Meta-Stable Brane Configuration of Product Gauge Groups
Meta-Stable Brane Configuration of Product Gauge Groups
Changhyun Ahn
Department of Physics, Kyungpook National University, Taegu 702-701, Korea
ahn@knu.ac.kr
Abstract
Starting from the N = 1 SU(Nc) × SU(N ′c) gauge theory with fundamental and bifun-
damental flavors, we apply the Seiberg dual to the first gauge group and obtain the N = 1
dual gauge theory with dual matters including the gauge singlets. By analyzing the F-term
equations of the superpotential, we describe the intersecting type IIA brane configuration for
the meta-stable nonsupersymmetric vacua of this gauge theory. By introducing an orientifold
6-plane, we generalize to the case for N = 1 SU(Nc)×SO(N ′c) gauge theory with fundamental
and bifundamental flavors. Finally, the N = 1 SU(Nc)× Sp(N ′c) gauge theory with matters
is also described very briefly.
http://arxiv.org/abs/0704.0121v3
1 Introduction
It is well-known that the N = 1 SU(Nc) QCD with fundamental flavors has a vanishing
superpotential before we deform this theory by mass term for quarks. The vanishing su-
perpotential in the electric theory makes it easier to analyze its nonvanishing dual magnetic
superpotential. Sometimes by tuning the various rotation angles between NS5-branes and
D6-branes appropriately, even if the electric theory has nonvanishing superpotential, one can
make the nonzero superpotential to vanish in the electric theory. Two procedures, deforming
the electric gauge theory by adding the mass for the quarks and taking the Seiberg dual
magnetic theory from the electric theory, are crucial to find out meta-stable supersymmetry
breaking vacua in the context of dynamical supersymmetry breaking [1, 2]. Some models
of dynamical supersymmetry breaking can be studied by gauging the subgroup of the flavor
symmetry group by either field theory analysis or using the brane configuration 1.
In this paper, starting from the known N = 1 supersymmetric electric gauge theories, we
construct the N = 1 supersymmetric magnetic gauge theories by brane motion and linking
number counting. The dual gauge group appears only on the first gauge group. Based on
their particular limits of corresponding magnetic brane configurations in the sense that their
electric theories do not have any superpotentials except the mass deformations for the quarks,
we describe the intersecting brane configurations of type IIA string theory for the meta-stable
nonsupersymmetric vacua of these gauge theories.
We focus on the cases where the whole gauge group is given by a product of two gauge
groups. One example can be realized by three NS5-branes with D4- and D6-branes, and the
other by four NS5-branes with D4- and D6-branes. For the latter, the appropriate orientifold
6-plane should be located at the center of this brane configuration in order to have two gauge
groups. Of course, it is also possible, without changing the number of gauge groups, to have
the brane configuration consisting of five NS5-branes and orientifold 6-plane, at which the
extra NS5-brane is located, with D4- and D6-branes, but we’ll not do this particular case in
this paper.
In section 2, we review the type IIA brane configuration that contains three NS5-branes,
corresponding to the electric theory based on the N = 1 SU(Nc) × SU(N ′c) gauge theory
[4, 5, 6] with matter contents and deform this theory by adding the mass term for the quarks.
Then we construct the Seiberg dual magnetic theory which is N = 1 SU(Ñc)×SU(N ′c) gauge
theory with corresponding dual matters as well as various gauge singlets, by brane motion
and linking number counting. We do not touch the part of second gauge group SU(N ′c) in
1For the type IIA brane configuration description of N = 1 supersymmetric gauge theory, see the review
paper [3].
this dual process.
In section 3, we consider the nonsupersymmetric meta-stable minimum by looking at
the magnetic brane configuration we obtained in section 2 and present the corresponding
intersecting brane configuration of type IIA string theory, along the line of [7, 8, 9, 10, 11](see
also [12, 13, 14]) and we describe M-theory lift of this supersymmetry breaking type IIA brane
configuration.
In section 4, we describe the type IIA brane configuration that contains four NS5-branes,
corresponding to the electric theory based on the N = 1 SU(Nc) × SO(N ′c) gauge theory
[15] with matter contents and deform this theory by adding the mass term for the quarks.
Then we take the Seiberg dual magnetic theory which is N = 1 SU(Ñc) × SO(N ′c) gauge
theory with corresponding dual matters as well as various gauge singlets, by brane motion
and linking number counting. The part of second gauge group SO(N ′c) in this dual process is
not changed under this process.
In section 5, the nonsupersymmetric meta-stable minimum by looking at the magnetic
brane configuration we obtained in section 4 is constructed and we present the corresponding
intersecting brane configuration of type IIA string theory and describe M-theory lift of this
supersymmetry breaking type IIA brane configuration, as we did in section 3.
In section 6, we describe the similar application to the N = 1 SU(Nc) × Sp(N ′c) gauge
theory [15] briefly and make some comments for the future directions.
2 The N = 1 supersymmetric brane configuration of
SU(Nc)× SU(N ′c) gauge theory
After reviewing the type IIA brane configuration corresponding to the electric theory based
on the N = 1 SU(Nc)×SU(N ′c) gauge theory [4, 5, 6], we construct the Seiberg dual magnetic
theory which is N = 1 SU(Ñc)× SU(N ′c) gauge theory.
2.1 Electric theory with SU(Nc)× SU(N ′c) gauge group
The gauge group is given by SU(Nc)×SU(N ′c) and the matter contents [4, 5, 6] are given by
• Nf chiral multiplets Q are in the fundamental representation under the SU(Nc), Nf
chiral multiplets Q̃ are in the antifundamental representation under the SU(Nc) and then Q
are in the representation (Nc, 1) while Q̃ are in the representation (Nc, 1) under the gauge
group
• N ′f chiral multiplets Q′ are in the fundamental representation under the SU(N ′c), N ′f
chiral multiplets Q̃′ are in the antifundamental representation under the SU(N ′c) and then Q
are in the representation (1,N′
) while Q̃′ are in the representation (1,N′
) under the gauge
group
• The flavor singlet field X is in the bifundamental representation (Nc,N′c) under the
gauge group and its complex conjugate field X̃ is in the bifundamental representation (Nc,N
under the gauge group
In the electric theory, since there exist Nf quarks Q, Nf quarks Q̃, one bifundamental
field X which will give rise to the contribution of N ′c and its complex conjugate X̃ which will
give rise to the contribution of N ′c, the coefficient of the beta function of the first gauge group
factor is
bSU(Nc) = 3Nc −Nf −N ′c
and similarly since there exist N ′f quarks Q
′, N ′f quarks Q̃
′, one bifundamental field X which
will give rise to the contribution of Nc and its complex conjugate X̃ which will give rise to
the contribution of Nc, the coefficient of the beta function of the second gauge group factor is
bSU(N ′c) = 3N
c −N ′f −Nc.
The anomaly free global symmetry is given by [SU(Nf ) × SU(N ′f )]2 × U(1)3 × U(1)R
[4, 5, 6] and let us denote the strong coupling scales for SU(Nc) as Λ1 and for SU(N
c) as Λ2.
The theory is asymptotically free when bSU(Nc) = 3Nc − Nf − N ′c > 0 for the SU(Nc) gauge
theory and when bSU(N ′c) = 3N
c −N ′f −Nc > 0 for the SU(N ′c) gauge theory.
The type IIA brane configuration for this theory can be described by Nc color D4-branes
(01236) suspended between a middle NS5-brane (012345) and the right NS5’-brane (012389)
(denoted by NS5′R-brane) along x
6 direction, together with Nf D6-branes (0123789) which are
parallel to NS5′R-brane and have nonzero (45) directions. Moreover, an extra left NS5’-brane
(denoted by NS5′L-brane) is located at the left hand side of a middle NS5-brane along the
x6 direction and there exist N ′c color D4-branes suspended between them, with N
f D6-branes
which have zero (45) directions. These are shown in Figure 1 explicitly. See also [3] for the
brane configuration.
By realizing that the two outer NS5′L,R-branes are perpendicular to a middle NS5-brane
and the fact that Nf D6-branes are parallel to NS5
R-brane and N
f D6-branes are parallel
to NS5′L-brane, the classical superpotential vanishes. However, one can deform this theory.
Then the classical superpotential by deforming this theory by adding the mass term for the
quarks Q and Q̃, along the lines of [1, 11, 10, 9, 8, 7], is given by
W = mQQ̃ (2.1)
and this type IIA brane configuration can be summarized as follows 2:
• One middle NS5-brane with worldvolume (012345).
• Two NS5’-branes with worldvolume (012389).
• Nf D6-branes with worldvolume (0123789) located at the positive region in v direction.
• Nc color D4-branes with worldvolume (01236). These are suspended between a middle
NS5-brane and NS5′R-brane.
• N ′c color D4-branes with worldvolume (01236). These are suspended between NS5′L-
brane and a middle NS5-brane.
Now we draw this electric brane configuration in Figure 1 and we put the coincident Nf
D6-branes in the nonzero v direction. If we ignore the left NS5′L-brane, N
c D4-branes and
N ′f D6-branes, then this brane configuration corresponds to the standard N = 1 SQCD with
the gauge group SU(Nc) with Nf massive flavors. The electric quarks Q and Q̃ correspond
to strings stretching between the Nc color D4-branes with Nf D6-branes, the electric quarks
Q′ and Q̃′ correspond to strings between the N ′c color D4-branes with N
f D6-branes and the
bifundamentals X and X̃ correspond to strings stretching between the Nc color D4-branes
with N ′c color D4-branes.
Figure 1: The N = 1 supersymmetric electric brane configuration of SU(Nc)×SU(N ′c) with
Nf chiral multiplets Q, Nf chiral multiplets Q̃, N
f chiral multiplets Q
′, N ′f chiral multiplets
Q̃′, the flavor singlet bifundamental field X and its complex conjugate bifundamental field X̃ .
The Nf D6-branes have nonzero v coordinates where v = m for equal massive case of quarks
Q, Q̃ while Q′ and Q̃′ are massless.
2We introduce two complex coordinates v ≡ x4 + ix5 and w ≡ x8 + ix9 for simplicity.
2.2 Magnetic theory with SU(Ñc)× SU(N ′c) gauge group
One can consider dualizing one of the gauge groups regarding as the other gauge group as a
spectator. One takes the Seiberg dual for the first gauge group factor SU(Nc) while remaining
the second gauge group factor SU(N ′c) unchanged. Also we consider the case where Λ1 >> Λ2,
in other words, the dualized group’s dynamical scale is far above that of the other spectator
group.
Let us move a middle NS5-brane to the right all the way past the right NS5′R-brane. For
example, see [12, 13, 14, 11, 10, 9, 8, 7]. After this brane motion, one arrives at the Figure 2.
Note that there exists a creation of Nf D4-branes connecting Nf D6-branes and NS5
R-brane.
Recall that the Nf D6-branes are perpendicular to a middle NS5-brane in Figure 1. The
linking number [16] of NS5-brane from Figure 2 is L5 =
− Ñc. On the other hand, the
linking number of NS5-brane from Figure 1 is L5 = −Nf2 +Nc−N
c. Due to the connection of
N ′c D4-branes with NS5
R-brane, the presence of N
c in the linking number arises. From these
two relations, one obtains the number of colors of dual magnetic theory
Ñc = Nf +N
c −Nc. (2.2)
The linking number counting looks similar to the one in [7] where there exists a contribution
from O4-plane.
Let us draw this magnetic brane configuration in Figure 2 and recall that we put the
coincident Nf D6-branes in the nonzero v directions in the electric theory, along the lines of
[12, 13, 14, 11, 10, 9, 8, 7]. The Nf created D4-branes connecting between D6-branes and
NS5′R-brane can move freely in the w direction. Moreover since N
c D4-branes are suspending
between two equal NS5′L,R-branes located at different x
6 coordinate, these D4-branes can
slide along the w direction also. If we ignore the left NS5′L-brane, N
c D4-branes and N
D6-branes(detaching these from Figure 2), then this brane configuration corresponds to the
standard N = 1 SQCD with the magnetic gauge group SU(Ñc = Nf −Nc) with Nf massive
flavors [12, 13, 14].
The dual magnetic gauge group is given by SU(Ñc) × SU(N ′c) and the matter contents
are given by
• Nf chiral multiplets q are in the fundamental representation under the SU(Ñc), Nf
chiral multiplets q̃ are in the antifundamental representation under the SU(Ñc) and then q
are in the representation (Ñc, 1) while q̃ are in the representation (Ñc, 1) under the gauge
group
• N ′f chiral multiplets Q′ are in the fundamental representation under the SU(N ′c), N ′f
chiral multiplets Q̃′ are in the antifundamental representation under the SU(N ′c) and then Q
Figure 2: The N = 1 supersymmetric magnetic brane configuration of SU(Ñc = Nf +N ′c −
Nc) × SU(N ′c) with Nf chiral multiplets q, Nf chiral multiplets q̃, N ′f chiral multiplets Q′,
N ′f chiral multiplets Q̃
′, the flavor singlet bifundamental field Y and its complex conjugate
bifundamental field Ỹ as well as Nf fields F
′, its complex conjugate Nf fields F̃ ′, N
f fields
M and the gauge singlet Φ. There exist Nf flavor D4-branes connecting D6-branes and
NS5′R-brane.
are in the representation (1,N′
) while Q̃′ are in the representation (1,N′
) under the gauge
group
• The flavor singlet field Y is in the bifundamental representation (Ñc,N′c) under the gauge
group and its complex conjugate field Ỹ is in the bifundamental representation (Ñc,N
) under
the gauge group
There are (Nf +N
2 gauge singlets in the first dual gauge group factor as follows:
• Nf -fields F ′ are in the fundamental representation under the SU(N ′c), its complex con-
jugate Nf -fields F̃ ′ are in the antifundamental representation under the SU(N
c) and then F
are in the representation (1,N′
) under the gauge group while F̃ ′ are in the representation
(1,N′
) under the gauge group
These additional Nf SU(N
c) fundamentals and Nf SU(N
c) antifundamentals are origi-
nating from the SU(Nc) chiral mesons X̃Q and XQ̃ respectively. It is clear to see that from
the Figure 2, since the Nf D6-branes are parallel to the NS5
R-brane, the newly created Nf
D4-branes can slide along the plane consisting of these D6-branes and NS5′R-brane arbitrar-
ily. Then strings stretching between the Nf D6-branes and N
c D4-branes will give rise to
these additional Nf SU(N
c) fundamentals and Nf SU(N
c) antifundamentals.
• N2f -fields M are in the representation (1, 1) under the gauge group
This corresponds to the SU(Nc) chiral meson QQ̃ and the fluctuations of the singlet M
correspond to the motion of Nf flavor D4-branes along (789) directions in Figure 2.
• The N ′2c -fields Φ is in the representation (1,N′2c − 1)⊕ (1, 1) under the gauge group
This corresponds to the SU(Nc) chiral meson XX̃ and note that X has a representation
of SU(N ′c) while X̃ has a representation N
of SU(N ′c). The fluctuations of the singlet
Φ correspond to the motion of N ′c D4-branes suspended two NS5
L,R-branes along the (789)
directions in Figure 2.
In the dual theory, since there exist Nf quarks q, Nf quarks q̃, one bifundamental field Y
which will give rise to the contribution of N ′c and its complex conjugate Ỹ which will give rise
to the contribution of N ′c, the coefficient of the beta function for the first gauge group factor
[6] is
SU( eNc)
= 3Ñc −Nf −N ′c = 2Nf + 2N ′c − 3Nc
where we inserted the number of colors given in (2.2) in the second equality and since there
exist N ′f quarks Q
′, N ′f quarks Q̃
′, one bifundamental field Y which will give rise to the
contribution of Ñc, its complex conjugate Ỹ which will give rise to the contribution of Ñc, Nf
fields F ′, its complex conjugate Nf fields F̃ ′ and the singlet Φ which will give rise to N
c, the
coefficient of the beta function of second gauge group factor [6] is
SU(N ′c)
= 3N ′c −N ′f − Ñc −Nf −N ′c = N ′c +Nc − 2Nf −N ′f .
Therefore, both SU(Ñc) and SU(N
c) gauge couplings are IR free by requiring the negativeness
of the coefficients of beta function. One can rely on the perturbative calculations at low energy
for this magnetic IR free region b
SU( eNc)
< 0 and b
SU(N ′c)
< 0. Note that the SU(N ′c) fields in
the magnetic theory are different from those of the electric theory. Since bSU(N ′c)−b
SU(N ′c)
SU(N ′c) is more asymptotically free than SU(N
mag [6]. Neglecting the SU(N ′c) dynamics,
the magnetic SU(Ñc) is IR free when Nf +N
Nc [6].
The dual magnetic superpotential, by adding the mass term (2.1) for Q and Q̃ in the
electric theory which is equal to put a linear term in M in the dual magnetic theory, is given
Wdual =
Mqq̃ + Y F ′q̃ + Ỹ qF̃ ′ + ΦY Ỹ
+mM (2.3)
where the mesons in terms of the fields defined in the electric theory are
M ≡ QQ̃, Φ ≡ XX̃, F ′ ≡ X̃Q, F̃ ′ ≡ XQ̃.
By orientifolding procedure(O4-plane) into this brane configuration, the q(Q) and q̃(Q̃) are
equivalent to each other, the Y (X) and Ỹ (X̃) become identical and F ′ and F̃ ′ become the
same. Then the reduced superpotential is identical with the one in [7]. Here q and q̃ are fun-
damental and antifundamental for the gauge group index respectively and antifundamentals
for the flavor index. Then, qq̃ has rank Ñc while m has a rank Nf . Therefore, the F-term
condition, the derivative the superpotential Wdual with respect to M , cannot be satisfied if
the rank Nf exceeds Ñc. This is so-called rank condition and the supersymmetry is broken.
Other F-term equations are satisfied by taking the vacuum expectation values of Y, Ỹ , F ′ and
F̃ ′ to vanish.
The classical moduli space of vacua can be obtained from F-term equations
qq̃ +m = 0, q̃M + F̃ ′Ỹ = 0,
Mq + Y F ′ = 0, F ′q̃ + Ỹ Φ = 0,
q̃Y = 0, qF̃ ′ + ΦY = 0,
Ỹ q = 0, Y Ỹ = 0.
Then, it is easy to see that there exist three reduced equations
q̃M = 0 = Mq, qq̃ +m = 0
and other F-term equations are satisfied if one takes the zero vacuum expectation values for
the fields Y, Ỹ , F ′ and F̃ ′. Then the solutions can be written as follows:
< q > =
meφ1 eNc
, < q̃ >=
me−φ1 eNc 0
, < M >=
0 Φ01Nf− eNc
< Y > = < Ỹ >=< F ′ >=< F̃ ′ >= 0. (2.4)
Let us expand around a point on (2.4), as done in [1]. Then the remaining relevant terms of
superpotential are given by
W reldual = Φ0 (δϕ δϕ̃+m) + δZ δϕ q̃0 + δZ̃ q0δϕ̃
by following the same fluctuations for the various fields as in [9]:
q01 eNc +
(δχ+ + δχ−)1 eNc
, q̃ =
q̃01 eNc +
(δχ+ − δχ−)1 eNc δϕ̃
δY δZ
δZ̃ Φ01Nf− eNc
as well as the fluctuations of Y, Ỹ , F ′ and F̃ ′. Note that there exist also three kinds of terms,
the vacuum < q > multiplied by δỸ δF̃ ′, the vacuum < q̃ > multiplied by δF ′δY , and the
vacuum < Φ > multiplied by δY δỸ . However, by redefining these, they do not enter the
contributions for the one loop result, up to quadratic order. As done in [17], one gets that
m2Φ0 will contain (log 4− 1) > 0 implying that these are stable.
3 Nonsupersymmetric meta-stable brane configuration
of SU(Nc)× SU(N ′c) gauge theory
Now we recombine Ñc D4-branes among Nf flavor D4-branes connecting between D6-branes
and NS5′R-brane with those connecting between NS5
R-brane and NS5-brane and push them
in +v direction from Figure 2. After this procedure, there are no color D4-branes between
NS5′R-brane and NS5-brane. For the flavor D4-branes, we are left with only (Nf − Ñc) flavor
D4-branes.
Then the minimal energy supersymmetry breaking brane configuration is shown in Figure
3, along the lines of [12, 13, 14, 11, 10, 9, 8, 7]. If we ignore the left NS5′L-brane, N
c D4-
branes and N ′f D6-branes(detaching these from Figure 3), as observed already, then this brane
configuration corresponds to the minimal energy supersymmetry breaking brane configuration
for the N = 1 SQCD with the magnetic gauge group SU(Ñc = Nf − Nc) with Nf massive
flavors [12, 13, 14].
Figure 3: The nonsupersymmetric minimal energy brane configuration of SU(Ñc = Nf +
N ′c −Nc)× SU(N ′c) with Nf chiral multiplets q, Nf chiral multiplets q̃, N ′f chiral multiplets
Q′, N ′f chiral multiplets Q̃
′, the flavor singlet bifundamental field Y and its complex conjugate
bifundamental field Ỹ and various gauge singlets.
The type IIA/M-theory brane construction for the N = 2 gauge theory was described
by [18] and after lifting the type IIA description to M-theory, the corresponding magnetic
M5-brane configuration 3 with equal mass for the quarks where the gauge group is given by
3The M5-brane lives in (0123) directions and is wrapping on a Riemann surface inside (4568910) directions.
The Taub-NUT space in (45610) directions is parametrized by two complex variables v and y and the flat two
dimensions in (89) directions by a complex variable w. See [14] for the relevant discussions.
SU(Ñc)×SU(N ′c), in a background space of xt = vN
k=1(v−ek) where this four dimensional
space replaces (45610) directions, is described by
t3 + (v
eNc + · · · )t2 + (vN ′c + · · · )t+ vN ′f
(v − ek) = 0 (3.1)
where ek is the position of the D6-branes in the v direction(for equal massive case, we can
write ek = m) and we have ignored the lower power terms in v in t
2 and t denoted by · · · and
the scales for the gauge groups in front of the first term and the last term, for simplicity. For
fixed x, the coordinate t corresponds to y.
From this curve (3.1) of cubic equation for t above, the asymptotic regions for three NS5-
branes can be classified by looking at the first two terms providing NS5-brane asymptotic
region, next two terms providing NS5′R-brane asymptotic region and the final two terms
giving NS5′L-brane asymptotic region as follows
1. v → ∞ limit implies
w → 0, y ∼ v eNc + · · · NS asymptotic region.
2. w → ∞ limit implies
v → m, y ∼ wNf+N ′f−N ′c + · · · NS ′L asymptotic region,
v → m, y ∼ wN ′c− eNc + · · · NS ′R asymptotic region.
Here the two NS5′L,R-branes are moving in the +v direction holding everything else fixed
instead of moving D6-branes in the +v direction, in the spirit of [14]. The harmonic function
sourced by the D6-branes can be written explicitly by summing over two contributions from
the Nf and N
f D6-branes and similar analysis to both solve the differential equation and
find out the nonholomorphic curve can be done [14, 10, 9, 8, 7]. An instability from a new
M5-brane mode arises.
4 The N = 1 supersymmetric brane configuration of
SU(Nc)× SO(N ′c) gauge theory
After reviewing the type IIA brane configuration corresponding to the electric theory based
on the N = 1 SU(Nc) × SO(N ′c) gauge theory [15], we describe the Seiberg dual magnetic
theory which is N = 1 SU(Ñc)× SO(N ′c) gauge theory.
4.1 Electric theory with SU(Nc)× SO(N ′c) gauge group
The gauge group is given by SU(Nc)× SO(N ′c) and the matter contents [15](similar matter
contents are found in [4]) are given by
• Nf chiral multiplets Q are in the fundamental representation under the SU(Nc), Nf
chiral multiplets Q̃ are in the antifundamental representation under the SU(Nc) and then Q
are in the representation (Nc, 1) while Q̃ are in the representation (Nc, 1) under the gauge
group
• 2N ′f chiral multiplets Q′ are in the fundamental representation under the SO(N ′c) and
then Q′ are in the representation (1,N′
) under the gauge group
• The flavor singlet field X is in the bifundamental representation (Nc,N′c) under the
gauge group and the flavor singlet X̃ is in the bifundamental representation (Nc,N
) under
the gauge group
In the electric theory, since there exist Nf quarks Q, Nf quarks Q̃, one bifundamental
field X which will give rise to the contribution of N ′c and its complex conjugate X̃ which will
give rise to the contribution of N ′c, the coefficient of the beta function of the first gauge group
factor is
bSU(Nc) = 3Nc −Nf −N ′c
and similarly, since there exist 2N ′f quarks Q
′, one bifundamental field X which will give rise
to the contribution of Nc and its complex conjugate X̃ which will give rise to the contribution
of Nc, the coefficient of the beta function of the second gauge group factor is
bSO(N ′c) = 3(N
c − 2)− 2N ′f − 2Nc.
The anomaly free global symmetry is given by SU(Nf )
2 × SU(2N ′f)×U(1)2 ×U(1)R and
let us denote the strong coupling scales for SU(Nc) as Λ1 and for SO(N
c) as Λ2, as in previous
section. The theory is asymptotically free when bSU(Nc) > 0 for the SU(Nc) gauge theory and
when bSO(N ′c) > 0 for the SO(N
c) gauge theory.
The type IIA brane configuration of N = 2 gauge theory [19] consists of four NS5-branes
(012345) which have different x6 values, Nc and N
c D4-branes (01236) suspended between
them, 2Nf and 2N
f D6-branes (0123789) and an orientifold 6 plane (0123789) of positive
Ramond charge 4. According to Z2 symmetry of orientifold 6-plane(O6-plane) sitting at
v = 0 and x6 = 0, the coordinates (x4, x5, x6) transform as −(x4, x5, x6), as usual. See also
[3] for the discussion of O6-plane.
4There are many different brane configurations with O6-plane in the literature and some of them are
present in [20, 21, 22, 23, 24].
By rotating the third and fourth NS5-branes which are located at the right hand side of
O6-plane, from v direction toward −w and +w directions respectively, one obtains N = 1
theory. Their mirrors, the first and second NS5-branes which are located at the left hand
side of O6-plane, can be rotated in a Z2 symmetric manner due to the presence of O6-plane
simultaneously. That is, if the first NS5-brane rotates by an angle −ω in (v, w) plane, denoted
by NS5−ω-brane [3], then the mirror image of this NS5-brane, the fourth NS5-brane, is rotated
by an angle ω in the same plane, denoted by NS5ω-brane. If the second NS5-brane rotates
by an angle θ in (v, w) plane, denoted by NS5θ-brane [3], then the mirror image of this
NS5-brane, the third NS5-brane, is rotated by an angle −θ in the same plane, denoted by
NS5−θ-brane. For more details, see the Figure 4
We also rotate the N ′f D6-branes which are located between the second NS5-brane and
an O6-plane and make them be parallel to NS5θ-brane and denote them as D6θ-brane with
zero v coordinate(the angle between the unrotated D6-branes and D6θ-branes is equal to
− θ) and its mirrors N ′f D6-branes appear as D6−θ-branes between the O6-plane and third
NS5-brane. There is no coupling between the adjoint field and the quarks since the rotated
D6θ-branes are parallel to the rotated NS5θ-brane [5, 3]. Similarly, the Nf D6-branes which
are located between the third NS5-brane and the fourth NS5-brane can be rotated and we
can make them be parallel to NS5ω-brane and denote them as D6ω-branes with nonzero v
coordinate(the angle between the unrotated D6-branes and D6ω-branes is equal to
−ω) and
its mirrors Nf D6-branes appear as D6−ω-branes between the first NS5-brane and the second
NS5-brane.
Moreover the Nc D4-branes are suspended between the first NS5-brane and the second
NS5-brane(and its mirrors) and the N ′c D4-branes are suspended between the second NS5-
brane and the third NS5-brane.
For this brane setup 6, the classical superpotential is given by [15]
W = −1
4 tan(ω − θ) +
tan 2θ
tr(XX̃)2 +
trXX̃X̃X
4 sin 2θ
(trXX̃)2
4Nc tan(ω − θ)
. (4.1)
It is easy to see that when θ approaches 0 and ω approaches π
, then this superpotential
vanishes.
5The angles of θ1 and θ2 in [15] are related to the angles θ and ω as follows: θ = θ1 and ω = θ2.
6For arbitrary angles θ and ω, the superpotential for the SU(Nc) sector is given by W = XφX̃ + tan(ω −
θ) trφ2 where φ ia an adjoint field for SU(Nc). There is no coupling between φ and Nf quarks because
D6±ω-branes are parallel to NS5±ω-branes. The superpotential for the SO(N
c) sector is given by W =
XφAX̃ +XφSX̃ + tan θ trφ
A − 1tan θ trφ
S where φA and φS are an adjoint field and a symmetric tensor for
SO(N ′c) [25]. After integrating out φ, φA and φS , the whole superpotential can be written as in (4.1).
Now one summarizes the supersymmetric electric brane configuration with their worldvol-
umes in type IIA string theory as follows.
• NS5−ω-brane with worldvolume by both (0123) and two spatial dimensions in (v, w)
plane and with negative x6.
• NS5θ-brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane
and with negative x6.
• NS5−θ-brane with worldvolume by both (0123) and two spatial dimensions in (v, w)
plane and with positive x6.
• NS5ω-brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane
and with positive x6.
• N ′f D6θ-branes with worldvolume by both (01237) and two spatial dimensions in (v, w)
plane and with negative x6 and v = 0.
• N ′f D6−θ-branes with worldvolume by both (01237) and two space dimensions in (v, w)
plane and with positive x6 and v = 0.
• Nf D6ω-branes with worldvolume by both (01237) and two spatial dimensions in (v, w)
plane and with positive x6. Before the rotation, the distance from Nc color D4-branes in the
+v direction is nonzero.
• Nf D6−ω-branes with worldvolume by both (01237) and two space dimensions in (v, w)
plane and with negative x6. Before the rotation, the distance from Nc color D4-branes in the
−v direction is nonzero.
• O6-plane with worldvolume (0123789) with v = 0 = x6.
•Nc D4-branes connecting NS5−ω-brane and NS5θ-brane, with worldvolume (01236) with
v = 0 = w(and its mirrors).
• N ′c D4-branes connecting NS5θ-brane and NS5−θ-brane, with worldvolume (01236) with
v = 0 = w.
We draw the type IIA electric brane configuration in Figure 4 which was basically given
in [15] already but the only difference is to put Nf D6-branes in the nonzero v direction in
order to obtain nonzero masses for the quarks which are necessary to obtain the meta-stable
vacua.
4.2 Magnetic theory with SU(Ñc)× SO(N ′c) gauge group
One takes the Seiberg dual for the first gauge group factor SU(Nc) while remaining the second
gauge group factor SO(N ′c), as in previous case. Also we consider the case where Λ1 >> Λ2,
in other words, the dualized group’s dynamical scale is far above that of the other spectator
group.
Figure 4: The N = 1 supersymmetric electric brane configuration of SU(Nc)×SO(N ′c) with
Nf chiral multiplets Q, Nf chiral multiplets Q̃, 2N
f chiral multiplets Q
′, the flavor singlet
bifundamental field X and its complex conjugate bifundamental field X̃ . The Nf D6ω-branes
have nonzero v coordinates where v = m(and its mirrors) for equal massive case of quarks
Q, Q̃ while Q′ is massless.
Let us move the NS5−θ-brane to the right all the way past the right NS5ω-brane(and
its mirrors to the left). After this brane motion, one arrives at the Figure 5. Note that
there exists a creation of Nf D4-branes connecting Nf D6ω-branes and NS5ω-brane(and its
mirrors). Recall that the Nf D6ω-branes are not parallel to the NS5−θ-brane in Figure 4(and
its mirrors). The linking number of NS5−θ-brane from Figure 5 is L5 =
− Ñc. On the
other hand, the linking number of NS5−θ-brane from Figure 4 is L5 = −Nf2 +Nc −N
c. From
these, one gets the number of colors in dual magnetic theory
Ñc = Nf +N
c −Nc. (4.2)
Let us draw this magnetic brane configuration in Figure 5 and remember that we put the
coincident Nf D6ω-branes in the nonzero v directions(and its mirrors). The Nf created D4-
branes connecting between D6ω-branes and NS5ω-brane can move freely in the w direction,
as in previous case. Moreover, since N ′c D4-branes are suspending between two unequal
NS5±ω-branes located at different x
6 coordinate, these D4-branes cannot slide along the w
direction, for arbitrary rotation angles. If we are detaching all the branes except NS5ω-brane,
NS5−θ-brane, D6ω-branes, Nf D4-branes and Ñc D4-branes from Figure 5, then this brane
configuration corresponds to N = 1 SQCD with the magnetic gauge group SU(Ñc = Nf−Nc)
with Nf massive flavors with tilted NS5-branes.
The dual magnetic gauge group is given by SU(Ñc) × SO(N ′c) and the matter contents
are given by
Figure 5: The N = 1 supersymmetric magnetic brane configuration of SU(Ñc = Nf +N ′c −
Nc) × SO(N ′c) with Nf chiral multiplets q, Nf chiral multiplets q̃, 2N ′f chiral multiplets Q′,
the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Ỹ as
well as Nf fields F
′, its complex conjugate Nf fields F̃ ′, N
f fields M and the gauge singlet Φ.
There exist Nf flavor D4-branes connecting D6ω-branes and NS5ω-brane(and its mirrors).
• Nf chiral multiplets q are in the fundamental representation under the SU(Ñc), Nf
chiral multiplets q̃ are in the antifundamental representation under the SU(Ñc) and then q
are in the representation (Ñc, 1) while q̃ are in the representation (Ñc, 1) under the gauge
group
• 2N ′f chiral multiplets Q′ are in the fundamental representation under the SO(N ′c) and
then Q′ are in the representation (1,N′
) under the gauge group
• The flavor singlet field Y is in the bifundamental representation (Ñc,N′c) under the gauge
group and its complex conjugate field Ỹ is in the bifundamental representation (Ñc,N
) under
the gauge group
There are (Nf +N
2 gauge singlets in the first dual gauge group factor as follows:
• Nf -fields F ′ are in the fundamental representation under the SO(N ′c), Nf -fields F̃ ′ are
in the fundamental representation under the SO(N ′c) and then F
′ are in the representation
(1,N′
) under the gauge group while F̃ ′ are in the representation (1,N′
) under the gauge
group
These additional 2Nf SO(N
c) vectors are originating from the SU(Nc) chiral mesons X̃Q
and XQ̃ respectively. It is easy to see that from the Figure 5, since the D6−ω-branes are par-
allel to the NS5−ω-brane, the newly created Nf D4-branes can slide along the plane consisting
of D6−ω-branes and NS5−ω-brane arbitrarily(and its mirrors). Then strings connecting the
Nf D6−ω-branes and N
c D4-branes will give rise to these additional 2Nf SO(N
c) vectors.
• N2f -fields M are in the representation (1, 1) under the gauge group
This corresponds to the SU(Nc) chiral meson QQ̃ and the fluctuations of the singlet M
correspond to the motion of Nf flavor D4-branes along (789) directions in Figure 5.
• The N ′2c singlet Φ is in the representation (1, adj)⊕ (1, symm) under the gauge group
This corresponds to the SU(Nc) chiral meson XX̃ and note that both X and X̃ have
representation N′
of SO(N ′c). In general, the fluctuations of the singlet Φ correspond to the
motion of N ′c D4-branes suspended two NS5±ω-branes along the (789) directions in Figure 5.
In the dual theory, since there exist Nf quarks q, Nf quarks q̃, one bifundamental field Y
which will give rise to the contribution of N ′c and its complex conjugate Ỹ which will give rise
to the contribution of N ′c, the coefficient of the beta function of the first gauge group factor
with (4.2) is
SU( eNc)
= 3Ñc −Nf −N ′c = 2Nf + 2N ′c − 3Nc
and since there exist 2N ′f quarks Q
′, one bifundamental field Y which will give rise to the
contribution of Ñc, its complex conjugate Ỹ which will give rise to the contribution of Ñc, Nf
fields F ′, its complex conjugate Nf fields F̃ ′ and the singlet Φ which will give rise to N
c, the
coefficient of the beta function is
SO(N ′c)
= 3(N ′c − 2)− 2N ′f − 2Ñc − 2Nf − 2N ′c = −N ′c + 2Nc − 4Nf − 2N ′f − 6.
Therefore, both SU(Ñc) and SO(N
c) gauge couplings are IR free by requiring the negativeness
of the coefficients of beta function. One can rely on the perturbative calculations at low energy
for this magnetic IR free region b
SU( eNc)
< 0 and b
SO(N ′c)
< 0. Note that the SO(N ′c) fields in
the magnetic theory are different from those of the electric theory. Since bSO(N ′c)−b
SO(N ′c)
SO(N ′c) is more asymptotically free than SO(N
mag. Neglecting the SO(N ′c) dynamics, the
magnetic SU(Ñc) is IR free when Nf +N
Nc, as in previous case.
The dual magnetic superpotential, by adding the mass term for Q and Q̃ in the electric
theory which is equal to put a linear term in M in the dual magnetic theory, is given by 7
Wdual =
(Φ2 + · · · ) +Q′ΦQ′ +Mqq̃ + Y F̃ ′q̃ + Ỹ qF ′ + ΦY Ỹ
+mM (4.3)
where the mesons in terms of the fields defined in the electric theory are
M ≡ QQ̃, Φ ≡ XX̃, F ′ ≡ X̃Q, F̃ ′ ≡ XQ̃.
7There appears a mismatch between the number of colors from field theory analysis and those from brane
motion when we take the full dual process on the two gauge group factors simultaneously [15]. By adding
D4-branes to the dual brane configuration without affecting the linking number counting, this mismatch
can be removed. Similar phenomena occurred in [5, 26]. Then this turned out that there exists a deformation
∆W generated by the meson Q′XX̃Q′. This is exactly the second term, Q′ΦQ′, in (4.3). In previous example,
there is no such deformation term in (2.3).
We abbreviated all the relevant terms and coefficients appearing in the quartic superpotential
for the bifundamentals in electric theory (4.1) and denote them here by Φ2 + · · · . Here
q and q̃ are fundamental and antifundamental for the gauge group index respectively and
antifundamentals for the flavor index. Then, qq̃ has rank Ñc and m has a rank Nf . Therefore,
the F-term condition, the derivative the superpotential Wdual with respect to M , cannot be
satisfied if the rank Nf exceeds Ñc and the supersymmetry is broken. Other F-term equations
are satisfied by taking the vacuum expectation values of Y, Ỹ , F ′, F̃ ′ and Q′ to vanish.
The classical moduli space of vacua can be obtained from F-term equations and one gets
qq̃ +m = 0, q̃M + F ′Ỹ = 0,
Mq + Y F̃ ′ = 0, F̃ ′q̃ + Ỹ Φ = 0,
q̃Y = 0, qF ′ + ΦY = 0,
Ỹ q = 0, Q′Q′ + Y Ỹ = 0,
ΦQ′ = 0.
Then, it is easy to see that there exists a solution
q̃M = 0 = Mq, qq̃ +m = 0.
Other F-term equations are satisfied if one takes the zero vacuum expectation values for the
fields Y, Ỹ , F ′, Q′ and F̃ ′. Then the solutions can be written as
< q > =
meφ1 eNc
, < q̃ >=
me−φ1 eNc 0
, < M >=
0 Φ01Nf− eNc
< Y > = < Ỹ >=< F ′ >=< F̃ ′ >=< Q′ >= 0. (4.4)
Let us expand around a point on (4.4), as done in [1]. Then the remaining relevant terms of
superpotential are given by
W reldual = Φ0 (δϕ δϕ̃+m) + δZ δϕ q̃0 + δZ̃ q0δϕ̃
by following the similar fluctuations for the various fields as in [9]. Note that there exist also
four kinds of terms, the vacuum < q > multiplied by δỸ δF ′, the vacuum < q̃ > multiplied
by δF̃ ′δY , the vacuum < Φ > multiplied by δY δỸ , and the vacuum < Φ > multiplied by
δQ′δQ′. However, by redefining these, they do not enter the contributions for the one loop
result, up to quadratic order. As done in [17], one gets that m2Φ0 will contain (log 4 − 1) > 0
implying that these are stable.
5 Nonsupersymmetric meta-stable brane configuration
of SU(Nc)× SO(N ′c) gauge theory
Since the electric superpotential (4.1) vanishes for θ = 0 and ω = π
, the corresponding
magnetic superpotential in (4.3) does not contain the terms Φ2 + · · · and it becomes
Wdual =
Q′ΦQ′ +Mqq̃ + Y F̃ ′q̃ + Ỹ qF ′ + ΦY Ỹ
Now we recombine Ñc D4-branes among Nf flavor D4-branes connecting between D6ω=π
D6-branes and NS5ω=π
= NS5′R-brane with those connecting between NS5
R-brane and
NS5−θ=0 = NS5R-brane(and its mirrors) and push them in +v direction from Figure 5. Of
course their mirrors will move to−v direction in a Z2 symmetric manner due to the O6+-plane.
After this procedure, there are no color D4-branes between NS5′R-brane and NS5R-brane.
For the flavor D4-branes, we are left with only (Nf − Ñc) D4-branes(and its mirrors).
Then the minimal energy supersymmetry breaking brane configuration is shown in Figure
6. If we ignore all the branes except NS5′R-brane, NS5R-brane, D6-branes, (Nf − Ñc) D4-
branes and Ñc D4-branes, as observed already, then this brane configuration corresponds
to the minimal energy supersymmetry breaking brane configuration for the N = 1 SQCD
with the magnetic gauge group SU(Ñc) with Nf massive flavors [12, 13, 14]. Note that N
D4-branes can slide w direction for this brane configuration.
The type IIA/M-theory brane construction for the N = 2 gauge theory was described by
[19] and after lifting the type IIA description we explained so far to M-theory, the correspond-
ing magnetic M5-brane configuration with equal mass for the quarks where the gauge group
is given by SU(Ñc)×SO(N ′c), in a background space of xt = (−1)Nf+N
k=1(v
2− e2k)
where this four dimensional space replaces (45610) directions, is characterized by
t4 + (v
eNc + · · · )t3 + (vN ′c + · · · )t2 + (v eNc + · · · )t+ v2N ′f+4
(v2 − e2k) = 0.
From this curve of quartic equation for t above, the asymptotic regions can be classified
by looking at the first two terms providing NS5R-brane asymptotic region, next two terms
providing NS5′R-brane asymptotic region, next two terms providing NS5
L-brane asymptotic
region, and the final two terms giving NS5L-brane asymptotic region as follows:
1. v → ∞ limit implies
w → 0, y ∼ v eNc + · · · NS5R asymptotic region,
w → 0, y ∼ v2Nf+2N ′f− eNc+4 + · · · NS5L asymptotic region.
Figure 6: The nonsupersymmetric minimal energy brane configuration of SU(Ñc = Nf +
N ′c −Nc)× SO(N ′c) with Nf chiral multiplets q, Nf chiral multiplets q̃, 2N ′f chiral multiplets
Q′, the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Ỹ
and gauge singlets. The N ′c D4-branes and 2(Nf − Ñc) D4-branes can slide w direction freely
in a Z2 symmetric way.
2. w → ∞ limit implies
v → −m, y ∼ w eNc−N ′c + · · · NS5′L asymptotic region,
v → +m, y ∼ wN ′c− eNc + · · · NS5′R asymptotic region.
Now the two NS5′L,R-branes are moving in the ±v direction holding everything else fixed
instead of moving D6-branes in the ±v direction. Then the mirrors of D4-branes are moved
appropriately. The harmonic function sourced by the D6-branes can be written explicitly by
summing of three contributions from the Nf and N
f D6-branes(and its mirrors) plus an O6-
plane, and similar analysis to solve the differential equation and find out the nonholomorphic
curve can be done [14, 10, 9, 8, 7]. In this case also, we expect an instability from a new
M5-brane mode.
6 Discussions
So far, we have dualized only the first gauge group factor in the gauge group SU(Nc)×SO(N ′c).
What happens if we dualize the second gauge group factor SO(N ′c)?(For the case SU(Nc)×
SU(N ′c), the behavior of dual for the second gauge group will be the same as when we take
the dual for the first gauge group factor.) This can be done by moving the NS5θ-brane and
N ′f D6θ-branes that can be located at the nonzero v coordinate for massive quarks Q
′, to
the right passing through O6-plane(and their mirrors to the left). According to the linking
number counting, one obtains the dual gauge group SU(Nc)×SO(Ñ ′c = 2Nc+2N ′f −N ′c+4).
One can easily see that there is a creation of N ′f D4-branes connecting NS5θ-brane and
D6θ-branes(and its mirrors). Then from the brane configuration, there exist the additional
2N ′f SU(Nc) quarks originating from the SO(N
c) chiral mesons Q
′X ≡ F̃ ′ and Q′X̃ ≡ F ′.
The deformed superpotential ∆W = Q′XX̃Q′ can be interpreted as the mass term of F ′F̃ ′.
Then one can write dual magnetic superpotential in this case. However, it is not clear how
the recombination of color and flavor D4-branes and splitting procedure between them in
the construction of meta-stable vacua arises since there is no extra NS5-brane between two
NS5±θ-branes. If there exists an extra NS5-brane at the origin of our brane configuration(then
the gauge group and matter contents will change), it would be possible to construct the
corresponding meta-stable brane configuration. It would be interesting to study these more
in the future.
As already mentioned in [8] and section 4, the matter contents in [4] are different from the
ones in section 4 with the same gauge group. In other words, the theory of SU(Nc)×SO(N ′c)
with X , which transform as fundamental in SU(Nc) and vector in SO(N
c), a antisymmetric
tensor A in SU(Nc), as well as fundamentals for SU(Nc) and vectors for SO(N
c) can confine
either SU(Nc) factor or SO(N
c) factor. This theory can be described by the web of branes in
the presence of O4−-plane and orbifold fixed points. With two NS5-branes and O4−-plane, by
modding out Z3 symmetry acting on (v, w) as (v, w) → (v exp(2πi3 ), w exp(
)), the resulting
gauge group will be SU(Nc)×SO(Nc+4) with above matter contents [27]. Similar analysis for
SU(Nc)×Sp(Nc2 −2) gauge group with opposite O4
+-plane can be done. Then in this case, the
matter in SU(Nc) will be a symmetric tensor S and other matter contents are present also. It
would be interesting to see whether this gauge theory and corresponding brane configuration
will provide a meta-stable vacuum.
Let us comment on other possibility where the gauge group is given by SU(Nc)× Sp(N ′c)
and the matter contents are given by
• Nf chiral multiplets Q are in the fundamental representation under the SU(Nc), Nf
chiral multiplets Q̃ are in the antifundamental representation under the SU(Nc) and then Q
are in the representation (Nc, 1) while Q̃ are in the representation (Nc, 1) under the gauge
group
• 2N ′f chiral multiplets Q′ are in the fundamental representation under the Sp(N ′c) and
then Q′ are in the representation (1, 2N′
) under the gauge group
• The flavor singlet field X is in the bifundamental representation (Nc, 2N′c) under the
gauge group and the flavor singlet X̃ is in the bifundamental representation (Nc, 2N
) under
the gauge group
One can compute the coefficients of beta functions of the each gauge group factor, as we
did for previous examples.
The type IIA brane configuration of an electric theory is exactly the same as the Figure
4 except the RR charge O6-plane with negative sign. The classical superpotential 8 is given
by [15]
W = −1
4 tan(ω − θ) +
tan 2θ
tr(XX̃)2 − trXX̃X̃X
4 sin 2θ
(trXX̃)2
4Nc tan(ω − θ)
. (6.1)
In this case, when θ approaches π
and ω approaches 0, then this superpotential vanishes.
The dual magnetic gauge group is given by SU(Ñc = Nf + 2N
c −Nc)× Sp(N ′c) with the
same number of colors of dual theory as those in previous cases and the matter contents are
given by
• Nf chiral multiplets q are in the fundamental representation under the SU(Ñc), Nf
chiral multiplets q̃ are in the antifundamental representation under the SU(Ñc) and then q
are in the representation (Ñc, 1) while q̃ are in the representation (Ñc, 1) under the gauge
group
• 2N ′f chiral multiplets Q′ are in the fundamental representation under the Sp(N ′c) and
then Q′ are in the representation (1, 2N′
) under the gauge group
• The flavor singlet field Y is in the bifundamental representation (Ñc, 2N′c) under the
gauge group and its complex conjugate field Ỹ is in the bifundamental representation (Ñc, 2N
under the gauge group
There are (Nf + 2N
2 gauge singlets in the first dual gauge group factor
• Nf -fields F ′ are in the fundamental representation under the Sp(N ′c), Nf -fields F̃ ′ are
in the fundamental representation under the Sp(N ′c) and then F
′ are in the representation
(1, 2N′
) under the gauge group while F̃ ′ are in the representation (1, 2N′
) under the gauge
group
• N2f -fields M are in the representation (1, 1) under the gauge group
• The 4N ′2c singlet Φ is in the representation (1, adj)⊕ (1, antisymm) under the gauge
group
The dual magnetic superpotential for arbitrary angles is given by (4.3) with appropriate
Sp(N ′c) invariant metric J . The stability analysis can be done similarly.
8The superpotential for the Sp(N ′c) sector is given by W = XφAX̃+XφSX̃+tan θ trφ
S− 1tan θ trφ
A where
φS and φA are an adjoint field(symmetric tensor) and an antisymmetric tensor for Sp(N
c) [25]. Note that
there is a sign change in the second trace term of the superpotential in (6.1), compared to (4.1).
After following the procedure from Figure 4 to Figure 5 with opposite RR charge for O6-
plane and by taking the limit where θ → π
and ω → 0, the minimal energy supersymmetry
breaking brane configuration is shown in Figure 7.
Figure 7: The nonsupersymmetric minimal energy brane configuration of SU(Ñc = Nf +
2N ′c −Nc)×Sp(N ′c) with Nf chiral multiplets q, Nf chiral multiplets q̃, 2N ′f chiral multiplets
Q′, the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Ỹ
and gauge singlets. Note the RR charge of O6-plane is negative and its charge is equivalent
to −4 D6-branes. The 2N ′c D4-branes and 2(Nf − Ñc) D4-branes can slide w direction freely
in a Z2 symmetric way.
Compared to the previous nonsupersymmetric brane configuration in Figure 6, the role
of NS5-brane and NS5’-brane is interchanged to each other: undoing the Seiberg dual in the
context of [13]. This kind of feature of recombination and splitting between color D4-branes
and flavor D4-branes occurs in [8]. At the electric brane configuration, Nf D6-branes are
perpendicular to NS5-brane and this leads to the coupling between the quarks and adjoint
in the superpotential. However, the overall coefficient function including this extra terms
vanishes and eventually the whole electric superpotential will vanish according to the above
limit we take.
From the quartic equation with the presence of opposite RR charge for O6-plane, in a
background space of xt = (−1)Nf+N ′fv2N ′f−4
k=1(v
2 − e2k),
t4 + (v
eNc + · · · )t3 + (vN ′c + · · · )t2 + (v eNc + · · · )t+ v2N ′f−4
(v2 − e2k) = 0,
the asymptotic regions can be classified as follows:
1. v → ∞ limit implies
w → 0, y ∼ vN ′c− eNc · · · NS5R asymptotic region,
w → 0, y ∼ v eNc−N ′c + · · · NS5L asymptotic region.
2. w → ∞ limit implies
v → −m, y ∼ w2Nf+2N ′f− eNc−4 + · · · NS5′L asymptotic region,
v → +m, y ∼ w eNc + · · · NS5′R asymptotic region.
In [28], the SU(7)×S̃p(1) model and SU(9)×S̃p(2) model can be obtained by dualizing the
SU(7)× SU(2) model with a bifundamental and two antifundamentals for SU(7) and a fun-
damental for SU(2) and the SU(9)×SU(2) with a bifundamental and two antifundamentals
for SU(9) and a fundamental for Sp(1) respectively(Note that Sp(1) ∼ SU(2)). The matter
contents in an electric theory are different from those in previous paragraph. The matter
contents in the magnetic description are given by an antisymmetric tensor and a fundamen-
tal in the first gauge group as well as a bifundamental, a fundamental in the second gauge
group and two antifundamentals in the first gauge group. There exists a nonzero dual mag-
netic superpotential. Also the dual description the SU(7)× S̃p(1) model and SU(9)× S̃p(2)
model can be constructed from the antisymmetric models of Affleck-Dine-Seiberg by gauging
a maximal flavor symmetry and adding the extra matter to cancel all anomalies and extra
flavor.
On the other hand, the models SU(2Nc + 1)× SU(2) have its brane box model descrip-
tion in [29] where the above examples correspond to Nc = 3 and Nc = 4 respectively. In
particular, the case where Nc = 1(the gauge group is SU(3) × SU(2), i.e., (3, 2) model [30])
was described by brane box model with superpotential or without superpotential. Then it
would be interesting to obtain the Seiberg dual for these models using brane box model
and look for the possibility of having meta-stable vacua for these models. Moreover, this
gauge theory was generalized to SU(2Nc + 1) × Sp(N ′c) model with a bifundamental and
2N ′c antifundamentals for SU(2Nc + 1) and a fundamental for Sp(N
c) and its dual descrip-
tion SU(2Nc + 1)× Sp(Ñ ′c = Nc −N ′c − 1) with a bifundamental and 2N ′c antifundamentals
for SU(2Nc + 1) and a fundamental for Sp(N
c) as well as two gauge singlets [28]. For the
particular range of Nc, the dual theory is IR free, not asymptotically free.
According to [31], SU(2Nc) with antisymmetric tensor and antifundamentals can be de-
scribed by two gauge groups Sp(2Nc−4)×SU(2Nc) with bifundamental and antifundamentals
for SU(2Nc). Some of the brane realization with zero superpotential was given in the brane
box model in [29]. Similarly from the result of [32] by following the method of [31], the dual
description for SU(2Nc +1) with antisymmetric tensor and fundamentals can be represented
by two gauge group factors. This dual theory breaks the supersymmetry at the tree level.
Similar discussions are present in [33]. Then it would be interesting to construct the corre-
sponding Seigerg dual and see how the electric theory and its magnetic theory can be mapped
into each other in the brane box model.
Ther are also different directions concerning on the meta-stable vacua in different contexts
and some of the relevant works are present in [34]-[43] where some of them use anti D-branes
and some of them describe the type IIB theory and it would be interesting to find out how
similarities if any appear and what are the differences in what sense between the present work
and those works.
Acknowledgments
I would like to thank A. Hanany and K. Landsteiner for discussions. This work was
supported by grant No. R01-2006-000-10965-0 from the Basic Research Program of the Korea
Science & Engineering Foundation.
References
[1] K. Intriligator, N. Seiberg and D. Shih, “Dynamical SUSY breaking in meta-stable
vacua,” JHEP 0604, 021 (2006) [arXiv:hep-th/0602239].
[2] K. Intriligator and N. Seiberg, “Lectures on supersymmetry breaking,”
[arXiv:hep-ph/0702069].
[3] A. Giveon and D. Kutasov, “Brane dynamics and gauge theory,” Rev. Mod. Phys. 71,
983 (1999) [arXiv:hep-th/9802067].
[4] K. A. Intriligator, R. G. Leigh and M. J. Strassler, “New examples of duality in chi-
ral and nonchiral supersymmetric gauge theories,” Nucl. Phys. B 456, 567 (1995)
[arXiv:hep-th/9506148].
[5] J. H. Brodie and A. Hanany, “Type IIA superstrings, chiral symmetry, and N = 1 4D
gauge theory dualities,” Nucl. Phys. B 506, 157 (1997) [arXiv:hep-th/9704043].
[6] E. Barnes, K. Intriligator, B. Wecht and J. Wright, “N = 1 RG flows, product groups,
and a-maximization,” Nucl. Phys. B 716, 33 (2005) [arXiv:hep-th/0502049].
http://arxiv.org/abs/hep-th/0602239
http://arxiv.org/abs/hep-ph/0702069
http://arxiv.org/abs/hep-th/9802067
http://arxiv.org/abs/hep-th/9506148
http://arxiv.org/abs/hep-th/9704043
http://arxiv.org/abs/hep-th/0502049
[7] C. Ahn, “Meta-Stable Brane Configuration and Gauged Flavor Symmetry,”
[arXiv:hep-th/0703015].
[8] C. Ahn, “More on meta-stable brane configuration,” [arXiv:hep-th/0702038].
[9] C. Ahn, “Meta-stable brane configuration with orientifold 6 plane,”
[arXiv:hep-th/0701145], to appear in JHEP.
[10] C. Ahn, “M-theory lift of meta-stable brane configuration in symplectic and orthogonal
gauge groups,” Phys. Lett. B 647, 493 (2007) [arXiv:hep-th/0610025].
[11] C. Ahn, “Brane configurations for nonsupersymmetric meta-stable vacua in SQCD with
adjoint matter,” Class. Quant. Grav. 24, 1359 (2007) [arXiv:hep-th/0608160].
[12] H. Ooguri and Y. Ookouchi, “Meta-stable supersymmetry breaking vacua on intersecting
branes,” Phys. Lett. B 641, 323 (2006) [arXiv:hep-th/0607183].
[13] S. Franco, I. Garcia-Etxebarria and A. M. Uranga, “Non-supersymmetric meta-stable
vacua from brane configurations,” JHEP 0701, 085 (2007) [arXiv:hep-th/0607218].
[14] I. Bena, E. Gorbatov, S. Hellerman, N. Seiberg and D. Shih, “A note on (meta)stable
brane configurations in MQCD,” JHEP 0611, 088 (2006) [arXiv:hep-th/0608157].
[15] E. Lopez and B. Ormsby, “Duality for SU x SO and SU x Sp via branes,” JHEP 9811,
020 (1998) [arXiv:hep-th/9808125].
[16] A. Hanany and E. Witten, “Type IIB superstrings, BPS monopoles, and three-
dimensional gauge dynamics,” Nucl. Phys. B 492, 152 (1997) [arXiv:hep-th/9611230].
[17] D. Shih, “Spontaneous R-Symmetry Breaking in O’Raifeartaigh Models,”
[arXiv:hep-th/0703196].
[18] E. Witten, “Solutions of four-dimensional field theories via M-theory,” Nucl. Phys. B
500, 3 (1997) [arXiv:hep-th/9703166].
[19] K. Landsteiner and E. Lopez, “New curves from branes,” Nucl. Phys. B 516, 273 (1998)
[arXiv:hep-th/9708118].
[20] K. Landsteiner, E. Lopez and D. A. Lowe, “Supersymmetric gauge theories from branes
and orientifold six-planes,” JHEP 9807, 011 (1998) [arXiv:hep-th/9805158].
http://arxiv.org/abs/hep-th/0703015
http://arxiv.org/abs/hep-th/0702038
http://arxiv.org/abs/hep-th/0701145
http://arxiv.org/abs/hep-th/0610025
http://arxiv.org/abs/hep-th/0608160
http://arxiv.org/abs/hep-th/0607183
http://arxiv.org/abs/hep-th/0607218
http://arxiv.org/abs/hep-th/0608157
http://arxiv.org/abs/hep-th/9808125
http://arxiv.org/abs/hep-th/9611230
http://arxiv.org/abs/hep-th/0703196
http://arxiv.org/abs/hep-th/9703166
http://arxiv.org/abs/hep-th/9708118
http://arxiv.org/abs/hep-th/9805158
[21] C. Ahn, K. Oh and R. Tatar, “Comments on SO/Sp gauge theories from brane configu-
rations with an O(6) plane,” Phys. Rev. D 59, 046001 (1999) [arXiv:hep-th/9803197].
[22] K. Landsteiner, E. Lopez and D. A. Lowe, “Duality of chiral N = 1 supersymmetric
gauge theories via branes,” JHEP 9802, 007 (1998) [arXiv:hep-th/9801002].
[23] I. Brunner, A. Hanany, A. Karch and D. Lust, “Brane dynamics and chiral non-chiral
transitions,” Nucl. Phys. B 528, 197 (1998) [arXiv:hep-th/9801017].
[24] S. Elitzur, A. Giveon, D. Kutasov and D. Tsabar, “Branes, orientifolds and chiral gauge
theories,” Nucl. Phys. B 524, 251 (1998) [arXiv:hep-th/9801020].
[25] C. Csaki, M. Schmaltz, W. Skiba and J. Terning, “Gauge theories with tensors from
branes and orientifolds,” Phys. Rev. D 57, 7546 (1998) [arXiv:hep-th/9801207].
[26] C. Ahn, K. Oh and R. Tatar, “Branes, geometry and N = 1 duality with product gauge
groups of SO and Sp,” J. Geom. Phys. 31, 301 (1999) [arXiv:hep-th/9707027].
[27] J. D. Lykken, E. Poppitz and S. P. Trivedi, “M(ore) on chiral gauge theories from D-
branes,” Nucl. Phys. B 520, 51 (1998) [arXiv:hep-th/9712193].
[28] K. A. Intriligator and S. D. Thomas, “Dual descriptions of supersymmetry breaking,”
[arXiv:hep-th/9608046].
[29] A. Hanany and A. Zaffaroni, “On the realization of chiral four-dimensional gauge theories
using branes,” JHEP 9805, 001 (1998) [arXiv:hep-th/9801134].
[30] I. Affleck, M. Dine and N. Seiberg, “Dynamical Supersymmetry Breaking In Four-
Dimensions And Its Phenomenological Implications,” Nucl. Phys. B 256, 557 (1985).
[31] M. Berkooz, “The Dual of supersymmetric SU(2k) with an antisymmetric tensor and
composite dualities,” Nucl. Phys. B 452, 513 (1995) [arXiv:hep-th/9505067].
[32] P. Pouliot, “Duality in SUSY SU(N) with an Antisymmetric Tensor,” Phys. Lett. B
367, 151 (1996) [arXiv:hep-th/9510148].
[33] P. Pouliot and M. J. Strassler, “Duality and Dynamical Supersymmetry Breaking in
Spin(10) with a Spinor,” Phys. Lett. B 375, 175 (1996) [arXiv:hep-th/9602031].
[34] S. Murthy, “On supersymmetry breaking in string theory from gauge theory in a throat,”
[arXiv:hep-th/0703237].
http://arxiv.org/abs/hep-th/9803197
http://arxiv.org/abs/hep-th/9801002
http://arxiv.org/abs/hep-th/9801017
http://arxiv.org/abs/hep-th/9801020
http://arxiv.org/abs/hep-th/9801207
http://arxiv.org/abs/hep-th/9707027
http://arxiv.org/abs/hep-th/9712193
http://arxiv.org/abs/hep-th/9608046
http://arxiv.org/abs/hep-th/9801134
http://arxiv.org/abs/hep-th/9505067
http://arxiv.org/abs/hep-th/9510148
http://arxiv.org/abs/hep-th/9602031
http://arxiv.org/abs/hep-th/0703237
[35] R. Argurio, M. Bertolini, S. Franco and S. Kachru, “Metastable vacua and D-branes at
the conifold,” [arXiv:hep-th/0703236].
[36] A. Giveon and D. Kutasov, “Gauge symmetry and supersymmetry breaking from inter-
secting branes,” [arXiv:hep-th/0703135].
[37] Y. E. Antebi and T. Volansky, “Dynamical supersymmetry breaking from simple quiv-
ers,” [arXiv:hep-th/0703112].
[38] M. Wijnholt, “Geometry of particle physics,” [arXiv:hep-th/0703047].
[39] J. J. Heckman, J. Seo and C. Vafa, “Phase structure of a brane/anti-brane system at
large N,” [arXiv:hep-th/0702077].
[40] R. Tatar and B. Wetenhall, “Metastable vacua, geometrical engineering and MQCD
transitions,” JHEP 0702, 020 (2007) [arXiv:hep-th/0611303].
[41] H. Verlinde, “On metastable branes and a new type of magnetic monopole,”
[arXiv:hep-th/0611069].
[42] M. Aganagic, C. Beem, J. Seo and C. Vafa, “Geometrically induced metastability and
holography,” [arXiv:hep-th/0610249].
[43] R. Argurio, M. Bertolini, S. Franco and S. Kachru, “Gauge / gravity dual-
ity and meta-stable dynamical supersymmetry breaking,” JHEP 0701, 083 (2007)
[arXiv:hep-th/0610212].
http://arxiv.org/abs/hep-th/0703236
http://arxiv.org/abs/hep-th/0703135
http://arxiv.org/abs/hep-th/0703112
http://arxiv.org/abs/hep-th/0703047
http://arxiv.org/abs/hep-th/0702077
http://arxiv.org/abs/hep-th/0611303
http://arxiv.org/abs/hep-th/0611069
http://arxiv.org/abs/hep-th/0610249
http://arxiv.org/abs/hep-th/0610212
	Introduction
	The N=1 supersymmetric brane configuration of SU(Nc) SU(Nc') gauge theory
	Electric theory with SU(Nc) SU(Nc') gauge group
	Magnetic theory with SU(N"0365Nc) SU(Nc') gauge group
	Nonsupersymmetric meta-stable brane configuration of SU(Nc) SU(Nc') gauge theory 
	The N=1 supersymmetric brane configuration of SU(Nc) SO(Nc') gauge theory 
	Electric theory with SU(Nc) SO(Nc') gauge group 
	Magnetic theory with SU(N"0365Nc) SO(Nc') gauge group
	Nonsupersymmetric meta-stable brane configuration of SU(Nc) SO(Nc') gauge theory 
	Discussions
