intersecting brane configurations of typeIIA string theory. We also comment on the case of N = 1 SO (2 N c ) × Sp ( N ′ c ) × SO (2 N ′′ c )gauge theory with matters.In section 4, we discuss the type IIA brane configuration corresponding to the electrictheory based on the N = 1 Sp ( N c ) × SU ( N ′ c ) × SU ( N ′′ c ) gauge theory with matters anddeform this theory by adding the mass term for the bifundamental. Then we constructthe two different dual magnetic theories by taking the Seiberg dual for each unitary gaugegroup factor. We consider the nonsupersymmetric meta-stable minimum and present thecorresponding intersecting brane configurations of type IIA string theory. Moreover, we alsodiscuss the meta-stable brane configurations corresponding to the electric theory based on the N = 1 SO ( N c ) × SU ( N ′ c ) × SU ( N ′′ c ) gauge theory by changing the orientifold 6-plane charge.In section 5, we make some comments for the future directions.3 Meta-stable brane configurations with four NS-branes
The type IIA brane configuration [10, 11] corresponding to N = 1 supersymmetric gaugetheory with gauge group SU ( N c ) × SU ( N ′ c ) × SU ( N ′′ c ) (2.1)and with a field F charged under ( N c , N ′ c ), a field G charged under ( N ′ c , N ′′ c ), and theirconjugates e F and e G can be described by the left N S ′ L -brane(012389), the left N S L -brane(012345), the right N S ′ R -brane(012389), the right N S R -brane(012345), N c -, N ′ c - and N ′′ c -color D4-branes(01236). The fields F and e F correspond to 4-4 strings connecting the N c -color D4-branes with N ′ c -color D4-branes while the fields G and e G correspond to 4-4 stringsconnecting the N ′ c -color D4-branes with N ′′ c -color D4-branes.The left N S L -brane is located at x = 0 and let us denote the x coordinates for the N S ′ L -brane, the N S ′ R -brane and the N S R -brane by x = − y , y , y + y respectively.The N c D4-branes are suspended between the
N S ′ L -brane and the N S L -brane, the N ′ c D4-branes are suspended between the
N S L -brane and the N S ′ R -brane, and the N ′′ c D4-branes are suspended between the
N S ′ R -brane and the N S R -brane. We draw this braneconfiguration in Figure 1A for the vanishing mass case .The gauge couplings of SU ( N c ), SU ( N ′ c ) and SU ( N ′′ c ) are given by a string couplingconstant g s , a string scale ℓ s and the x coordinates y i for three NS-branes through g = g s ℓ s y , g = g s ℓ s y , g = g s ℓ s y . For example, as y goes to ∞ , the SU ( N ′′ c ) gauge group becomes a global symmetry andthe theory leads to SQCD with the gauge group SU ( N c ) × SU ( N ′ c ) and N ′′ c flavors in thefundamental representation.There is no superpotential in Figure 1A since the N S L -brane is perpendicular to twoNS5’-branes and the N S ′ R -brane is perpendicular to two NS5-branes. Let us deform thistheory. Displacing the two NS5’-branes relative each other in the v ≡ x + ix direction corresponds to turning on a quadratic mass-deformed superpotential for the fields F and e F as follows: W = mF e F ≡ m Φ ′ (2.2) There are similar brane configurations in the context of quiver gauge theory [12, 13]. N = 1 supersymmetric electric brane configuration for the gauge group SU ( N c ) × SU ( N ′ c ) × SU ( N ′′ c ) and bifundamentals F, e F , G and e G with vanishing(1A) and non-vanishing(1B) mass for the bifundamentals F and e F . The N ′ c D4-branes in 1A are decomposedinto ( N ′ c − N ′′ c ) D4-branes which are moving to + v direction in 1B and N ′′ c D4-branes whichare recombined with those D4-branes connecting between
N S ′ R -brane and N S R -brane in1B.where the first gauge group indices in F and e F are contracted, each second gauge group indexin them is encoded in Φ ′ and the mass m is given by m = ∆ x πα ′ = ∆ xℓ s . (2.3)The gauge-singlet Φ ′ for the first dual gauge group is in the adjoint representation forthe second dual gauge group, i.e., ( , ( N ′ c − N ′′ c ) − , ) ⊕ ( , , ) under the dual gaugegroup (2.4). Then the Φ ′ is a ( N ′ c − N ′′ c ) × ( N ′ c − N ′′ c ) matrix. The N S ′ R -brane together with( N ′ c − N ′′ c )-color D4-branes is moving to the + v direction for fixed other branes during this massdeformation. In other words, the N ′′ c D4-branes among N ′ c D4-branes are not participatingin the mass deformation. Then the x coordinate( ≡ x ) of N S ′ L -brane is equal to zero whilethe x coordinate of N S ′ R -brane is given by ∆ x . Giving an expectation value to the mesonfield Φ ′ corresponds to recombination of N c - and N ′ c - color D4-branes, which will become N c -color D4-branes in Figure 1A such that they are suspended between the N S ′ L -brane andthe N S ′ R -brane and pushing them into the w ≡ x + ix direction. We assume that the number of colors satisfies N ′ c ≥ N c ≥ N ′′ c . Now we draw this brane configuration in Figure 1B for nonvanishing mass for the fields F and e F . 5 .2 Magnetic theory By applying the Seiberg dual to the SU ( N c ) factor in (2.1), the two N S ′ L,R -branes can belocated at the inside of the two NS5-branes, as in Figure 2. Starting from Figure 1A andmoving the
N S ′ R -brane with ( N ′ c − N ′′ c ) D4-branes to the + v direction leading to Figure 1Band interchanging the N S ′ L -brane and the N S L -brane, one obtains the Figure 2A. Beforearriving at the Figure 2A, there exists an intermediate step where the ( N ′ c − N c ) D4-branes areconnecting between the N S L -brane and the N S ′ L -brane, ( N ′ c − N ′′ c ) D4-branes connectingbetween the N S ′ L -brane and N S ′ R -brane, and N ′′ c D4-branes between the
N S ′ L -brane andthe N S R -brane. By introducing − N ′′ c D4-branes and − N ′′ c anti-D4-branes between the N S L -brane and N S ′ L -brane, reconnecting the former with the N ′ c D4-branes connectingbetween
N S L -brane and the N S ′ L -brane (therefore N ′ c − N ′′ c D4-branes) and moving thosecombined ( N ′ c − N ′′ c ) D4-branes to + v -direction, one gets the final Figure 2A where we areleft with ( N c − N ′′ c ) anti-D4-branes between the N S L -brane and N S ′ L -brane.When two NS5’-branes in Figure 2A are close to each other, then it leads to Figure 2B byrealizing that the number of ( N ′ c − N ′′ c ) D4-branes connecting between N S L -brane and N S ′ R -brane can be rewritten as ( N c − N ′′ c ) plus e N c . If we ignore N ′′ c D4-branes and
N S R -branefrom Figure 2B, then the brane configuration becomes the one in [3].Figure 2: The N = 1 magnetic brane configuration for the gauge group SU ( e N c = N ′ c − N c ) × SU ( N ′ c ) × SU ( N ′′ c ) corresponding to Figure 1B with D4- and D N c − N ′′ c = ( N ′ c − N ′′ c ) − e N c .The dual gauge group is given by SU ( e N c = N ′ c − N c ) × SU ( N ′ c ) × SU ( N ′′ c ) . (2.4)6he matter contents are the field f charged under ( e N c , N ′ c , ), a field g charged under( , N ′ c , N ′′ c ), and their conjugates e f and e g under the dual gauge group (2.4) and the gauge-singlet Φ ′ for the first dual gauge group in the adjoint representation for the second dualgauge group, i.e., ( , ( N ′ c − N ′′ c ) − , ) ⊕ ( , , ) under the dual gauge group (2.4).The cubic superpotential with the mass term (2.2) in the dual theory is given by W dual = Φ ′ f e f + m Φ ′ . (2.5)Here the magnetic fields f and e f correspond to 4-4 strings connecting the e N c -color D4-branes(that are connecting between the N S L -brane and the N S ′ R -brane in Figure 2B)with N ′ c -flavor D4-branes(that are a combination of three different D4-branes in Figure 2B).Among these N ′ c -flavor D4-branes, only the strings ending on the upper ( N ′ c − N c ) D4-branesand on the tilted middle ( N c − N ′′ c ) D4-branes in Figure 2B enter the cubic superpotentialterm. Although the ( N ′ c − N ′′ c ) D4-branes in Figure 2A cannot move any directions, the tilted( N c − N ′′ c )-flavor D4-branes can move w direction in Figure 2B. The remaining upper e N c D4-branes are fixed also and cannot move any direction. Note that there is a decomposition( N ′ c − N ′′ c ) = ( N c − N ′′ c ) + e N c . The brane configuration for zero mass for the bifundamental, which has only a cubicsuperpotential, can be obtained from Figure 2A by moving the upper
N S ′ R -brane togetherwith ( N ′ c − N ′′ c ) color D4-branes into the origin v = 0. Then the number of dual colors for D4-branes becomes e N c between N S L -brane and N S ′ L -brane and N ′ c between two NS5’-branesas well as N ′′ c D4-branes between
N S ′ R -brane and N S R -brane. Or starting from Figure 1Aand moving the N S L -brane to the left all the way past the N S ′ L -brane, one also obtainsthe corresponding magnetic brane configuration for massless case.The brane configuration in Figure 2A is stable as long as the distance ∆ x between theupper NS5’-brane and the lower NS5’-brane is large, as in [3]. If they are close to each other,then this brane configuration is unstable to decay and leads to the brane configuration inFigure 2B. One can regard these brane configurations as particular states in the magneticgauge theory with the gauge group (2.4) and superpotential (2.5). The ( N ′ c − N ′′ c − e N c )flavor D4-branes of straight brane configuration of Figure 2B bend due to the fact that thereexists an attractive gravitational interaction between those flavor D4-branes and N S L -branefrom the DBI action, by following the procedure of [3], as long as the distance y goes to ∞ because the presence of an extra N S R -brane does not affect the DBI action. For the finiteand small y , the careful analysis for DBI action is needed in order to obtain the bendingcurve connecting two NS5’-branes. 7hen the upper NS5’-brane(or N S ′ R -brane) is replaced by coincident ( N ′ c − N ′′ c ) D6-branes and the N S R is rotated by an angle π in the ( v, w ) plane in Figure 2B, this braneconfiguration reduces to the one found in [14] where the gauge group was given by SU ( n f + n ′ c − n c ) × SU ( n ′ c ) with n f multiplets, n ′ f multiplets, flavor singlets and gauge singlets. Thenthe present number ( N ′ c − N ′′ c ) corresponds to the n f , the number N c corresponds to n c andthe number N ′′ c corresponds to the n ′ c of [14]. Note that the number of D4-branes touching N S ′ R -brane in Figure 2B is equal to ( N ′ c − N ′′ c ).The quantum corrections can be understood for small ∆ x by using the low energy fieldtheory on the branes. The low energy dynamics of the magnetic brane configuration can bedescribed by the N = 1 supersymmetric gauge theory with gauge group (2.4) and the gaugecouplings for the three gauge group factors are given by g ,mag = g s ℓ s y , g ,mag = g s ℓ s ( y − y ) , g ,mag = g s ℓ s y . The dual gauge theory has an adjoint Φ ′ of SU ( N ′ c ) and bifundamentals f, e f , g and e g underthe dual gauge group (2.4) and the superpotential corresponding to Figures 2A and 2B isgiven by W dual = h Φ ′ f e f − hµ Φ ′ , h = g ,mag , µ = − ∆ x πg s ℓ s . Then f e f is a e N c × e N c matrix where the second gauge group indices for f and e f are contractedwith those of Φ ′ while µ is a ( N ′ c − N ′′ c ) × ( N ′ c − N ′′ c ) matrix. Although the field f itselfis an antifundamental in the second gauge group which is a different representation for theusual standard quark coming from D6-branes, the product f e f has the same representationfor the product of quarks and moreover, the second gauge group indices for the field Φ ′ playthe role of the flavor indices, as in comparison with the brane configuration in the presenceof D6-branes before.Therefore, the F-term equation, the derivative W dual with respect to the meson field Φ ′ cannot be satisfied if the ( N ′ c − N ′′ c ) exceeds e N c . So the supersymmetry is broken. That is,there exist three equations from F-term conditions: f e f − µ = 0 , and Φ ′ f = 0 = e f Φ ′ . Then the solutions for these are given by < f > = (cid:18) µ e N c (cid:19) , < e f > = (cid:0) µ e N c (cid:1) , < Φ ′ > = (cid:18) ′ ( N ′ c − N ′′ c − e N c ) (cid:19) (2.6)8here the zero of < f > is a ( N ′ c − N ′′ c − e N c ) × e N c matrix, the zero of < e f > is a e N c × ( N ′ c − N ′′ c − e N c ) matrix and the zeros of < Φ ′ > are e N c × e N c , e N c × ( N ′ c − N ′′ c − e N c ), and( N ′ c − N ′′ c − e N c ) × e N c matrices. Then one can expand these fields around on a point (2.6), asin [4, 15, 16, 17, 18] and one arrives at the relevant superpotential up to quadratic order inthe fluctuation. At one loop, the effective potential V (1) eff for Φ ′ leads to the positive value for m ′ implying that these vacua are stable. Let us consider other magnetic theory for the same undeformed electric theory given in thesubsection 2.1. Here we consider the different mass deformation. By applying the Seibergdual to the SU ( N ′ c ) factor in (2.1), the two N S ′ L,R -branes can be located at the left handside of the two NS5-branes, as in Figure 4.Figure 3: The N = 1 supersymmetric electric brane configuration for the gauge group SU ( N c ) × SU ( N ′ c ) × SU ( N ′′ c ) and bifundamentals F, e F , G and e G with vanishing(3A) whichis identical to Figure 1A and nonvanishing(3B) mass for the bifundamentals F and e F . Thisdeformation is different from the one (2.2) given previously. In 3B, the N S ′ L -brane togetherwith N c D4-branes is moving to + v direction.By starting from Figure 3A which is the same as Figure 1A and moving the N S ′ L -brane with N c D4-branes to the + v direction leading to Figure 3B and interchanging the N S L -brane and the N S ′ R -brane, one obtains the Figure 4A. Before arriving at the Figure4A, there exists an intermediate step where the N c D4-branes are connecting between the
N S ′ L -brane and the N S ′ R -brane, ( N ′′ c − N ′ c + N c ) D4-branes are connecting between the N S ′ R -brane and N S L -brane, and N ′′ c D4-branes are suspended between the
N S L -braneand the N S R -brane. By moving the combined N c D4-branes, obtained from reconnectionof those D4-branes between
N S ′ L -brane and the N S ′ R -brane and those D4-branes between9he N S ′ R -brane and N S L -brane(therefore between the N S ′ L -brane and the N S L -brane),to + v -direction, one gets the final Figure 4A where we are left with ( N ′ c − N ′′ c ) anti-D4-branesbetween the N S ′ R -brane and N S L -brane. We assume that the number of colors satisfies N c + N ′′ c ≥ N ′ c ≥ N ′′ c . When two NS5’-branes in Figure 4A are close to each other, then it leads to Figure 4B byrealizing that the number of N c D4-branes connecting between
N S ′ L -brane and N S L -branecan be rewritten as ( N ′ c − N ′′ c ) plus e N ′ c .Figure 4: The N = 1 magnetic brane configuration for the gauge group SU ( N c ) × SU ( e N ′ c = N c + N ′′ c − N ′ c ) × SU ( N ′′ c ) corresponding to Figure 3B with D4- and D N ′ c − N ′′ c = N c − e N ′ c in 4B.The dual gauge group is SU ( N c ) × SU ( e N ′ c = N c + N ′′ c − N ′ c ) × SU ( N ′′ c ) . (2.7)The matter contents are the field f charged under ( N c , e N ′ c , ), a field g charged under( , e N ′ c , N ′′ c ), and their conjugates e f and e g under the dual gauge group (2.7) and the gauge-singlet Φ for the second dual gauge group in the adjoint representation for the first dualgauge group, i.e., ( N − , , ) ⊕ ( , , ) under the dual gauge group (2.7). Then the Φ isa N c × N c matrix. All the N c D4-branes participate in the mass deformation.The cubic superpotential with the mass term in the dual theory is given by W dual = Φ f e f + m Φ (2.8) One can also construct the mass deformation by rotating
N S R -brane and moving it to + v direction.Then the brane configuration will look like as the Figure 15. ≡ F e F , the second gauge group indices in F and e F are contracted,each first gauge group index in them is encoded in Φ and the mass m is given by (2.3) where∆ x refers to the distance between two NS5’-branes along the x direction in Figure 4A. Letus emphasize that although the Φ which has first gauge group indices looks similar to theprevious Φ ′ which has second gauge group indices in (2.2), the group indices are different. Herethe magnetic fields f and e f correspond to 4-4 strings connecting the e N ′ c -color D4-branes(thatare connecting between the N S ′ L -brane and the N S L -brane in Figure 4B) with N c -flavorD4-branes(which are realized as corresponding D4-branes in Figure 4A). Although the N c D4-branes in Figure 4A cannot move any directions, the tilted ( N ′ c − N ′′ c )-flavor D4-branescan move w direction in Figure 4B. The remaining upper e N ′ c D4-branes are fixed also andcannot move any direction. Note that there is a decomposition N c = ( N ′ c − N ′′ c ) + e N ′ c . The brane configuration for zero mass for the bifundamental, which has only a cubicsuperpotential, can be obtained from Figure 4A by moving the upper NS5’-brane(or
N S ′ L -brane) together with N c color D4-branes into the origin v = 0. Then the number of dualcolors for D4-branes becomes N c between two NS5’-branes, e N ′ c between N S ′ R -brane and N S L -brane and N ′′ c between N S L -brane and N S R -brane. Or starting from Figure 3A andmoving the N S L -brane to the right all the way past the N S ′ R -brane, one also obtains thecorresponding magnetic brane configuration for massless case.The brane configuration in Figure 4A is stable as long as the distance ∆ x between theupper NS5’-brane and the lower NS5’-brane is large. If they are close to each other, thenthis brane configuration is unstable to decay to the brane configuration in Figure 4B. Onecan regard these brane configurations as particular states in the magnetic gauge theory withthe gauge group (2.7) and superpotential (2.8). The ( N c − e N ′ c ) flavor D4-branes of straightbrane configuration of Figure 4B bend since there exists an attractive gravitational interactionbetween those flavor D4-branes and N S L -brane from the DBI action, as long as the distance y is large because the presence of an extra N S R -brane does not affect the DBI action. Forthe finite and small y , the careful analysis for DBI action is needed in order to obtain thebending curve connecting two NS5’-branes. Or if y goes to zero, then this extra N S R -braneplays the role of enhancing the strength for the NS5-branes and will affect both the energyof bending curve, E curved , which is proportional to l with l ≡ √ kℓ s where k is the number ofNS5-branes and ∆ x which depends on both l and l [3].When the upper NS5’-brane(or N S ′ L -brane) is replaced by coincident N c (that is equal tothe number of D4-branes touching the N S ′ L -brane) D6-branes, this brane configuration looks11imilar to the one found in [14] where the gauge group was given by SU ( n ′ f + n c − n ′ c ) × SU ( n c )with n f multiplets, n ′ f multiplets, flavor singlets and gauge singlets. Then the present N c corresponds to the n ′ f , the number N ′ c corresponds to n ′ c , and N ′′ c corresponds to the n c of[14].The low energy dynamics of the magnetic brane configuration can be described by the N = 1 supersymmetric gauge theory with gauge group (2.7) and the gauge couplings for thethree gauge group factors are given by g ,mag = g s ℓ s ( y − y ) , g ,mag = g s ℓ s y , g ,mag = g s ℓ s ( y + y ) . The dual gauge theory has an adjoint Φ of SU ( N c ) and bifundamentals f, e f , g and e g under the dual gauge group (2.7) and the superpotential corresponding to Figures 4A and 4Bis given by W dual = h Φ f e f − hµ Φ , h = g ,mag , µ = − ∆ x πg s ℓ s . Then f e f is a e N ′ c × e N ′ c matrix where the first gauge group indices for f and e f are contractedwith those of Φ while µ is a N c × N c matrix. The product f e f has the same representationfor the product of quarks and moreover, the first gauge group indices for the field Φ play therole of the flavor indices when there are D6-branes before.Therefore, the F-term equation, the derivative W dual with respect to the meson field Φcannot be satisfied if the N c exceeds e N ′ c . So the supersymmetry is broken. That is, there existthree equations from F-term conditions: f e f − µ = 0 and Φ f = 0 = e f Φ. Then the solutionsfor these are given by < f > = (cid:18) µ e N ′ c (cid:19) , < e f > = (cid:0) µ e N ′ c (cid:1) , < Φ > = (cid:18) ( N c − e N ′ c ) (cid:19) (2.9)where the zero of < f > is a ( N c − e N ′ c ) × e N ′ c matrix, the zero of < e f > is a e N ′ c × ( N c − e N ′ c )matrix and the zeros of < Φ > are e N ′ c × e N ′ c , e N ′ c × ( N c − e N ′ c ) and ( N c − e N ′ c ) × e N ′ c matrices.Then one can expand these fields around on a point (2.9), as in [4, 15] and one arrives at therelevant superpotential up to quadratic order in the fluctuation. At one loop, the effectivepotential V (1) eff for Φ leads to the positive value for m implying that these vacua are stable. One can think of the following dual gauge group SU ( N c ) × SU ( N ′ c ) × SU ( e N ′′ c = N ′ c − N ′′ c ) (2.10)12y performing the magnetic dual for the last gauge group in (2.1). The electric brane configu-ration can be given in terms of Figure 1A or Figure 1A with an exchange between NS5-braneand NS5’-brane. Then for the latter, the resulting brane configuration is given by N S L -brane, N S ′ L -brane, N S R -brane, and N S ′ R -brane from the left to the right in the x direction.In order to obtain the above dual gauge group, we need to interchange between the N S R -brane and the N S ′ R -brane, as we did before. One can do this either by following the previousprocedure or by looking at the Figure 2 from the negative w direction which is an oppositeviewpoint, compared with Figure 2. In other words, we are looking at the Figure 2 from theother side of w . Then the resulting brane configuration in this case can be obtained by takinga reflection for all the NS-branes, D4-branes and anti D4-branes with respect to the N S L -brane(rotating them to the left for fixed N S L -brane) in Figure 2A and Figure 2B. Thenthe N = 1 magnetic brane configuration for the gauge group SU ( N c ) × SU ( N ′ c ) × SU ( e N ′′ c = N ′ c − N ′′ c ) corresponds to the Figure 5A’ with D4- and D N ′′ c − N c = ( N ′ c − N c ) − e N ′′ c . We do not presentthe Figures 5A’ and 5B’ here.Let us consider other magnetic theory for the same electric theory given in the subsection2.1 with Figure 1A. By applying the Seiberg dual to the SU ( N ′′ c ) factor in (2.1) and inter-changing the N S ′ R -brane and the N S R -brane, one obtains the Figure 5A”. Before arrivingat the Figure 5A”, there exists an intermediate step where N c D4-branes between
N S ′ L -brane and the N S R -brane, the N ′ c D4-branes are connecting between the
N S L -brane andthe N S R -brane, and ( N ′ c − N ′′ c ) D4-branes are connecting between the N S R -brane and N S ′ R -brane. By rotating N S L -brane by an angle π which will become N S ′ M -brane, mov-ing it with the ( N ′ c − N c ) D4-branes to + v direction where we introduce ( N ′ c − N c ) D4-branesand ( N ′ c − N c ) anti D4-branes between the N S R -brane and the N S ′ R -brane, one gets thefinal Figure 5A” where we are left with ( N ′′ c − N c ) anti-D4-branes between the NS5-braneand the N S ′ R -brane. When two NS5’-branes in Figure 5A” are close to each other, then itleads to Figure 5B” by realizing that the number of ( N ′ c − N c ) D4-branes connecting between N S ′ M -brane and NS5-brane can be rewritten as ( N ′′ c − N c ) plus e N ′′ c .The brane configuration in Figure 5A” is stable as long as the distance ∆ x between theupper NS5’-brane and the lower NS5’-brane(or N S ′ R -brane) is large. If they are close to eachother, then this brane configuration is unstable to decay to the brane configuration in Figure5B”. One can regard these brane configurations as particular states in the magnetic gaugetheory with the gauge group and superpotential. The ( N ′ c − N c − e N ′′ c ) flavor D4-branes ofstraight brane configuration of Figure 5B” bend since there exists an attractive gravitational13igure 5: The N = 1 magnetic brane configuration for the gauge group SU ( N c ) × SU ( N ′ c ) × SU ( e N ′′ c = N ′ c − N ′′ c ) with D4- and D N ′′ c − N c = ( N ′ c − N c ) − e N ′′ c . The deformation is related to thebifundamentals G and e G .interaction between those flavor D4-branes and NS5-brane from the DBI action. As mentionedin [9], the two NS5’-branes are located at different side of NS5-brane in Figure 5B” and theDBI action computation for this bending curve should be taken into account.The matter contents are the field f charged under ( N c , N ′ c , ), a field g charged under( , N ′ c , e N ′′ c ) and their conjugates e f and e g under the dual gauge group (2.10) and the gauge-singlet Φ ′ which is in the adjoint representation for the second dual gauge group, in otherwords, ( , ( N ′ c − N c ) − , ) ⊕ ( , , ) under the dual gauge group (2.10). Then the Φ ′ isa ( N ′ c − N c ) × ( N ′ c − N c ) matrix. Only ( N ′ c − N c ) D4-branes can participate in the massdeformation.The cubic superpotential with the mass term is given by W dual = Φ ′ g e g + m Φ ′ (2.11)where we define Φ ′ as Φ ′ ≡ G e G and the third gauge group indices in G and e G are contracted,each second gauge group index in them is encoded in Φ ′ . Here the magnetic fields g and e g correspond to 4-4 strings connecting the e N ′′ c -color D4-branes(that are connecting between the N S ′ M -brane and the NS5-brane in Figure 5B”) with N ′ c -flavor D4-branes. Among these N ′ c -flavor D4-branes, only the strings ending on the upper ( N ′ c − N ′′ c ) D4-branes and on the tiltedmiddle ( N ′′ c − N c ) D4-branes in Figure 5B” enter the cubic superpotential term. Althoughthe ( N ′ c − N c ) D4-branes in Figure 5A” cannot move any directions, the tilted ( N ′′ c − N c )-flavor D4-branes can move w direction. The remaining upper e N ′′ c D4-branes are fixed also14nd cannot move any direction. Note that there is a decomposition( N ′ c − N c ) = ( N ′′ c − N c ) + e N ′′ c . The brane configuration for zero mass for the bifundamental, which has only a cubicsuperpotential, can be obtained from Figure 5A” by moving the upper NS5’-brane togetherwith ( N ′ c − N c ) color D4-branes into the origin v = 0. Then the number of dual colorsfor D4-branes becomes N c between the N S ′ L -brane and the N S ′ M -brane, N ′ c between the N S ′ M -brane and the NS5-brane and e N ′′ c between NS5-brane and N S ′ R -brane. Or startingfrom Figure 1A and moving the N S ′ R -brane to the right all the way past the N S R -brane,one also obtains the corresponding magnetic brane configuration for massless case.The low energy dynamics of the magnetic brane configuration can be described by the N = 1 supersymmetric gauge theory with gauge group (2.10) and the gauge couplings for thethree gauge group factors are given by g ,mag = g s ℓ s y , g ,mag = g s ℓ s ( y − y ) , g ,mag = g s ℓ s y . The dual gauge theory has an adjoint Φ ′ of SU ( N ′ c ) and bifundamentals f, e f , g and e g underthe dual gauge group (2.10) and the superpotential corresponding to Figures 5A” and 5B” isgiven by W dual = h Φ ′ g e g − hµ Φ ′ , h = g ,mag , µ = − ∆ x πg s ℓ s . Then g e g is a e N ′′ c × e N ′′ c matrix where the second gauge group indices for g and e g are contractedwith those of Φ ′ while µ is a ( N ′ c − N c ) × ( N ′ c − N c ) matrix. The product g e g has the samerepresentation for the product of quarks and moreover, the second gauge group indices forthe field Φ ′ play the role of the flavor indices.Therefore, the F-term equation, the derivative W dual with respect to the meson field Φ ′ cannot be satisfied if the ( N ′ c − N c ) exceeds e N ′′ c . So the supersymmetry is broken. That is,there exist three equations from F-term conditions: g e g − µ = 0 and Φ ′ g = 0 = e g Φ ′ . Thenthe solutions for these are given by < g > = (cid:18) µ e N ′′ c (cid:19) , < e g > = (cid:0) µ e N ′′ c (cid:1) , < Φ ′ > = (cid:18) ′ ( N ′ c − N c ) − e N ′′ c (cid:19) where the zero of < g > is a ( N ′ c − N c − e N ′′ c ) × e N ′′ c matrix, the zero of < e g > is a e N ′′ c × ( N ′ c − N c − e N ′′ c ) matrix and the zeros of < Φ ′ > are e N ′′ c × e N ′′ c , e N ′′ c × ( N ′ c − N c − e N ′′ c ) and( N ′ c − N c − e N ′′ c ) × e N ′′ c matrices. Then one can expand these fields around on a point, as in154, 15] and one arrives at the relevant superpotential up to quadratic order in the fluctuation.At one loop, the effective potential V (1) eff for Φ ′ leads to the positive value for m ′ implyingthat these vacua are stable. In this section, we add an orientifold 4-plane to the previous brane configurations and findout new meta-stable brane configurations. Or one can realize these brane configurations byinserting the extra NS-brane and O4-planes into the brane configuration [16].
The type IIA brane configuration [19] corresponding to N = 1 supersymmetric gauge theorywith gauge group Sp ( N c ) × SO (2 N ′ c ) × Sp ( N ′′ c ) (3.1)and with a field F charged under ( c , ′ c ), a field G charged under ( ′ c , ′′ c ) can bedescribed by the left N S ′ L -brane, the left N S L -brane, the right N S ′ R -brane, the right N S R -brane, 2 N c -, 2 N ′ c - and 2 N ′′ c -color D4-branes as well as an O ± -plane(01236) we shouldadd. The O ± -planes act as ( x , x , x , x , x ) → ( − x , − x , − x , − x , − x ) as usual and theyhave RR charge ± ± v direction corresponds to turning on a quadraticmass-deformed superpotential for the field F as follows: W = mF F ≡ m Φ ′ (3.2)where the first gauge group indices in F are contracted, each second gauge group indexin F is encoded in Φ ′ and the mass m is given by (2.3). The gauge-singlet Φ ′ for thefirst dual gauge group is in the adjoint representation for the second dual gauge group,i.e., ( , ( N ′ c − N ′′ c )( ′ c − ′′ c − ) , ) under the dual gauge group (3.3). Then the Φ ′ isa 2( N ′ c − N ′′ c ) × N ′ c − N ′′ c ) matrix. The half N S ′ R -brane [20] together with ( N ′ c − N ′′ c )-colorD4-branes is moving to the + v direction(and their mirrors to − v direction) for fixed otherbranes during this mass deformation. The 2 N ′′ c D4-branes among 2 N ′ c D4-branes are not16igure 6: The N = 1 supersymmetric electric brane configuration for the gauge group Sp ( N c ) × SO (2 N ′ c ) × Sp ( N ′′ c ) and bifundamentals F and G with vanishing(6A) and nonva-nishing(6B) mass for the bifundamental F . The 2 N ′ c D4-branes in 6A are decomposed into2( N ′ c − N ′′ c ) D4-branes which are moving to ± v direction in Z symmetric way in 6B and 2 N ′′ c D4-branes which are recombined with those D4-branes connecting between
N S ′ R -brane and N S R -brane in 6B.participating in the mass deformation. Then the x coordinate of N S ′ L -brane is equal tozero while the x coordinates of half N S ′ R -brane are given by ± ∆ x .Giving an expectation value to the meson field Φ ′ corresponds to recombination of 2 N c -and 2 N ′ c - color D4-branes, which will become 2 N c -color D4-branes in Figure 6A such thatthey are suspended between the N S ′ L -brane and the N S ′ R -brane and pushing them into the w direction. We assume that the number of colors satisfies N ′ c ≥ N c + 2 ≥ N ′′ c . Now we draw this brane configuration in Figure 6B for nonvanishing mass for the field F . By applying the Seiberg dual to the Sp ( N c ) factor in (3.1), the two N S ′ L,R -branes can belocated at the inside of the two NS5-branes, as in Figure 7. Starting from Figure 6B andinterchanging the
N S ′ L -brane and the N S L -brane, one obtains the Figure 7A.Before arriving at the Figure 7A, there exists an intermediate step where the 2( N ′ c − N c − N S L -brane and the N S ′ L -brane, ( N ′ c − N ′′ c ) D4-branesconnecting between the N S ′ L -brane and N S ′ R -brane(and their mirrors), and 2 N ′′ c D4-branesbetween the
N S ′ L -brane and the N S R -brane. By introducing − N ′′ c D4-branes and − N ′′ c anti-D4-branes between the N S L -brane and N S ′ L -brane, reconnecting the former with the N ′ c D4-branes connecting between
N S L -brane and the N S ′ L -brane (therefore ( N ′ c − N ′′ c ) D4-17igure 7: The N = 1 magnetic brane configuration for the gauge group Sp ( e N c = N ′ c − N c − × SO (2 N ′ c ) × Sp ( N ′′ c ) corresponding to Figure 6B with D4- and D N c − N ′′ c + 2 = ( N ′ c − N ′′ c ) − e N c .branes) and moving those combined ( N ′ c − N ′′ c ) D4-branes to + v -direction(and their mirrorsto − v direction), one gets the final Figure 7A where we are left with 2( N c − N ′′ c + 2) anti-D4-branes between the N S L -brane and N S ′ L -brane. When two NS5’-branes in Figure 7A areclose to each other, then it leads to Figure 7B by realizing that the number of ( N ′ c − N ′′ c ) D4-branes connecting between N S L -brane and N S ′ R -brane can be rewritten as ( N c − N ′′ c + 2)plus e N c .The dual gauge group is Sp ( e N c = N ′ c − N c − × SO (2 N ′ c ) × Sp ( N ′′ c ) . (3.3)The matter contents are the field f charged under ( e N c , ′ c , ), a field g charged under( , ′ c , ′′ c ) under the dual gauge group (3.3) and the gauge-singlet Φ ′ that is in the adjointrepresentation for the second dual gauge group, i.e., ( , ( N ′ c − N ′′ c )( ′ c − ′′ c − ) , ) underthe dual gauge group. That is, the Φ ′ is an 2( N ′ c − N ′′ c ) × N ′ c − N ′′ c ) antisymmetric matrix.The cubic superpotential with the mass term (3.2) in the dual theory is given by W dual = Φ ′ f f + m Φ ′ . (3.4)Here the magnetic field f corresponds to 4-4 strings connecting the 2 e N c -color D4-branes(thatare connecting between the N S L -brane and the N S ′ R -brane including the mirrors) with2 N ′ c -flavor D4-branes(that is a combination of three different D4-branes including the mirrorsin Figure 7B). Among these 2 N ′ c -flavor D4-branes, only the strings ending on the upper2( N ′ c − N c −
2) D4-branes and on the tilted middle 2( N c − N ′′ c + 2) D4-branes including themirrors in Figure 7B enter the cubic superpotential term. Although the ( N ′ c − N ′′ c ) D4-branes18n Figure 7A cannot move any directions, the tilted 2( N c − N ′′ c + 2)-flavor D4-branes includingthe mirrors can move w direction. The remaining upper e N c D4-branes(and its mirrors) arefixed also and cannot move any direction. Note that there is a decomposition( N ′ c − N ′′ c ) = ( N c − N ′′ c + 2) + e N c . The brane configuration for zero mass for the bifundamental, which has only a cubicsuperpotential, can be obtained from Figure 7A by moving the upper and lower NS5’-branestogether with ( N ′ c − N ′′ c ) color D4-branes into the origin v = 0. Then the number of dualcolors for D4-branes becomes 2 e N c between N S L -brane and N S ′ L -brane and 2 N ′ c betweentwo NS5’-branes as well as 2 N ′′ c between N S ′ R -brane and N S R -brane. Or starting fromFigure 6A and moving the N S L -brane to the left all the way past the N S ′ L -brane, one alsoobtains the corresponding magnetic brane configuration for massless case.The brane configuration in Figure 7A is stable as long as the distance ∆ x between theupper NS5’-brane and the middle NS5’-brane is large. If they are close to each other, thenthis brane configuration is unstable to decay and leads to the brane configuration in Figure7B. One can regard these brane configurations as particular states in the magnetic gaugetheory with the gauge group (3.3) and superpotential (3.4). The upper ( N ′ c − N ′′ c − e N c ) flavorD4-branes of straight brane configuration of Figure 7B bend due to the fact that there existsan attractive gravitational interaction between those flavor D4-branes and N S L -brane fromthe DBI action, as long as the distance y goes to ∞ because the presence of an extra N S R -brane does not affect the DBI action. For the finite and small y , the careful analysis for DBIaction is needed in order to obtain the bending curve connecting two NS5’-branes. Of course,their mirrors, the lower ( N ′ c − N ′′ c − e N ′ c ) flavor D4-branes of straight brane configuration ofFigure 7B can bend and their trajectories connecting two NS5’-branes should be preservedunder the O4-plane, i.e., Z symmetric way.When the upper and lower half N S ′ R -branes are replaced by coincident ( N ′ c − N ′′ c ) D6-branes and the N S R is rotated by an angle π in the ( v, w ) plane in Figure 7B, this braneconfiguration reduces to the one found in [16] where the gauge group was given by Sp ( n f + n ′ c − n c − × SO (2 n ′ c ) with 2 n f multiplets, flavor singlet and gauge singlets. Then the present( N ′ c − N ′′ c ) corresponds to the n f , the number N c corresponds to n c and N ′′ c corresponds tothe n ′ c of [16]. However, the gauge group Sp ( N ′′ c ) corresponds to the different gauge group SO (2 n ′ c ). When we discuss the subsection 3.5 and take the Seiberg dual for the middle gaugegroup, then it becomes SO (2 N c ) × Sp ( e N ′ c = N c + N ′′ c − N ′ c − × SO (2 N ′′ c ). Then the N c corresponds to the n f , the number N ′ c corresponds to n c and N ′′ c corresponds to the n ′ c of [16].If we ignore 2 N ′′ c D4-branes and
N S R -brane from Figure 7B, then the brane configuration19ecomes the one in [21, 6].The dual gauge theory has an adjoint Φ ′ of SO (2 N ′ c ) and bifundamentals f and g underthe dual gauge group (3.3) and the superpotential corresponding to Figures 7A and 7B isgiven by W dual = h Φ ′ f f − hµ Φ ′ , h = g ,mag , µ = − ∆ x πg s ℓ s . Then f f is a 2 e N c × e N c matrix where the second gauge group indices for f are contractedwith those of Φ ′ while µ is a 2( N ′ c − N ′′ c ) × N ′ c − N ′′ c ) matrix. The product f f has the samerepresentation for the product of quarks and moreover, the first gauge group indices for thefield Φ ′ play the role of the flavor indices, as we observed above for the comparison with thebrane configuration in the presence of D6-branes.Therefore, the F-term equation, the derivative W dual with respect to the meson field Φ ′ cannot be satisfied if the 2( N ′ c − N ′′ c ) exceeds 2 e N c . So the supersymmetry is broken. Thatis, there exist two equations from F-term conditions: f f − µ = 0 and Φ ′ f = 0. Then thesolutions for these are given by < f > = (cid:18) µ e N c (cid:19) , < Φ ′ > = (cid:18) ′ ( N ′ c − N ′′ c − e N c ) ⊗ iσ (cid:19) (3.5)where the zero of < f > is a 2( N ′ c − N ′′ c − e N c ) × e N c matrix and the zeros of < Φ ′ > are2 e N c × e N c , 2 e N c × N ′ c − N ′′ c − e N c ), and 2( N ′ c − N ′′ c − e N c ) × e N c matrices. Then one can expandthese fields around on a point (3.5), as in [4] and one arrives at the relevant superpotentialup to quadratic order in the fluctuation. At one loop, the effective potential V (1) eff for Φ ′ leadsto the positive value for m ′ implying that these vacua are stable. Let us consider other magnetic theory for the same electric theory given in the subsection 3.1.By applying the Seiberg dual to the SO (2 N ′ c ) factor in (3.1), the two N S ′ L,R -branes can belocated at the left hand side of the two NS5-branes, as in Figure 9.The Figure 8A is the same as the one in Figure 6A and one moves half
N S ′ L -branetogether with N c D4-branes to + v direction(and its mirrors to − v direction) and is given byFigure 8B. Starting from Figure 8B and interchanging the N S L -brane and the N S ′ R -brane,one obtains the Figure 9A. Before arriving at the Figure 9A, there exists an intermediate stepwhere the N c D4-branes are connecting between the
N S ′ L -brane and the N S ′ R -brane(andtheir mirrors), 2( N ′′ c − N ′ c + N c + 2) D4-branes are connecting between the N S ′ R -brane and20igure 8: The N = 1 supersymmetric electric brane configuration for the gauge group Sp ( N c ) × SO (2 N ′ c ) × Sp ( N ′′ c ) and bifundamentals F and G with vanishing(8A) which is thesame as Figure 6A and nonvanishing(8B) mass for the bifundamental F . This deformation isdifferent from the previous case (3.2). In 8B, the 2 N c D4-branes are moving to ± v directionsin Z symmetric way. N S L -brane, and 2 N ′′ c D4-branes are suspended between the
N S L -brane and the N S R -brane. By moving the combined N c D4-branes, obtained from the reconnection of thoseD4-branes between the
N S ′ L -brane and the N S ′ R -brane and those D4-branes between the N S ′ R -brane and the N S L -brane(therefore between the N S ′ L -brane and the N S L -brane),to + v -direction(and their mirrors to − v direction), one gets the final Figure 9A where weare left with 2( N ′ c − N ′′ c −
2) anti-D4-branes between the
N S ′ R -brane and N S L -brane. Weassume that the number of colors satisfies N c + N ′′ c ≥ N ′ c − ≥ N ′′ c . When two NS5’-branes in Figure 9A are close to each other, then it leads to Figure 9B byrealizing that the number of N c D4-branes connecting between
N S ′ L -brane and N S L -branein Figure 9A can be rewritten as ( N ′ c − N ′′ c −
2) plus e N ′ c . If we ignore 2 N ′′ c D4-branes and
N S R -brane and change the O4-plane charge(corresponding to change the symplectic gauge groupinto the orthogonal gauge group and vice versa) from Figure 9B, then the brane configurationbecomes the one in [9].The dual gauge group is Sp ( N c ) × SO (2 e N ′ c = 2 N c + 2 N ′′ c − N ′ c + 4) × Sp ( N ′′ c ) . (3.6)The matter contents are the field f charged under ( c , e N ′ c , ), a field g charged un-der ( , e N ′ c , ′′ c ) under the dual gauge group (3.6) and the gauge-singlet Φ for the sec-ond dual gauge group in the adjoint representation for the first dual gauge group, i.e.,21igure 9: The N = 1 magnetic brane configuration for the gauge group Sp ( N c ) × SO (2 e N ′ c =2 N c + 2 N ′′ c − N ′ c + 4) × Sp ( N ′′ c ) corresponding to Figure 8B with D4- and D N ′ c − N ′′ c − N c − e N ′ c in 9B.( N c ( c + ) , , ) under the dual gauge group. Then the Φ is a 2 N c × N c matrix. Allthe 2 N c D4-branes are participating in the mass deformation.The cubic superpotential with the mass term in the dual theory is given by W dual = Φ f f + m Φ (3.7)where we define Φ as Φ ≡ F F and the second gauge group indices in F are contracted,each first gauge group index in them is encoded in Φ. Although the Φ that has first gaugegroup indices looks similar to the previous Φ ′ that has second gauge group indices, the groupindices are different. Here the magnetic field f corresponds to 4-4 strings connecting the 2 e N ′ c -color D4-branes(that are connecting between the N S ′ L -brane and the N S L -brane in Figure9B) with 2 N c -flavor D4-branes including the mirrors(which are realized as corresponding D4-branes in Figure 9A). Although the N c D4-branes(and its mirrors) in Figure 9A cannot moveany directions, the tilted ( N ′ c − N ′′ c − w directionin Figure 9B. The remaining upper e N ′ c D4-branes(and its mirrors) are fixed also and cannotmove any direction. Note that there is a decomposition N c = ( N ′ c − N ′′ c −
2) + e N ′ c . The brane configuration for zero mass for the bifundamental, which has only a cubicsuperpotential, can be obtained from Figure 9A by moving the upper and lower NS5’-branestogether with N c color D4-branes into the origin v = 0. Then the number of dual colors for One can also construct the mass deformation by rotating
N S R -brane and moving it to ± v direction, asin previous case in subsection 2.3. The brane configuration can be obtained easily. N c between two NS5’-branes, 2 e N ′ c between N S ′ R -brane and N S L -braneand 2 N ′′ c between N S L -brane and N S R -brane. Or starting from Figure 8A and moving the N S L -brane to the right all the way past the N S ′ R -brane, one also obtains the correspondingmagnetic brane configuration for massless case.The brane configuration in Figure 9A is stable as long as the distance ∆ x between theupper NS5’-brane and the middle NS5’-brane is large. If they are close to each other, thenthis brane configuration is unstable to decay and leads to the brane configuration in Figure9B. One can regard these brane configurations as particular states in the magnetic gaugetheory with the gauge group (3.6) and superpotential (3.7). The ( N c − e N ′ c ) flavor D4-branesof straight brane configuration of Figure 9B bend due to the fact that there exists an attrac-tive gravitational interaction between those flavor D4-branes and N S L -brane from the DBIaction, as long as the distance y goes to ∞ because the presence of an extra N S R -branedoes not affect the DBI action. For the finite and small y , the careful analysis for DBI actionis needed in order to obtain the bending curve connecting two NS5’-branes. Or if y goes tozero, then this extra N S R -brane plays the role of enhancing the strength for the NS5-branesand will affect both the energy of bending curve, E curved , and ∆ x [3]. Of course, their mirrors,the lower ( N c − e N ′ c ) flavor D4-branes of straight brane configuration of Figure 9B can bendand their trajectories connecting two NS5’-branes should be preserved under the O4-plane,i.e., Z symmetric way.When the NS5’-brane(or N S ′ L -brane) is replaced by coincident N c D6-branes, this braneconfiguration looks similar to the one found in [16] where the gauge group was given by SO (2 n ′ f + 2 n c − n ′ c + 4) × Sp ( n c ) with 2 n ′ f multiplets, flavor singlet and gauge singlets. Thenthe present N c corresponds to the n ′ f , the number N ′ c corresponds to n ′ c and N ′′ c correspondsto the n c of [16].The dual gauge theory has an adjoint Φ of Sp ( N c ) and bifundamentals f and g under thedual gauge group (3.6) and the superpotential corresponding to Figures 9A and 9B is givenby W dual = h Φ f f − hµ Φ , h = g ,mag , µ = − ∆ x πg s ℓ s . Then f f is a 2 e N ′ c × e N ′ c matrix where the first gauge group indices for f are contracted withthose of Φ while µ is a 2 N c × N c matrix. The product f f has the same representation forthe product of quarks and moreover, the first gauge group indices for the field Φ play the roleof the flavor indices as we observed above.Therefore, the F-term equation, the derivative W dual with respect to the meson field Φcannot be satisfied if the 2 N c exceeds 2 e N ′ c . So the supersymmetry is broken. That is, there23xist two equations from F-term conditions: f f − µ = 0 and Φ f = 0. Then the solutions forthese are given by < f > = (cid:18) µ e N ′ c (cid:19) , < Φ > = (cid:18) N c − e N ′ c ) (cid:19) (3.8)where the zero of < f > is a 2( N c − e N ′ c ) × e N ′ c matrix and the zeros of < Φ > are 2 e N ′ c × e N ′ c ,2 e N ′ c × N c − e N ′ c ) and 2( N c − e N ′ c ) × e N ′ c matrices. Then one can expand these fields aroundon a point (3.8), as in [4] and one arrives at the relevant superpotential up to quadratic orderin the fluctuation. At one loop, the effective potential V (1) eff for Φ leads to the positive valuefor m implying that these vacua are stable. One can think of the following dual gauge group Sp ( N c ) × SO (2 N ′ c ) × Sp ( e N ′′ c = N ′ c − N ′′ c −
2) (3.9)by performing the magnetic dual for the last gauge group in (3.1). The electric brane configu-ration can be given in terms of Figure 6A or Figure 6A with an exchange between NS5-braneand NS5’-brane. For the latter, the resulting brane configuration is given by
N S L -brane, N S ′ L -brane, N S R -brane, and N S ′ R -brane from the left to the right in the x direction.One can take the magnetic dual either by following the previous procedure or by looking atthe Figure 7 from the negative w direction which is an opposite viewpoint, compared withFigure 7. In other words, we are looking at the Figure 7 from the other side of w .Then the resulting brane configuration in this case can be obtained by taking a reflectionfor all the NS-branes, D4-branes and anti D4-branes with respect to the N S L -brane(rotatingthem to the left for fixed N S L -brane) in Figure 7A and Figure 7B. Then the N = 1 magneticbrane configuration for the gauge group Sp ( N c ) × SO (2 N ′ c ) × Sp ( e N ′′ c = N ′ c − N ′′ c −
2) correspondsto the Figure 10A’ with D4- and D N ′′ c − N c + 2 = ( N ′ c − N c ) − e N ′′ c . We do not present the Figures 10A’and 10B’ here.We turn to the other case. Let us consider other magnetic theory for the same electrictheory given in the subsection 3.1 with Figure 6A. By applying the Seiberg dual to the Sp ( N ′′ c )factor in (3.1) from Figure 6A and interchanging the N S ′ R -brane and the N S R -brane, oneobtains the Figure 10A”. Before arriving at the Figure 10A”, there exists an intermediatestep where 2 N c D4-branes between
N S ′ L -brane and the N S L -brane, the 2 N ′ c D4-branes24re connecting between the
N S L -brane and the N S R -brane, ( N ′ c − N ′′ c −
2) D4-branesare connecting between the
N S R -brane and N S ′ R -brane(and their mirrors). By rotating N S L -brane by an angle π , moving it with the ( N ′ c − N c ) D4-branes to + v direction where weintroduce 2( N ′ c − N c ) D4-branes and 2( N ′ c − N c ) anti D4-branes between the N S R -brane andthe N S ′ R -brane, one gets the final Figure 10A” where we are left with 2( N ′′ c − N c + 2) anti-D4-branes between the N S R -brane and the N S ′ R -brane. When two NS5’-branes in Figure10A” are close to each other, then it leads to Figure 10B” by realizing that the number of( N ′ c − N c ) D4-branes connecting between N S ′ M -brane and NS5-brane can be rewritten as( N ′′ c − N c + 2) plus e N ′′ c .The brane configuration in Figure 10A” is stable as long as the distance ∆ x between theupper NS5’-brane and the middle NS5’-brane(or N S ′ R -brane) is large. If they are close toeach other, then this brane configuration is unstable to decay to the brane configuration inFigure 10B”. One can regard these brane configurations as particular states in the magneticgauge theory with the gauge group and superpotential. The upper ( N ′ c − N c − e N ′′ c ) flavorD4-branes of straight brane configuration of Figure 10B” bend since there exists an attractivegravitational interaction between those flavor D4-branes and NS5-brane from the DBI action.As mentioned in [9], the two NS5’-branes are located at different side of NS5-brane in Figure10B” and the DBI action computation for this bending curve should be taken into account. Ofcourse, their mirrors, the lower ( N ′ c − N c − e N ′′ c ) flavor D4-branes of straight brane configurationof Figure 10B” can bend and their trajectories connecting two NS5’-branes should be preservedunder the O4-plane, i.e., Z symmetric way.Figure 10: The N = 1 magnetic brane configuration for the gauge group Sp ( N c ) × SO (2 N ′ c ) × Sp ( e N ′′ c = N ′ c − N ′′ c −
2) with D4- and D N ′′ c − N c + 2 = ( N ′ c − N c ) − e N ′′ c . The deformation is rleated to thebifundamentals G . 25he matter contents are the field f charged under ( c , ′ c , ), a field g charged under( , ′ c , e N ′′ c ) under the dual gauge group (3.9) and the gauge-singlet Φ ′ that is in the adjointrepresentation for the second dual gauge group, i.e., ( , ( N ′ c − N c )( ′ c − c − ) , ) underthe dual gauge group. That is, the Φ ′ is an 2( N ′ c − N c ) × N ′ c − N c ) antisymmetric matrix.The cubic superpotential with the mass term in the dual theory is given by W dual = Φ ′ gg + m Φ ′ (3.10)where we define Φ ′ as Φ ′ ≡ GG and the third gauge group indices in G are contracted, eachseond gauge group index in G is encoded in Φ ′ . Although the Φ ′ that has second gaugegroup indices looks similar to the previous Φ that has first gauge group indices, the groupindices are different. Here the magnetic field g correspond to 4-4 strings connecting the 2 e N ′′ c -color D4-branes including the mirrors(that are connecting between the N S ′ M -brane and theNS5-brane in Figure 10B”) with 2 N ′ c -flavor D4-branes. Among these 2 N ′ c -flavor D4-branes,only the strings ending on the upper 2( N ′ c − N ′′ c −
2) D4-branes and on the tilted middle2( N ′′ c − N c + 2) D4-branes in Figure 10B” enter the cubic superpotential term. Although the( N ′ c − N c ) D4-branes(and its mirrors) in Figure 10A” cannot move any directions, the tilted( N ′′ c − N c + 2)-flavor D4-branes(and its mirrors) can move w direction. The remaining upperand lower e N ′′ c D4-branes are fixed also and cannot move any direction. Note that there is adecomposition ( N ′ c − N c ) = ( N ′′ c − N c + 2) + e N ′′ c . The brane configuration for zero mass for the bifundamental, which has only a cubicsuperpotential, can be obtained from Figure 10A” by moving the upper and lower NS5’-branes together with ( N ′ c − N c ) color D4-branes into the origin v = 0. Then the number ofdual colors for D4-branes becomes 2 N c between two NS5’-branes and 2 N ′ c between the N S ′ M -brane and the NS5-brane and 2 e N ′′ c between the NS5-brane and the N S ′ R -brane. Or startingfrom Figure 6A and moving the N S R -brane to the left all the way past the N S ′ R -brane,one also obtains the corresponding magnetic brane configuration for massless case.The dual gauge theory has an adjoint Φ ′ of SO (2 N ′ c ) and bifundamentals f and g underthe dual gauge group (3.9) and the superpotential corresponding to Figures 10A” and 10B”is given by W dual = h Φ ′ gg − hµ Φ ′ , h = g ,mag , µ = − ∆ x πg s ℓ s . Then gg is a 2 e N ′′ c × e N ′′ c matrix where the second gauge group indices for g are contractedwith those of Φ ′ while µ is a 2( N ′ c − N c ) × N ′ c − N c ) matrix. The product gg has the same26epresentation for the product of quarks and moreover, the first gauge group indices for thefield Φ ′ play the role of the flavor indices.Therefore, the F-term equation, the derivative W dual with respect to the meson field Φ ′ cannot be satisfied if the 2( N ′ c − N c ) exceeds 2 e N ′′ c . So the supersymmetry is broken. Thatis, there exist two equations from F-term conditions: gg − µ = 0 and Φ ′ g = 0. Then thesolutions for these are given by < g > = (cid:18) µ e N ′′ c (cid:19) , < Φ ′ > = (cid:18) ′ ( N ′ c − N c − e N ′′ c ) ⊗ iσ (cid:19) (3.11)where the zero of < g > is a 2( N ′ c − N c − e N ′′ c ) × e N ′′ c matrix and the zeros of < Φ ′ > are2 e N ′′ c × e N ′′ c , 2 e N ′′ c × N ′ c − N c − e N ′′ c ), and 2( N ′ c − N c − e N ′′ c ) × e N ′′ c matrices. Then one can expandthese fields around on a point (3.11), as in [4] and one arrives at the relevant superpotentialup to quadratic order in the fluctuation. At one loop, the effective potential V (1) eff for Φ ′ leadsto the positive value for m ′ implying that these vacua are stable. By changing the charges of O4-plane in previous brane configuration of Figure 6A, the typeIIA brane configuration is realized by an N = 1 supersymmetric gauge theory with SO (2 N c ) × Sp ( N ′ c ) × SO (2 N ′′ c )and corresponding matter contents. Then by deforming the theory by mass term and takingthe magnetic dual on each gauge group factor, one gets meta-stable brane configurations.There exists an N = 1 magnetic supersymmetric gauge theory with SO (2 e N c = 2 N ′ c − N c +4) × Sp ( N ′ c ) × SO (2 N ′′ c ) with matters which corresponds to Figure 7 with opposite O4-planecharges. Also there is an N = 1 magnetic supersymmetric gauge theory with SO (2 N c ) × Sp ( e N ′ c = N c + N ′′ c − N ′ c − × SO (2 N ′′ c ) with matters which corresponds to Figure 9 withopposite O4-plane charges. Finally, there exists an N = 1 magnetic supersymmetric gaugetheory with SO (2 N c ) × Sp ( N ′ c ) × SO (2 e N ′′ c = 2 N ′ c − N ′′ c + 4) with matters which correspondsto Figure 10 with opposite O4-plane charges. The remaining analysis can be done easilywithout any difficulty. In this section, we add an orientifold 6-plane to the previous brane configuration for theproduct gauge group [10] realized by three NS-branes, together with the extra mirrors for27hem, and find out new meta-stable brane configurations. Or one can realize these braneconfigurations by inserting the two outer NS-branes into the brane configuration [22, 14].
The type IIA brane configuration corresponding to N = 1 supersymmetric gauge theory withgauge group Sp ( N c ) × SU ( N ′ c ) × SU ( N ′′ c ) (4.1)and with a field F charged under ( c , N ′ c ), a field G charged under ( N ′ c , N ′′ c ), and theirconjugates e F and e G can be described by the left N S ′ L -brane, the NS5-brane, the right N S ′ R -brane(and their mirrors), 2 N c -, N ′ c - and N ′′ c -color D4-branes as well as O6-plane(0123789) .The O − -plane acts as ( x , x , x ) → ( − x , − x , − x ) and has RR charge − x = 0 and let us denote the x coordinates for the N S ′ L -brane, the NS5-brane and the N S ′ R -brane by x = y , y + y , y + y + y respectively.Their mirrors can be understood similarly. The 2 N c D4-branes are suspended between the
N S ′ L -brane and its mirror, the N ′ c D4-branes are suspended between the
N S ′ L -brane andthe NS5-brane(and their mirrors), and the N ′′ c D4-branes are suspended between the NS5-brane and the
N S ′ R -brane(and their mirrors). We draw this brane configuration in Figure11A for the vanishing mass for the field G .The gauge couplings of Sp ( N c ), SU ( N ′ c ) and SU ( N ′′ c ) are given by a string coupling con-stant g s , a string scale ℓ s and the x coordinates y i for three NS-branes through g = g s ℓ s y , g = g s ℓ s y , g = g s ℓ s y . As y goes to ∞ , the SU ( N ′′ c ) gauge group becomes a global symmetry and the theoryleads to SQCD with the gauge group Sp ( N c ) × SU ( N ′ c ) and N ′′ c flavors in the fundamentalrepresentation.There is no superpotential in Figure 11A. Let us deform this theory. Displacing the twoNS5’-branes relative each other in the + v direction corresponds to turning on a quadraticmass-deformed superpotential for the field G as follows: W = mG e G ≡ m Φ ′′ (4.2) From now on, when we say about NS-branes(NS5-brane or NS5’-brane), they refer to those in positiveregion of x . Their mirrors in the negative region of x are understood with O6-plane while we are taking thebrane motion. In other words, there exist three NS-branes: N S ′ L -brane, NS5-brane and N S ′ R -brane fromFigure 11A. G and e G are contracted and the mass m is given by(2.3). The gauge-singlet Φ ′′ for the second dual gauge group is in the adjoint representationfor the third dual gauge group, i.e., ( , , N ′′ c2 − ) ⊕ ( , , ) under the dual gauge group(4.3). The Φ ′′ is a N ′′ c × N ′′ c matrix. The N S ′ R -brane together with N ′′ c -color D4-branes ismoving to the + v direction for fixed other branes during this mass deformation(and theirmirrors to − v direction). Then the x coordinate of N S ′ L -brane is equal to zero while the x coordinate of N S ′ R -branes is given by ∆ x . Giving an expectation value to the mesonfield Φ ′′ corresponds to recombination of N ′ c - and N ′′ c - color D4-branes, which will become N ′′ c or N ′ c -color D4-branes in Figure 11A such that they are suspended between the N S ′ L -braneand the N S ′ R -brane and pushing them into the w direction. We assume that the number ofcolors satisfies 2 N c + N ′′ c ≥ N ′ c ≥ N c . Now we draw this brane configuration in Figure 11B for nonvanishing mass for the fields G and e G . The geometry for three NS-branes in Figure 11B is the same as the one given byfirst three NS-branes in Figure 1B.Figure 11: The N = 1 supersymmetric electric brane configuration for the gauge group Sp ( N c ) × SU ( N ′ c ) × SU ( N ′′ c ) and bifundamental F, e F , G and e G with vanishing(11A) andnonvanishing(11B) mass for the bifundamental G and e G . In 11B, the N S ′ R -brane togetherwith N ′′ c D4-branes is moving to + v direction(and their mirrors to − v direction). By applying the Seiberg dual to the SU ( N ′ c ) factor in (4.1), the N S ′ L,R -branes can be lo-cated at the outside of the two NS5-branes, as in Figure 12. Starting from Figure 11B andinterchanging the
N S ′ L -brane and the NS5-brane(and their mirrors), one obtains the Figure12A. 29igure 12: The N = 1 magnetic brane configuration for the gauge group Sp ( N c ) × SU ( e N ′ c =2 N c + N ′′ c − N ′ c ) × SU ( N ′′ c ) corresponding to Figure 11B with D4- and D N ′ c − N c = N ′′ c − e N ′ c in 12B. The notation for theanti D4-branes is used for the bar on the number of those branes in 12A.Before arriving at the Figure 12A, there exists an intermediate step where the ( N ′′ c − N ′ c + 2 N c ) D4-branes are connecting between the NS5-brane and the N S ′ L -brane, N ′′ c D4-branes are connecting between the
N S ′ L -brane and N S ′ R -brane(and their mirrors) as wellas 2 N c D4-branes between the NS5-brane and its mirror. By reconnecting the N ′′ c D4-branesconnecting between the NS5-brane and the
N S ′ L -brane with the N ′′ c D4-branes connectingbetween
N S ′ L -brane and the N S ′ R -brane and moving those combined N ′′ c D4-branes to + v -direction(and their mirrors to − v direction), one gets the final Figure 12A where we are leftwith ( N ′ c − N c ) anti-D4-branes between the NS5-brane and N S ′ L -brane. When two NS5’-branes in Figure 12A are close to each other, it becomes Figure 12B by realizing that thenumber of N ′′ c D4-branes connecting between NS5-brane and
N S ′ R -brane in Figure 12A canbe rewritten as ( N ′ c − N c ) plus e N ′ c . The brane configuration consisting of NS5-brane and twoNS5’-branes in Figure 12B is exactly the same as those in Figure 2B.The dual gauge group is given by Sp ( N c ) × SU ( e N ′ c = 2 N c + N ′′ c − N ′ c ) × SU ( N ′′ c ) . (4.3)The matter contents are the field f charged under ( c , e N ′ c , ), a field g charged under( , e N ′ c , N ′′ c ), and their conjugates e f and e g under the dual gauge group (4.3) and the gauge-singlet Φ ′′ for the second dual gauge group in the adjoint representation for the third dualgauge group, i.e., ( , , N ′′ c2 − ) ⊕ ( , , ) under the dual gauge group. Then the Φ ′′ is a N ′′ c × N ′′ c matrix. 30he cubic superpotential with the mass term (4.2) is given by W dual = Φ ′′ g e g + m Φ ′′ . (4.4)Here the magnetic fields g and e g correspond to 4-4 strings connecting the e N ′ c -color D4-branes(that are connecting between the NS5-brane and the N S ′ R -brane in Figure 12B) with N ′′ c -flavor D4-branes(which are realized as corresponding D4-branes in Figure 12A). Althoughthe N ′′ c D4-branes in Figure 12A cannot move any directions, the tilted ( N ′ c − N c )-flavorD4-branes can move w direction in Figure 12B(and its mirrors). The remaining upper e N ′ c D4-branes are fixed also and cannot move any direction. Note that there is a decomposition N ′′ c = ( N ′ c − N c ) + e N ′ c . The brane configuration for zero mass for the bifundamental, which has only a cubicsuperpotential, can be obtained from Figure 12A by moving the upper NS5’-brane(or
N S ′ R -brane) together with N ′′ c color D4-branes into the origin v = 0(and their mirrors). Thenthe number of dual colors for D4-branes becomes 2 N c between the NS5-brane and its mirror, e N ′ c between NS5-brane and N S ′ L -brane and N ′′ c between N S ′ L -brane and N S ′ R -brane. Orstarting from Figure 11A and moving the NS5-brane to the left all the way past the N S ′ L -brane(and their mirrors), one also obtains the corresponding magnetic brane configurationfor massless case.The brane configuration in Figure 12A is stable as long as the distance ∆ x between theupper NS5’-brane and the lower NS5’-brane is large. If they are close to each other, then thisbrane configuration is unstable to decay and leads to the brane configuration in Figure 12B.One can regard these brane configurations as particular states in the magnetic gauge theorywith the gauge group (4.3) and superpotential (4.4).One can perform similar analysis in our brane configuration since one can take into accountthe behavior of parameters geometrically in the presence of O6-plane. Then the upper ( N ′′ c − e N ′ c ) flavor D4-branes of straight brane configuration of Figure 12B can bend due to the factthat there exists an attractive gravitational interaction between those flavor D4-branes andNS5-brane from the DBI action, by following the procedure of [3], as long as y is very large.Then the mirror of NS5-brane does not affect the flavor D4-branes. On the other hand, if y goes to zero, then the mirror of NS5-brane plays the role of enhancing the strength for theNS5-branes and will affect both the energy of bending curve, E curved , and ∆ x . Of course,their mirrors, the lower ( N ′′ c − e N ′ c ) flavor D4-branes of straight brane configuration of Figure12B can bend and their trajectories connecting two NS5’-branes should be preserved underthe O6-plane, i.e., Z symmetric way. 31he low energy dynamics of the magnetic brane configuration can be described by the N = 1 supersymmetric gauge theory with gauge group (4.3) and the gauge couplings for thethree gauge group factors are given by g ,mag = g s ℓ s y + y ) , g ,mag = g s ℓ s y , g ,mag = g s ℓ s ( y − y ) . The dual gauge theory has an adjoint Φ ′′ of SU ( N ′′ c ) and bifundamentals f, e f , g and e g under the dual gauge group (4.3) and the superpotential corresponding to Figures 12A and12B is given by W dual = h Φ ′′ g e g − hµ Φ ′′ , h = g ,mag , µ = − ∆ x πg s ℓ s . Then g e g is a e N ′ c × e N ′ c matrix where the third gauge group indices for g and e g are contractedwith those of Φ ′′ while µ is a N ′′ c × N ′′ c matrix. The product g e g has the same representationfor the product of quarks and moreover, the third gauge group indices for the field Φ ′′ playthe role of the flavor indices.When the upper NS5’-brane(or N S ′ R -brane) is replaced by coincident N ′′ c D6-branes inFigure 12B, this brane configuration looks similar to the one found in [14] where the gaugegroup was given by SU ( n f + 2 n ′ c − n c ) × Sp ( n ′ c ) with n f multiplets and singlets. Then thepresent 2 N c corresponds to the 2 n ′ c , N ′ c corresponds to n c , and N ′′ c corresponds to the n f of[14].Therefore, the F-term equation, the derivative W dual with respect to the meson field Φ ′′ cannot be satisfied if the N ′′ c exceeds e N ′ c . So the supersymmetry is broken. That is, thereexist three equations from F-term conditions: g e g − µ = 0 and Φ ′′ g = 0 = e g Φ ′′ . Then thesolutions for these are given by < g > = (cid:18) µ e N ′ c (cid:19) , < e g > = (cid:0) µ e N ′ c (cid:1) , < Φ ′′ > = (cid:18) ′′ ( N ′′ c − e N ′ c ) (cid:19) (4.5)where the zero of < g > is a ( N ′′ c − e N ′ c ) × e N ′ c matrix, the zero of < e g > is a e N ′ c × ( N ′′ c − e N ′ c )matrix and the zeros of < Φ ′′ > are e N ′ c × e N ′ c , e N ′ c × ( N ′′ c − e N ′ c ) and ( N ′′ c − e N ′ c ) × e N ′ c matrices.Then one can expand these fields around on a point (4.5), as in [4] and one arrives at therelevant superpotential up to quadratic order in the fluctuation. At one loop, the effectivepotential V (1) eff for Φ ′′ leads to the positive value for m ′′ implying that these vacua are stable. Let us consider other magnetic theory for the same electric theory given in the subsection4.1. By applying the Seiberg dual to the SU ( N ′′ c ) factor in (4.1), the N S ′ L,R -branes can be32ocated at the inside of the two NS5-branes, as in Figure 14. Starting from Figure 13B andinterchanging the NS5-brane and the
N S ′ R -brane(and their mirrors), one obtains the Figure14A. The geometry for three NS-branes in Figure 13B is the same as the one given by firstthree NS-branes in Figure 3B.Figure 13: The N = 1 supersymmetric electric brane configuration for the gauge group Sp ( N c ) × SU ( N ′ c ) × SU ( N ′′ c ) and the bifundamentals with vanishing(13A) which is the same asthe Figure 11A and nonvanishing(13B) mass for the bifundamental G and e G . This deformationis different from the one in (4.2). The N ′ c D4-branes in 13A are decomposed into ( N ′ c − N c )D4-branes which are moving to + v direction in 13B and 2 N c D4-branes which are recombinedwith those D4-branes connecting between
N S ′ L -brane and its mirror in 13B.Before arriving at the Figure 14A, there exists an intermediate step where the ( N ′ c − N c )D4-branes are connecting between the N S ′ L -brane and the N S ′ R -brane, ( N ′ c − N ′′ c ) D4-branesare connecting between the N S ′ R -brane and NS5-brane(and their mirrors) as well as 2 N c D4-branes between
N S ′ R -brane and its mirror. By reconnecting the ( N ′ c − N c ) D4-branesconnecting between the N S ′ L -brane and the N S ′ R -brane with the ( N ′ c − N c ) D4-branesconnecting between N S ′ R -brane and the NS5-brane where we introduce − N c D4-branesand − N c anti D4-branes and moving those combined D4-branes to + v -direction(and theirmirrors to − v direction), one gets the final Figure 14A where we are left with ( N ′′ c − N c )anti-D4-branes between the N S ′ R -brane and the NS5-brane. We assume that the number ofcolors satisfies N ′ c ≥ N ′′ c ≥ N c . When two NS5’-branes in Figure 14A are close to each other, then it leads to Figure 14Bby realizing that the number of ( N ′ c − N c ) D4-branes connecting between N S ′ L -brane andNS5-brane in Figure 14A can be rewritten as ( N ′′ c − N c ) plus e N ′′ c . The brane configurationconsisting of NS5-brane and two NS5’-branes in Figure 14B is exactly the same as those inFigure 4B. 33igure 14: The N = 1 magnetic brane configuration for the gauge group Sp ( N c ) × SU ( N ′ c ) × SU ( e N ′′ c = N ′ c − N ′′ c ) corresponding to Figure 13B with D4- and D N ′′ c − N c = ( N ′ c − N c ) − e N ′′ c .The dual gauge group is given by Sp ( N c ) × SU ( N ′ c ) × SU ( e N ′′ c = N ′ c − N ′′ c ) (4.6)The matter contents are the field f charged under ( c , N ′ c , ), a field g charged under( , N ′ c , e N ′′ c ) and their conjugates e f and e g under the dual gauge group (4.6) and the gauge-singlet Φ ′ which is in the adjoint representation for the second dual gauge group, in otherwords, ( , ( N ′ c − c ) − , ) ⊕ ( , , ) under the dual gauge group (4.6). Then the Φ ′ isa ( N ′ c − N c ) × ( N ′ c − N c ) matrix. Only ( N ′ c − N c ) D4-branes can participate in the massdeformation.The cubic superpotential with the mass term is given by W dual = Φ ′ g e g + m Φ ′ (4.7)where we define Φ ′ as Φ ′ ≡ G e G and the third gauge group indices in G and e G are contracted,each second gauge group index in them is encoded in Φ ′ . Although the Φ ′ that has secondgauge group indices looks similar to the previous Φ ′′ that has third gauge group indices,the group indices are different. Here the magnetic fields g and e g correspond to 4-4 stringsconnecting the e N ′′ c -color D4-branes(that are connecting between the N S ′ L -brane and theNS5-brane in Figure 14B) with N ′ c -flavor D4-branes. Among these N ′ c -flavor D4-branes, onlythe strings ending on the upper ( N ′ c − N ′′ c ) D4-branes and on the tilted ( N ′′ c − N c ) D4-branesin Figure 14B enter the cubic superpotential term. Although the ( N ′ c − N c ) D4-branes inFigure 14A cannot move any directions, the tilted ( N ′′ c − N c )-flavor D4-branes can move w e N ′′ c D4-branes are fixed also and cannot move any direction.Note that there is a decomposition( N ′ c − N c ) = ( N ′′ c − N c ) + e N ′′ c . The brane configuration for zero mass for the bifundamental, which has only a cubicsuperpotential, can be obtained from Figure 14A by moving the upper NS5’-brane togetherwith ( N ′ c − N c ) color D4-branes into the origin v = 0(and their mirrors). Then the number ofdual colors for D4-branes becomes 2 N c between the N S ′ L -brane and its mirror, N ′ c betweenthe N S ′ L -brane and the N S ′ R -brane and e N ′′ c between N S ′ R -brane and NS5-brane. Orstarting from Figure 13A and moving the NS5-brane to the right all the way past the N S ′ R -brane(and their mirrors), one also obtains the corresponding magnetic brane configurationfor massless case.The brane configuration in Figure 14A is stable as long as the distance ∆ x between theupper NS5’-brane and the lower NS5’-brane is large. If they are close to each other, thenthis brane configuration is unstable to decay and leads to the brane configuration in Figure14B. One can regard these brane configurations as particular states in the magnetic gaugetheory with the gauge group (4.6) and superpotential (4.7). Then the upper ( N ′ c − N c − e N ′′ c )flavor D4-branes of straight brane configuration of Figure 14B can bend due to the fact thatthere exists an attractive gravitational interaction between those flavor D4-branes and NS5-brane from the DBI action, as long as y is very large. Of course, their mirrors, the lower( N ′ c − N c − e N ′′ c ) flavor D4-branes of straight brane configuration of Figure 14B can bend andtheir trajectories connecting two NS5’-branes should be preserved under the O6-plane, i.e., Z symmetric way.When the upper NS5’-brane(or N S ′ L -brane) is replaced by coincident ( N ′ c − N c ) D6-branes in Figure 14B, this brane configuration looks similar to the one found in [14] wherethe gauge group was given by SU ( n f + n ′ c − n c ) × SO ( n ′ c ) with n f multiplets, bifundamentals,and singlets. Then the present 2 N c corresponds to the n ′ c , ( N ′ c − N c ) corresponds to n f , and N ′′ c corresponds to the n c of [14]. Note that Sp ( N c ) corresponds to SO ( n ′ c ). Moreover, thereis a meta-stable brane configuration for the gauge group given by SU ( n c ) × SU ( n ′ f + n c − n ′ c )with fundamentals, bifundamentals, an antisymmetric flavor, a conjugate symmetric flavor,and singlets where there are NS5’-brane, O ± -planes, and eight semi infinite D6-branes at x = 0. Then the our 2 N c corresponds to the n c , the number ( N ′ c − N c ) corresponds to n ′ f ,and our N ′′ c corresponds to the n ′ c of [23].The low energy dynamics of the magnetic brane configuration can be described by the N = 1 supersymmetric gauge theory with gauge group (4.6) and the gauge couplings for the35hree gauge group factors are given by g ,mag = g s ℓ s y , g ,mag = g s ℓ s ( y − y ) , g ,mag = g s ℓ s y . The dual gauge theory has an adjoint Φ ′ of SU ( N ′ c ) and bifundamentals f, e f , g and e g underthe dual gauge group (4.6) and the superpotential corresponding to Figures 14A and 14B isgiven by W dual = h Φ ′ g e g − hµ Φ ′ , h = g ,mag , µ = − ∆ x πg s ℓ s . Then g e g is a e N ′′ c × e N ′′ c matrix where the second gauge group indices for g and e g are contractedwith those of Φ ′ while µ is a ( N ′ c − N c ) × ( N ′ c − N c ) matrix. The product g e g has the samerepresentation for the product of quarks and moreover, the second gauge group indices forthe field Φ ′ play the role of the flavor indices, as above.Therefore, the F-term equation, the derivative W dual with respect to the meson field Φ ′ cannot be satisfied if the ( N ′ c − N c ) exceeds e N ′′ c . So the supersymmetry is broken. That is,there exist three equations from F-term conditions: g e g − µ = 0 and Φ ′ g = 0 = e g Φ ′ . Thenthe solutions for these are given by < g > = (cid:18) µ e N ′′ c (cid:19) , < e g > = (cid:0) µ e N ′′ c (cid:1) , < Φ ′ > = (cid:18) ′ ( N ′ c − N c ) − e N ′′ c (cid:19) (4.8)where the zero of < g > is a ( N ′ c − N c − e N ′′ c ) × e N ′′ c matrix, the zero of < e g > is a e N ′′ c × ( N ′ c − N c − e N ′′ c ) matrix and the zeros of < Φ ′ > are e N ′′ c × e N ′′ c , e N ′′ c × ( N ′ c − N c − e N ′′ c ) and( N ′ c − N c − e N ′′ c ) × e N ′′ c matrices. Then one can expand these fields around on a point (4.8), asin [4] and one arrives at the relevant superpotential up to quadratic order in the fluctuation.At one loop, the effective potential V (1) eff for Φ ′ leads to the positive value for m ′ implyingthat these vacua are stable. In this subsection, we add an orientifold 6-plane with positive charge to the previous braneconfiguration for the product gauge group [10] realized by three NS-branes, together with theextra mirrors for them, and find out new meta-stable brane configurations. Or one can realizethese brane configurations by inserting the two outer NS-branes into the brane configuration[22, 14]. 36 .4.1 Electric theory
The type IIA brane configuration corresponding to N = 1 supersymmetric gauge theory withgauge group SO ( N c ) × SU ( N ′ c ) × SU ( N ′′ c ) (4.9)and with a field F charged under ( N c , N ′ c ), a field G charged under ( N ′ c , N ′′ c ), and theirconjugates e F and e G can be described by the left N S L -brane, the NS5’-brane, the right N S R -brane(and their mirrors), N c -, N ′ c - and N ′′ c -color D4-branes as well as O + -plane(0123789).The O + -plane acts as ( x , x , x ) → ( − x , − x , − x ) and has RR charge +4.Let us place an O + -plane at the origin x = 0 and let us denote the x coordinatesfor the N S L -brane, the NS5’-brane and the N S R -brane by x = y , y + y , y + y + y respectively. Their mirrors can be understood similarly. The N c D4-branes are suspendedbetween the
N S L -brane and its mirror, the N ′ c D4-branes are suspended between the
N S L -brane and the NS5’-brane(and their mirrors), and the N ′′ c D4-branes are suspended betweenthe NS5’-brane and the
N S R -brane(and their mirrors). We assume that the number of colorssatisfies N c + N ′′ c ≥ N ′ c ≥ N c . By applying the Seiberg dual to the SU ( N ′ c ) factor in (4.9) and interchanging the N S L -braneand the NS5’-brane(and their mirrors), one obtains the Figure 15A.Before arriving at the Figure 15A, there exists an intermediate step where the ( N ′′ c − N ′ c + N c ) D4-branes are connecting between the NS5’-brane and the N S L -brane, N ′′ c D4-branes are connecting between the
N S L -brane and N S R -brane(and their mirrors) as wellas N c D4-branes between NS5’-brane and its mirror. By rotating
N S R -brane by an angle π ,moving it with N ′′ c D4-branes to + v -direction(and their mirrors to − v direction), one gets thefinal Figure 15A where we are left with ( N ′ c − N c ) anti-D4-branes between the N S ′ L -braneand NS5-brane. When two NS5’-branes in Figure 15A are close to each other, it becomesFigure 15B by realizing that the number of N ′′ c D4-branes connecting between NS5-brane and
N S ′ R -brane can be rewritten as ( N ′ c − N c ) plus e N ′ c .The dual gauge group is given by SO ( N c ) × SU ( e N ′ c = N c + N ′′ c − N ′ c ) × SU ( N ′′ c ) . (4.10)37igure 15: The N = 1 magnetic brane configuration for the gauge group SO ( N c ) × SU ( e N ′ c = N c + N ′′ c − N ′ c ) × SU ( N ′′ c ) with D4- and D SO ( N c ) is equal to N c not 2 N c . The number of tilted D4-branesis equal to N ′ c − N c = N ′′ c − e N ′ c in 15B. The deformation is rleated to the bifundamentals G and e G .The matter contents are the field f charged under ( N c , e N ′ c , ), a field g charged under( , e N ′ c , N ′′ c ), and their conjugates e f and e g under the dual gauge group (4.10) and the gauge-singlet Φ ′′ for the second dual gauge group in the adjoint representation for the third dualgauge group, i.e., ( , , N ′′ c2 − ) ⊕ ( , , ) under the dual gauge group. Then the Φ ′′ is a N ′′ c × N ′′ c matrix.The cubic superpotential with the mass term is given by (4.4) where we define Φ ′′ asΦ ′′ ≡ G e G and the second gauge group indices in G and e G are contracted, each third gaugegroup index in them is encoded in Φ ′′ . Although the Φ ′′ that has third gauge group indiceslooks similar to the previous Φ ′ that has second gauge group indices the group indices aredifferent. Here the magnetic fields g and e g correspond to 4-4 strings connecting the e N ′ c -color D4-branes(that are connecting between the NS5-brane and the N S ′ R -brane in Figure15B) with N ′′ c -flavor D4-branes(which are realized as corresponding D4-branes in Figure 15A).Although the N ′′ c D4-branes in Figure 15A cannot move any directions, the tilted ( N ′ c − N c )-flavor D4-branes can move w direction in Figure 15B(and its mirrors). The remaining upper e N ′ c D4-branes are fixed also and cannot move any direction. Note that there is a decomposition N ′′ c = ( N ′ c − N c ) + e N ′ c . The brane configuration for zero mass for the bifundamental, which has only a cubicsuperpotential, can be obtained from Figure 15A by moving the upper NS5’-brane togetherwith N ′′ c color D4-branes into the origin v = 0(and their mirrors). Then the number ofdual colors for D4-branes becomes N c between the N S ′ L -brane and its mirror, e N ′ c between38 S ′ L -brane and NS5-brane and N ′′ c between NS5-brane and N S ′ R -brane.The brane configuration in Figure 15A is stable as long as the distance ∆ x between theupper NS5’-brane and the lower NS5’-brane is large. If they are close to each other, thenthis brane configuration is unstable to decay to the brane configuration in Figure 15B. Onecan regard these brane configurations as particular states in the magnetic gauge theory withthe gauge group and superpotential. The upper ( N ′′ c − e N ′ c ) flavor D4-branes of straight braneconfiguration of Figure 15B bend since there exists an attractive gravitational interactionbetween those flavor D4-branes and NS5-brane from the DBI action. As mentioned in [9], thetwo NS5’-branes are located at different side of NS5-brane in Figure 15B and the DBI actioncomputation for this bending curve should be taken into account.The low energy dynamics of the magnetic brane configuration can be described by the N = 1 supersymmetric gauge theory with gauge group (4.10) and the gauge couplings for thethree gauge group factors are given by the expressions in subsection 4.2.The dual gauge theory has an adjoint Φ ′′ of SU ( N ′′ c ) and bifundamentals f, e f , g and e g under the dual gauge group (4.10) and the superpotential corresponding to Figures 15A and15B is given by the expressions in subsection 4.2. Then g e g is a e N ′ c × e N ′ c matrix where thethird gauge group indices for g and e g are contracted with those of Φ ′′ while µ is a N ′′ c × N ′′ c matrix. The product g e g has the same representation for the product of quarks and moreover,the third gauge group indices for the field Φ ′′ play the role of the flavor indices.When the upper NS5’-brane(or N S ′ R -brane) is replaced by coincident N ′′ c D6-branes inFigure 15B, this brane configuration looks similar to the one found in [14] where the gaugegroup was given by SU ( n f + n ′ c − n c ) × Sp ( n ′ c ) with n f multiplets and singlets. Then thepresent N c corresponds to the n ′ c , N ′ c corresponds to n c , and N ′′ c corresponds to the n f of [14].Therefore, the F-term equation, the derivative W dual with respect to the meson field Φ ′′ cannot be satisfied if the N ′′ c exceeds e N ′ c . So the supersymmetry is broken. That is, there existthree equations from F-term conditions: g e g − µ = 0 and Φ ′′ g = 0 = e g Φ ′′ . Then the solutionsfor these are given by the expressions in subsection 4.2. Then one can expand these fieldsaround on a point, as in [4] and one arrives at the relevant superpotential up to quadraticorder in the fluctuation. At one loop, the effective potential V (1) eff for Φ ′′ leads to the positivevalue for m ′′ implying that these vacua are stable. Let us consider other magnetic theory for the same electric theory given in the subsection4.4.1. By applying the Seiberg dual to the SU ( N ′′ c ) factor in (4.9) and interchanging theNS5’-brane and the N S R -brane(and their mirrors), one obtains the Figure 16A.39efore arriving at the Figure 16A, there exists an intermediate step where the N ′ c D4-branes are connecting between the
N S L -brane and the N S R -brane, ( N ′ c − N ′′ c ) D4-branesare connecting between the N S R -brane and NS5’-brane(and their mirrors) as well as N c D4-branes between
N S L -brane and its mirror. By rotating N S L -brane by an angle π , movingit with the ( N ′ c − N c ) D4-branes to + v direction where we introduce ( N ′ c − N c ) D4-branes and( N ′ c − N c ) anti D4-branes between the N S R -brane and the NS5’-brane(and their mirrors to − v direction), one gets the final Figure 16A where we are left with ( N ′′ c − N c ) anti-D4-branesbetween the NS5-brane and the N S ′ R -brane. We assume that the number of colors satisfies N ′ c ≥ N ′′ c ≥ N c . When two NS5’-branes in Figure 16A are close to each other, then it leads to Figure 16Bby realizing that the number of ( N ′ c − N c ) D4-branes connecting between N S ′ L -brane andNS5-brane can be rewritten as ( N ′′ c − N c ) plus e N ′′ c . The brane configuration consisting ofNS5-brane and two NS5’-branes in Figure 16B is exactly the same as those in Figure 5B”.Figure 16: The N = 1 magnetic brane configuration for the gauge group SO ( N c ) × SU ( N ′ c ) × SU ( e N ′′ c = N ′ c − N ′′ c ) with D4- and D N ′′ c − N c = ( N ′ c − N c ) − e N ′′ c . The deformation is different from theprevious one.The dual gauge group is given by SO ( N c ) × SU ( N ′ c ) × SU ( e N ′′ c = N ′ c − N ′′ c ) (4.11)The matter contents are the field f charged under ( N c , N ′ c , ), a field g charged under( , N ′ c , e N ′′ c ) and their conjugates e f and e g under the dual gauge group (4.11) and the gauge-singlet Φ ′ which is in the adjoint representation for the second dual gauge group, in otherwords, ( , ( N ′ c − N c ) − , ) ⊕ ( , , ) under the dual gauge group (4.11). Then the Φ ′ is40 ( N ′ c − N c ) × ( N ′ c − N c ) matrix. Only ( N ′ c − N c ) D4-branes are participating in the massdeformation.The cubic superpotential with the mass term is given by (4.7) where we define Φ ′ asΦ ′ ≡ G e G and the third gauge group indices in G and e G are contracted, each second gaugegroup index in them is encoded in Φ ′ . Although the Φ ′ that has second gauge group indiceslooks similar to the previous Φ ′′ that has third gauge group indices, the group indices aredifferent. Here the magnetic fields g and e g correspond to 4-4 strings connecting the e N ′′ c -colorD4-branes(that are connecting between the N S ′ L -brane and the NS5-brane in Figure 16B)with N ′ c -flavor D4-branes. Among these N ′ c -flavor D4-branes, only the strings ending on theupper ( N ′ c − N ′′ c ) D4-branes and on the tilted ( N ′′ c − N c ) D4-branes in Figure 16B enter thecubic superpotential term. Although the ( N ′ c − N c ) D4-branes in Figure 16A cannot moveany directions, the tilted ( N ′′ c − N c )-flavor D4-branes can move w direction. The remainingupper e N ′′ c D4-branes are fixed also and cannot move any direction. Note that there is adecomposition ( N ′ c − N c ) = ( N ′′ c − N c ) + e N ′′ c . The brane configuration for zero mass for the bifundamental, which has only a cubicsuperpotential, can be obtained from Figure 16A by moving the upper NS5’-brane togetherwith ( N ′ c − N c ) color D4-branes into the origin v = 0(and their mirrors). Then the number ofdual colors for D4-branes becomes N c between the N S ′ L -brane and its mirror, N ′ c betweenthe N S ′ L -brane and the NS5-brane and e N ′′ c between NS5-brane and N S ′ R -brane.When the upper NS5’-brane(or N S ′ L -brane) is replaced by coincident ( N ′ c − N c ) D6-branesin Figure 16B, this brane configuration looks similar to the one found in [14] where the gaugegroup was given by SU ( n f +2 n ′ c − n c ) × Sp ( n ′ c ) with n f multiplets, bifundamentals, and singlets.Then the present N c corresponds to the 2 n ′ c , ( N ′ c − N c ) corresponds to n f , and N ′′ c correspondsto the n c of [14]. Note that SO ( N c ) corresponds to Sp ( n ′ c ). Moreover, the meta-stablebrane configuration corresponding to gauge group given by SU ( n c ) × SU ( n ′ f + n c − n ′ c ) withfundamentals, bifundamentals, a symmetric flavor, a conjugate symmetric flavor, and singletswas given in [23] where there exists NS5-brane on the O6-plane. Then our N c corresponds tothe n c , our ( N ′ c − N c ) corresponds to n ′ f , and our N ′′ c corresponds to the n ′ c .The brane configuration in Figure 16A is stable as long as the distance ∆ x between theupper NS5’-brane and the lower NS5’-brane is large. If they are close to each other, then thisbrane configuration is unstable to decay to the brane configuration in Figure 16B. One canregard these brane configurations as particular states in the magnetic gauge theory with thegauge group and superpotential. The upper ( N ′ c − N c − e N ′′ c ) flavor D4-branes of straight brane41onfiguration of Figure 16B bend since there exists an attractive gravitational interactionbetween those flavor D4-branes and NS5-brane from the DBI action. As mentioned in [9], thetwo NS5’-branes are located at different side of NS5-brane in Figure 16B and the DBI actioncomputation for this bending curve should be taken into account.The low energy dynamics of the magnetic brane configuration can be described by the N = 1 supersymmetric gauge theory with gauge group (4.11) and the gauge couplings forthe three gauge group factors are given by the expressions in subsection 4.3. The dual gaugetheory has an adjoint Φ ′ of SU ( N ′ c ) and bifundamentals f, e f , g and e g under the dual gaugegroup (4.11) and the superpotential corresponding to Figures 16A and 16B is given by theone in subsection 4.3. Then g e g is a e N ′′ c × e N ′′ c matrix where the second gauge group indicesfor g and e g are contracted with those of Φ ′ while µ is a ( N ′ c − N c ) × ( N ′ c − N c ) matrix. Theproduct g e g has the same representation for the product of quarks and moreover, the secondgauge group indices for the field Φ ′ play the role of the flavor indices.Therefore, the F-term equation, the derivative W dual with respect to the meson field Φ ′ cannot be satisfied if the ( N ′ c − N c ) exceeds e N ′′ c . So the supersymmetry is broken. That is,there exist three equations from F-term conditions: g e g − µ = 0 and Φ ′ g = 0 = e g Φ ′ . Thenthe solutions for these are given by < g > = (cid:18) µ e N ′′ c (cid:19) , < e g > = (cid:0) µ e N ′′ c (cid:1) , < Φ ′ > = (cid:18) ′ ( N ′ c − N c ) − e N ′′ c (cid:19) (4.12)where the zero of < g > is a ( N ′ c − N c − e N ′′ c ) × e N ′′ c matrix, the zero of < e g > is a e N ′′ c × ( N ′ c − N c − e N ′′ c ) matrix and the zeros of < Φ ′ > are e N ′′ c × e N ′′ c , e N ′′ c × ( N ′ c − N c − e N ′′ c ) and( N ′ c − N c − e N ′′ c ) × e N ′′ c matrices. Then one can expand these fields around on a point (4.12), asin [4] and one arrives at the relevant superpotential up to quadratic order in the fluctuation.At one loop, the effective potential V (1) eff for Φ ′ leads to the positive value for m ′ implyingthat these vacua are stable. The meta-stable brane configurations we have found are summarized by Figures 2, 4, 5, 7, 9,10, 12, 14, 15 and 16. If we replace the NS5’-brane in Figures 2B, 7B with opposite O4-planecharge, 14B with opposite O6-plane charge, and 16B with opposite O6-plane charge, with thecoincident D6-branes, those brane configurations become nonsupersymmetic minimal energybrane configurations found in [14], in [16], in [14], and in [14], respectively.So far, we have considered the cases for even number of NS-branes, i.e., four and six. Forodd cases, i.e., three and five NS-branes, the construction of meta-stable brane configuration42as been done in [9]. So it is natural to ask what happens if there are seven NS-branes.When this extra seventh NS-brane is located at the O6-plane in section 4, then the gaugegroup will be the same as the one in section 2, i.e., SU ( N c ) × SU ( N ′ c ) × SU ( N ′′ c ) with differentmatter contents. This can be obtained also from the brane configuration of [23] by adding twoouter NS-branes. It would be interesting to find out how the meta-stable brane configurationsappear.Some different directions on the meta-stable vacua are present in recent relevant works[24]-[33] where some of them are described in the type IIB string theory. It would be veryinteresting to find out how the meta-stable brane configurations from type IIA string theoryincluding the present work are related to those brane configurations from type IIB stringtheory. Acknowledgments
I would like to thank D. Kutasov for discussions. I would like to thank Kyungho Oh, whopassed away from cancer, for ongoing collaboration and discussions during the last 10 yearsand, in memory of him, I would like to dedicate this work to him. This work was supportedby grant No. R01-2006-000-10965-0 from the Basic Research Program of the Korea Science& Engineering Foundation.
References [1] A. Giveon and D. Kutasov, “Brane dynamics and gauge theory,” Rev. Mod. Phys. ,983 (1999) [arXiv:hep-th/9802067].[2] C. G. . Callan, J. A. Harvey and A. Strominger, “Supersymmetric string solitons,”arXiv:hep-th/9112030.[3] A. Giveon and D. Kutasov, “Gauge symmetry and supersymmetry breaking from inter-secting branes,” [arXiv:hep-th/0703135].[4] K. Intriligator, N. Seiberg and D. Shih, “Dynamical SUSY breaking in meta-stablevacua,” JHEP , 021 (2006) [arXiv:hep-th/0602239].[5] H. Ooguri and Y. Ookouchi, “Meta-stable supersymmetry breaking vacua on intersectingbranes,” Phys. Lett. B , 323 (2006) [arXiv:hep-th/0607183].[6] S. Franco, I. Garcia-Etxebarria and A. M. Uranga, “Non-supersymmetric meta-stablevacua from brane configurations,” JHEP , 085 (2007) [arXiv:hep-th/0607218].437] I. Bena, E. Gorbatov, S. Hellerman, N. Seiberg and D. Shih, “A note on (meta)stablebrane configurations in MQCD,” JHEP , 088 (2006) [arXiv:hep-th/0608157].[8] C. Ahn, “Brane configurations for nonsupersymmetric meta-stable vacua in SQCD withadjoint matter,” Class. Quant. Grav. , 1359 (2007) [arXiv:hep-th/0608160].[9] C. Ahn, “Meta-Stable Brane Configurations by Adding an Orientifold-Plane to Giveon-Kutasov,” arXiv:0706.0042 [hep-th].[10] J. H. Brodie and A. Hanany, “Type IIA superstrings, chiral symmetry, and N = 1 4Dgauge theory dualities,” Nucl. Phys. B , 157 (1997) [arXiv:hep-th/9704043].[11] C. Ahn and R. Tatar, “Geometry, D-branes and N = 1 duality in four dimensions withproduct gauge groups,” Phys. Lett. B , 293 (1997) [arXiv:hep-th/9705106].[12] R. Argurio, M. Bertolini, S. Franco and S. Kachru, “Gauge / gravity dual-ity and meta-stable dynamical supersymmetry breaking,” JHEP , 083 (2007)[arXiv:hep-th/0610212].[13] R. Kitano, H. Ooguri and Y. Ookouchi, “Direct mediation of meta-stable supersymmetrybreaking,” Phys. Rev. D , 045022 (2007) [arXiv:hep-ph/0612139].[14] C. Ahn, “Meta-Stable Brane Configuration of Product Gauge Groups,” arXiv:0704.0121[hep-th].[15] D. Shih, “Spontaneous R-Symmetry Breaking in O’Raifeartaigh Models,”[arXiv:hep-th/0703196].[16] C. Ahn, “Meta-Stable Brane Configuration and Gauged Flavor Symmetry,”[arXiv:hep-th/0703015].[17] C. Ahn, “More on meta-stable brane configuration,” [arXiv:hep-th/0702038], to appearin CQG.[18] C. Ahn, “Meta-stable brane configuration with orientifold 6 plane,”[arXiv:hep-th/0701145], to appear in JHEP.[19] C. Ahn, K. Oh and R. Tatar, “Branes, geometry and N = 1 duality with product gaugegroups of SO and Sp,” J. Geom. Phys. , 301 (1999) [arXiv:hep-th/9707027].[20] G. Bertoldi, B. Feng and A. Hanany, “The splitting of branes on orientifold planes,”JHEP , 015 (2002) [arXiv:hep-th/0202090].4421] C. Ahn, “M-theory lift of meta-stable brane configuration in symplectic and orthogonalgauge groups,” Phys. Lett. B , 493 (2007) [arXiv:hep-th/0610025].[22] E. Lopez and B. Ormsby, “Duality for SU x SO and SU x Sp via branes,” JHEP ,020 (1998) [arXiv:hep-th/9808125].[23] C. Ahn, “Meta-stable brane configurations with five NS5-branes,” arXiv:0705.0056 [hep-th].[24] J. Marsano, K. Papadodimas and M. Shigemori, “Nonsupersymmetric brane / antibraneconfigurations in type IIA and M theory,” arXiv:0705.0983 [hep-th].[25] I. Garcia-Etxebarria, F. Saad and A. M. Uranga, “Supersymmetry breaking metastablevacua in runaway quiver gauge theories,” arXiv:0704.0166 [hep-th].[26] S. Murthy, “On supersymmetry breaking in string theory from gauge theory in a throat,”[arXiv:hep-th/0703237].[27] R. Argurio, M. Bertolini, S. Franco and S. Kachru, “Metastable vacua and D-branes atthe conifold,” [arXiv:hep-th/0703236].[28] Y. E. Antebi and T. Volansky, “Dynamical supersymmetry breaking from simple quiv-ers,” [arXiv:hep-th/0703112].[29] M. Wijnholt, “Geometry of particle physics,” [arXiv:hep-th/0703047].[30] J. J. Heckman, J. Seo and C. Vafa, “Phase structure of a brane/anti-brane system atlarge N,” [arXiv:hep-th/0702077].[31] R. Tatar and B. Wetenhall, “Metastable vacua, geometrical engineering and MQCDtransitions,” JHEP0702