Supersymmetry breaking metastable vacua in runaway quiver gauge theories
IFT-UAM/CSIC-07-14
CERN-PH-TH/2007-063
Supersymmetry breaking metastable vacua
in runaway quiver gauge theories
I. Garćıa-Etxebarria, F. Saad, A.M.Uranga
PH-TH Division, CERN CH-1211 Geneva 23, Switzerland
Instituto de F́ısica Teórica C-XVI,
Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
Abstract
In this paper we consider quiver gauge theories with fractional branes whose infrared dy-
namics removes the classical supersymmetric vacua (DSB branes). We show that addition
of flavors to these theories (via additional non-compact branes) leads to local meta-stable
supersymmetry breaking minima, closely related to those of SQCD with massive flavors.
We simplify the study of the one-loop lifting of the accidental classical flat directions by di-
rect computation of the pseudomoduli masses via Feynman diagrams. This new approach
allows to obtain analytic results for all these theories. This work extends the results for
the dP1 theory in hep-th/0607218. The new approach allows to generalize the computa-
tion to general examples of DSB branes, and for arbitrary values of the superpotential
couplings.
http://arxiv.org/abs/0704.0166v2
http://arxiv.org/abs/hep-th/0607218
1 Introduction
Systems of D-branes at singularities provide a very interesting setup to realize and study
diverse non-perturbative gauge dynamics phenomena in string theory. In the context
of N = 1 supersymmetric gauge field theories, systems of D3-branes at Calabi-Yau
singularities lead to interesting families of tractable 4d strongly coupled conformal field
theories, which extend the AdS/CFT correspondence [1, 2, 3] to theories with reduced
(super)symmetry [4, 5, 6] and enable non-trivial precision tests of the correspondence
(see for instance [7, 8]). Addition of fractional branes leads to families of non-conformal
gauge theories, with intricate RG flows involving cascades of Seiberg dualities [9, 10,
11, 12, 13], and strong dynamics effects in the infrared.
For instance, fractional branes associated to complex deformations of the singular
geometry (denoted deformation fractional branes in [12]), correspond to supersym-
metric confinement of one or several gauge factors in the gauge theory [9, 12]. The
generic case of fractional branes associated to obstructed complex deformations (de-
noted DSB branes in [12]), corresponds to gauge theories developing a non-perturbative
Affleck-Dine-Seiberg superpotential, which removes the classical supersymmetric vacua
[14, 15, 16]. As shown in [15] (see also [17, 18]), assuming canonical Kahler potential
leads to a runaway potential for the theory, along a baryonic direction. A natural
suggestion to stop this runaway has been proposed for the particular example of the
dP1 theory (the theory on fractional branes at the complex cone over dP1) in [19]. It
was shown that, upon the addition of D7-branes to the configuration (which intro-
duce massive flavors), the theory develops a meta-stable minimum (closely related to
the Intriligator-Seiberg-Shih (ISS) model [20]), parametrically long-lived against decay
to the runaway regime (see [21] for an alternative suggestion to stop the runaway, in
compact models).
In this paper we show that the appearance of meta-stable minima in gauge theories
on DSB fractional branes, in the presence of additional massless flavors, is much more
general (and possibly valid in full generality). We use the tools of [15] to introduce
D7-branes on general toric singularities, and give masses to the corresponding flavors.
Since quiver gauge theories are rather involved, we develop new techniques to efficiently
analyze the one-loop stability of the meta-stable minima, via the direct computation
of Feynman diagrams. These tools can be used to argue that the results plausibly
hold for general systems of DSB fractional branes at toric singularities. It is very
satisfactory to verify the correspondence between the existence of meta-stable vacua
and the geometric property of having obstructed complex deformations.
The present work thus enlarges the class of string models realizing dynamical su-
persymmetry breaking in meta-stable vacua (see [22, 23, 24, 25, 26] for other proposed
realizations, and [27, 28, 29] for models of dynamical supersymmetry breaking in ori-
entifold theories). Although we will not discuss it in the present paper, these results
can be applied to the construction of models of gauge mediation in string theory as
in [30] (based on the additional tools in [31]), in analogy with [32]. This is another
motivation for the present work.
The paper is organized as follows. In Section 2 we review the ISS model, evaluating
one-loop pseudomoduli masses directly in terms of Feynman diagrams. In Section 3
we study the theory of DSB branes at the dP1 and dP2 singularities upon the addition
of flavors, and we find that metastable vacua exist for these theories. In Section 4
we extend this analysis to the general case of DSB branes at toric singularities with
massive flavors, and we illustrate the results by showing the existence of metastable
vacua for DSB branes at some well known families of toric singularities. Finally, the
Appendix provides some technical details that we have omitted from the main text in
order to improve the legibility.
2 The ISS model revisited
In this Section we review the ISS meta-stable minima in SQCD, and propose that the
analysis of the relevant piece of the one-loop potential (the quadratic terms around the
maximal symmetry point) is most simply carried out by direct evaluation of Feynman
diagrams. This new tool will be most useful in the study of the more involved examples
of quiver gauge theories.
2.1 The ISS metastable minimum
The ISS model [20] (see also [33] for a review of these and other models) is given by
N = 1 SU(Nc) theory with Nf flavors, with small masses
Welectric = mTrφφ̃, (2.1)
where φ and φ̃ are the quarks of the theory. The number of colors and flavors are
chosen so as to be in the free magnetic phase:
Nc + 1 ≤ Nf <
Nc. (2.2)
This condition guarantees that the Seiberg dual is infrared free. This Seiberg dual is
the SU(N) theory (with N = Nf −Nc) with Nf flavors of dual quarks q and q̃ and the
meson M . The dual superpotential is given by rewriting (2.1) in terms of the mesons
and adding the usual coupling between the meson and the dual quarks:
Wmagnetic = h (Tr q̃Mq − µ
2TrM), (2.3)
where h and µ can be expressed in terms of the parameters m and Λ, and some
(unknown) information about the dual Kähler metric1. It was also argued in [20] that
it is possible to study the supersymmetry breaking minimum in the origin of (dual)
field space without taking into account the gauge dynamics (their main effect in this
discussion consists of restoring supersymmetry dynamically far in field space). In the
following we will assume that this is always the case, and we will forget completely
about the gauge dynamics of the dual.
Once we forget about gauge dynamics, studying the vacua of the dual theory be-
comes a matter of solving the F-term equations coming from the superpotential (2.3).
The mesonic F-term equation reads:
− FMij = hq̃i · qj − hµ2δij = 0, (2.4)
where i and j are flavor indices and the dot denotes color contraction. This has no
solution, since the identity matrix δij has rank Nf while q̃
i · qj has rank N = Nf −Nc.
Thus this theory breaks supersymmetry spontaneously at tree level. This mechanism
for F-term supersymmetry breaking is called the rank condition.
The classical scalar potential has a continuous set of minima, but the one-loop
potential lifts all of the non-Goldstone directions, which are usually called pseudomod-
uli. The usual approach to study the one-loop stabilization is the computation of the
complete one-loop effective potential over all pseudomoduli space via the Coleman-
Weinberg formula [34]:
M4B log
−M4F log
. (2.5)
This approach has the advantage that it allows the determination of the one-loop
minimum, without a priori information about its location, and moreover it provides
the full potential around it, including higher terms. However, it has the disadvantage
1The exact expressions can be found in (5.7) in [20], but we will not need them for our analysis.
We just take all masses in the electric description to be small enough for the analysis of the metastable
vacuum to be reliable.
of requiring the diagonalization of the mass matrix, which very often does not admit a
closed expression, e.g. for the theories we are interested in.
In fact, we would like to point out that to determine the existence of a meta-stable
minimum there exists a computationally much simpler approach. In our situation, we
have a good ansatz for the location of the one-loop minimum, and are interested just in
the one-loop pseudomoduli masses around such point. This information can be directly
obtained by computing the one-loop masses via the relevant Feynman diagrams. This
technique is extremely economical, and provides results in closed form in full generality,
e.g. for general values of the couplings, etc. The correctness of the original ansatz for
the vacuum can eventually be confirmed by the results of the computation (namely
positive one-loop squared masses, and negligible tadpoles for the classically massive
fields 2).
Hence, our strategy to study the one-loop stabilization in this paper is as follows:
• First we choose an ansatz for the classical minimum to become the one-loop
vacuum. It is natural to propose a point of maximal enhanced symmetry (in
particular, close to the origin in the space of vevs for M there exist and R-
symmetry, whose breaking by gauge interactions (via anomalies) is negligible in
that region). Hence the natural candidate for the one-loop minimum is
q = q̃T =
, (2.6)
with the rest of the fields set to 0. This initial ansatz for the one-loop minimum
is eventually confirmed by the positive square masses at one-loop resulting from
the computations described below. In our more general discussion of meta-stable
minima in runaway quiver gauge theories, our ansatz for the one-loop minimum
is a direct generalization of the above (and is similarly eventually confirmed by
the one-loop mass computation).
• Then we expand the field linearly around this vacuum, and identify the set of
classically massless fields. We refer to these as pseudomoduli (with some abuse
of language, since there could be massless fields which are not classically flat
directions due to higher potential terms)
2Since supersymmetry is spontaneously broken the effective potential will get renormalized by
quantum effects, and thus classically massive fields might shift slightly. This appears as a one loop
tadpole which can be encoded as a small shift of µ. This will enter in the two loop computation of
the pseudomoduli masses, which are beyond the scope of the present paper.
• As a final step we compute one-loop masses for these pseudomoduli by evaluating
their two-point functions via conventional Feynman diagrams, as explained in
more detail in appendix A.1 and illustrated below in several examples.
The ISS model is a simple example where this technique can be illustrated. Con-
sidering the above ansatz for the vacuum, we expand the fields around this point as:
µ+ 1√
(ξ+ + ξ−)
(ρ+ + ρ−)
, q̃T =
µ+ 1√
(ξ+ − ξ−)
(ρ+ − ρ−)
, M =
Z̃T Φ
(2.7)
where we have taken linear combinations of the fields in such a way that the bosonic
mass matrix is diagonal. This will also be convenient in section 2.2, where we discuss
the Goldstone bosons in greater detail.
We now expand the superpotential (2.3) to get
2µξ+Y +
µZρ+ +
µZρ− +
µρ+Z̃ −
µρ−Z̃
ρ2+Φ−
ρ2−Φ− µ2Φ+ . . . , (2.8)
where we have not displayed terms of order three or higher in the fluctuations, unless
they contain Φ, since they are irrelevant for the one loop computation we will perform.
Note also that we have set h = 1 and we have removed the trace (the matricial structure
is easy to restore later on, here we just set Nf = 2 for simplicity). The massless bosonic
fluctuations are given by Re ρ+, Im ρ−, Φ and ξ−. The first two together with Im ξ− are
Goldstone bosons, as explained in section 2.2. Thus the pseudomoduli we are interested
in are given by Φ and Re ξ−. Let us focus on Φ (the case of Re ξ− admits a similar
discussion). In this case the relevant terms in the superpotential simplify further, and
just the following superpotential contributes:
W = µZ
(ρ+ + ρ−) + µZ̃
(ρ+ − ρ−) +
ρ2+Φ−
ρ2−Φ− µ2Φ+ . . . ,
which we recognize, up to a field redefinition, as the symmetric model of appendix A.2.
We can thus directly read the result
δm2Φ =
|h|4µ2
(log 4− 1). (2.9)
This matches the value given in [20], which was found using the Coleman-Weinberg
potential.
2.2 The Goldstone bosons
One aspect of our technique that merits some additional explanation concerns the
Goldstone bosons. The one-loop computation of the masses for the fluctuations associ-
ated to the symmetries broken by the vacuum, using just the interactions described in
appendix A.1, leads to a non-vanishing result. This puzzle is however easily solved by
realizing that certain (classically massive) fields have a one-loop tadpole. This leads to
a new contribution to the one-loop Goldstone two-point amplitude, given by the dia-
gram in Figure 1. Adding this contribution the total one-loop mass for the Goldstone
bosons is indeed vanishing, as expected. This tadpole does not affect the computation
of the one-loop pseudomoduli masses (except for Re ξ+, but its mass remains positive)
as it is straightforward to check.
Re ξ+
Figure 1: Schematic tadpole contribution to the Im ξ− two point function. Both bosons and
fermions run in the loop.
The structure of this cancellation can be understood by using the derivation of the
Goldstone theorem for the 1PI effective potential, as we now discuss. The proof can
be found in slightly more detail, together with other proofs, in [35]. Let us denote by
V the 1PI effective potential. Invariance of the action under a given symmetry implies
∆φi = 0, (2.10)
where we denote by ∆φi the variation of the field φi under the symmetry, which will
in general be a function of all the fields in the theory. Taking the derivative of this
equation with respect to some other field φk
δφiδφk
∆φi +
· δ∆φi
= 0. (2.11)
Let us consider how this applies to our case. At tree level, there is no tadpole and
the above equation (truncated at tree level) states that for each symmetry generator
broken by the vacuum, the value of ∆φi gives a nonvanishing eigenvector of the mass
matrix with zero eigenvalue. This is the classical version of the Goldstone theorem,
which allows the identification of the Goldstone bosons of the theory.
For instance, in the ISS model in the previous section (for Nf = 2), there are
three global symmetry generators broken at the minimum described around (2.6). The
SU(2) × U(1) symmetry of the potential gets broken down to a U(1)′, which can be
understood as a combination of the original U(1) and the tz generator of SU(2). The
Goldstone bosons can be taken to be the ones associated to the three generators of
SU(2), and correspond (for µ real) to Im ξ−, Im ρ− and Re ρ+, in the parametrization
of the fields given by equation (2.7).
Even in the absence of tree-level tadpoles, there could still be a one-loop tadpole.
When this happens, there should also be a non-trivial contribution to the mass term
for the Goldstone bosons in the one-loop 1PI potential, related to the tadpole by the
one-loop version of (2.11). This relation guarantees that the mass term in the physical
(i.e. Wilsonian) effective potential, which includes the 1PI contribution, plus those of
the diagram in Figure 1, vanishes, as we described above.
In fact, in the ISS example, there is a non-vanishing one-loop tadpole for the real
part of ξ+ (and no tadpole for other fields). The calculation of the tadpole at one loop
is straightforward, and we will only present here the result
iM = −i|h|
(4π)2
(2 log 2). (2.12)
The 1PI one-loop contribution to the Goldstone boson mass is also simple to calculate,
giving the result
iM = −i|h|
(4π)2
(log 2). (2.13)
Using the variations of the relevant fields under the symmetry generator, e.g. for tz,
∆Re ξ+ = −Im ξ− (2.14)
∆Im ξ− = Re ξ+ + 2µ. (2.15)
we find that the (2.11) is satisfied at one-loop.
δφiδφk
∆φi +
· δ∆φi
= m2Im ξ− · 2µ+ (Re ξ+tadpole) · (−1) = 0. (2.16)
A very similar discussion applies to tx and ty.
The above discussion of Goldstone bosons can be similarly carried out in all ex-
amples of this paper. Hence, it will be enough to carry out the computation of the
1PI diagrams discussed in appendix A.1, and verify that they lead to positive squared
masses for all classically massless fields (with Goldstone bosons rendered massless by
the additional diagrams involving the tadpole).
3 Meta-stable vacua in quiver gauge theories with
DSB branes
In this section we show the existence of a meta-stable vacuum in a few examples
of gauge theories on DSB branes, upon the addition of massive flavors. As already
discussed in [19], the choice of fractional branes of DSB kind is crucial in the result.
The reason is that in order to have the ISS structure, and in particular supersymmetry
breaking by the rank condition, one needs a node such that its Seiberg dual satisfies
Nf > N , with N = Nf − Nc with Nc, Nf the number of colors, flavors of that gauge
factor. Denoting Nf,0, Nf,1 the number of massless and massive flavors (namely flavors
arising from bi-fundamentals of the original D3-brane quiver, or introduced by the D7-
branes), the condition is equivalent to Nf,0 < Nc. This is precisely the condition that
an ADS superpotential is generated, and is the prototypical behavior of DSB branes
[14, 15, 16, 18].
Another important general comment, also discussed in [19], is that theories on DSB
branes generically contain one or more chiral multiplets which do not appear in the
superpotential. Being decoupled, such fields remain as accidental flat directions at
one-loop, so that the one-loop minimum is not isolated. The proper treatment of these
flat directions is beyond the reach of present tools, so they remain an open question.
However, it is plausible that they do not induce a runaway behavior to infinity, since
they parametrize a direction orthogonal to the fields parametrizing the runaway of
DSB fractional branes.
3.1 The complex cone over dP1
In this section we describe the most familiar example of quiver gauge theory with DSB
fractional branes, the dP1 theory. In this theory, a non-perturbative superpotential
removes the classical supersymmetric vacua [14, 15, 16]. Assuming canonical Kähler
potential the theory has a runaway behavior [15, 17]. In this section, we revisit with
our techniques the result in [19] that the addition of massive flavors can induce the ap-
pearance of meta-stable supersymmetry breaking minima, long-lived against tunneling
to the runaway regime. As we show in coming sections, this behavior is prototypical
and extends to many other theories with DSB fractional branes. The example is also
representative of the computations for a general quiver coming from a brane at a toric
singularity, and illustrates the usefulness of the direct Feynman diagram evaluation of
one-loop masses.
Consider the dP1 theory, realized on a set ofM fractional D3-branes at the complex
cone over dP1. In order to introduce additional flavors, we introduce sets of Nf,1
D7-branes wrapping non-compact 4-cycles on the geometry and passing through the
singular point. We refer the reader to [19], and also to later sections, for more details on
the construction of the theory, and in particular on the introduction of the D7-branes.
Its quiver is shown in Figure 2, and its superpotential is
W = λ(X23X31Y12 −X23Y31X12)
+ λ′(Q3iQ̃i2X23 +Q2jQ̃j1X12 +Q1kQ̃k3X31)
+ m3Q3iQ̃k3δik +m2Q2jQ̃i2δji +m1Q1kQ̃j1δkj , (3.1)
where the subindices denote the groups under which the field is charged. The first
line is the superpotential of the theory of fractional brane, the second line describes
77-73-37 couplings between the flavor branes and the fractional brane, and the last line
gives the flavor masses. Note that there is a massless field, denoted Z12 in [19], that
does not appear in the superpotential. This is one of the decoupled fields mentioned
above, and we leave its treatment as an open question.
SU(3M)
SU(2M) SU(M)
PSfrag replacements Q3i
Q2j Q̃j1
Figure 2: Extended quiver diagram for a dP1 theory with flavors, from [19].
We are interested in gauge factors in the free magnetic phase. This is the case for
the SU(3M) gauge factor in the regime
M + 1 ≤ Nf,1 <
M. (3.2)
To apply Seiberg duality on node 3, we introduce the dual mesons:
M21 =
X23X31 ; Nk1 =
Q̃k3X31
M ′21 =
X23Y31 ; N
Q̃k3Y31
N2i =
X23Q3i ; Φki =
Q̃k3Q3i
(3.3)
and we also replace the electric quarks Q3i, Q̃k3, X23, X31, Y31 by their magnetic duals
Q̃i3, Q3k, X32, X13, Y13. The magnetic superpotential is given by rewriting the confined
fields in terms of the mesons and adding the coupling between the mesons and the dual
quarks,
W = h (M21X13X32 + M
21Y13X32 + N2iQ̃i3X32
+ Nk1X13Q3k + N
k1Y13Q3k + ΦkiQ̃i3Q3k )
+ hµ0 (M21Y12 − M ′21X12 ) + µ′Q1kNk1 + µ′N2iQ̃i2
− hµ 2TrΦ + λ′Q2jQ̃j1X12 + m2Q2iQ̃i2 + m1Q1iQ̃i1. (3.4)
This is the theory we want to study. In order to simplify the treatment of this example
we will disregard any subleading terms in mi/µ
′, and effectively integrate out Nk1 and
N2i by substituting them by 0. This is not necessary, and indeed the computations in
the next sections are exact. We do it here in order to compare results with [19].
As in the ISS model, this theory breaks supersymmetry via the rank condition. The
fields Q̃i3, Q3k and Φki are the analogs of q, q̃ and M in the ISS case discussed above.
This motivates a vacuum ansatz analogous to (2.6) and the following linear expansion:
φ00 φ01
φ10 φ11
; Q̃i3 =
µeθ +Q3,1
Q̃3,2
; QT3i =
µe−θ +Q3,1
Q̃k1 =
Q̃1,1
; Q2j =
Q2,11 x
Q2,21 x
; M21 =
M21,1
M21,2
Y13 = (Y13) ; X
X12,1
X12,2
; XT32 =
X32,1
X32,2
Y T12 =
Y12,1
Y12,2
; N ′k1 =
N ′k1,1
; M ′21 =
M ′21,1
M ′21,2
X13 = (X13) .
(3.5)
Note that we have chosen to introduce the nonlinear expansion in θ in order to re-
produce the results found in the literature in their exact form3. Note also that for
the sake of clarity we have not been explicit about the ranks of the different matrices.
They can be easily worked out (or for this case, looked up in [19]), and we will restrict
ourselves to the 2 flavor case where the matrix structure is trivial. As a last remark,
we are not being explicit either about the definitions of the different couplings in terms
of the electric theory. This can be done easily (and as in the ISS case they involve
3A linear expansion would lead to identical conclusions concerning the existence of the meta-stable
vacua, but to one-loop masses not directly amenable to comparison with results in the literature.
an unknown coefficient in the Kähler potential), but in any event, the existence of
the meta-stable vacua can be established for general values of the coefficients in the
superpotential. Hence we skip this more detailed but not very relevant discussion.
The next step consists in expanding the superpotential and identifying the massless
fields. We get the following quadratic contributions to the superpotential:
Wmass = 2hµφ00Q̃3,1 + hµφ01Q̃3,2 + hµφ10Q3,2
+ hµ0M21,1Y12,1 + hµ0M21,2Y12,2 − λ′M ′21,1X12,1 − λ′M ′21,2X12,2
+ hµN ′k1,1Y13 − h1µQ̃1,1X13 − h2µQ2,11X32,1 − h2µQ2,21X32,2. (3.6)
The fields massless at tree level are x, x′, y, z, φ11, θ, Q3,2 and Q̃3,2. Three of these
are Goldstone bosons as described in the previous section. For real µ they are Im θ,
Re (Q̃3,2 +Q3,2) and Im (Q̃3,2 −Q3,2). We now show that all other classically massless
fields get masses at one loop (with positive squared masses).
As a first step towards finding the one-loop correction, notice that the supersym-
metry breaking mechanism is extremely similar to the one in the ISS model before, in
particular it comes only from the following couplings in the superpotential:
Wrank = hQ3,2Q̃3,2φ11 − hµ2φ11 + . . . (3.7)
This breaks the spectrum degeneracy in the multiplets Q3,2 and Q̃3,2 at tree level, so
we refer to them as the fields with broken supersymmetry.
Let us compute now the correction for the mass of x, for example. For the one-loop
computation we just need the cubic terms involving one pseudomodulus and at least
one of the broken supersymmetry fields, and any quadratic term involving fields present
in the previous set of couplings. From the complete expansion one finds the following
supersymmetry breaking sector:
Wsymm. = hφ11Q3,2Q̃3,2 + hµφ01Q̃3,2 + hµφ10Q3,2 − hµ2φ11. (3.8)
The only cubic term involving the pseudomodulus x and the broken supersymmetry
fields is
Wcubic = −h2 x Q̃3,2X32,1, (3.9)
and there is a quadratic term involving the field X32,1
Wmass coupling = −h2µQ2,11X32,1. (3.10)
Assembling the three previous equations, the resulting superpotential corresponds to
the asymmetric model in appendix A.2, so we can directly obtain the one-loop mass
for x:
δm2x =
|h|4µ2C
|h2|2
. (3.11)
Proceeding in a similar way, the one-loop masses for φ11, x
′, y and z are:
δm2φ11 =
|h|4µ2(log 4− 1)
δm2x′ =
|h|4µ2C
|h2|2
δm2y =
|h|4µ2C
|h1|2
δm2z =
|h|4µ2(log 4− 1). (3.12)
There is just one pseudomodulus left, Re θ, which is qualitatively different to the
others. With similar reasoning, one concludes that it is necessary to study a superpo-
tential of the form
W = h(Xφ1φ2 + µe
θφ1φ3 + µe
−θφ2φ4 − µ2X). (3.13)
Due to the non-linear parametrization, the expansion in θ shows that there is a term
quadratic in θ which contributes to the one-loop mass via a vertex with two bosons and
two fermions, the relevant diagram is shown in Figure 16d. The result is a vanishing
mass for Im θ, as expected for a Goldstone boson (the one-loop tadpole vanishes in this
case), and a non-vanishing mass for Re θ
δm2Re θ =
|h|4µ4(log 4− 1). (3.14)
We conclude by mentioning that all squared masses are positive, thus confirming
that the proposed point in field space is the one-loop minimum. As shown in [19], this
minimum is parametrically long-lived against tunneling to the runaway regime.
3.2 Additional examples: The dP2 case
Let us apply these techniques to consider new examples. In this section we consider
a DSB fractional brane in the complex cone over dP2, which provides another quiver
theory with runaway behavior [15]. The quiver diagram for dP2 is given in Figure 3,
with superpotential
W = X34X45X53 −X53Y31X15 −X34X42Y23 + Y23X31X15X52
+ X42X23Y31X14 −X23X31X14X45X52 (3.15)
Figure 3: Quiver diagram for the dP2 theory.
We consider a set of M DSB fractional branes, corresponding to choosing ranks
(M, 0,M, 0, 2M) for the corresponding gauge factors. The resulting quiver is shown in
Figure 4, with superpotential
W = −λX53Y31X15 (3.16)
U(2M)
U(M)U(M)
Figure 4: Quiver diagram for the dP2 theory with M DSB fractional branes.
Following [19] and appendix B, one can introduce D7-branes leading to D3-D7
open strings providing (possibly massive) flavors for all gauge factors, and having cubic
couplings with diverse D3-D3 bifundamental chiral multiplets. We obtain the quiver
in Figure 5. Adding the cubic 33-37-73 coupling superpotential, and the flavor masses,
the complete superpotential reads
Wtotal = −λX53Y31X15 − λ′(Q1iQ̃i3Y31 +Q3jQ̃j5X53 +Q5kQ̃k1X15)
+ m1Q1iQ̃k1 +m2Q3jQ̃i3 +m5Q5kQ̃j5 (3.17)
where 1, 2, 3 are the gauge group indices and i, j, k are the flavor indices.
We consider the U(2M) node in the free magnetic phase, namely
M + 1 ≤ Nf,1 < 2M (3.18)
U(M) U(M)
U(2M)
PSfrag replacements
Q1i Qi3
Figure 5: Quiver for the dP2 theory with M fractional branes and flavors.
After Seiberg Duality the dual gauge factor is SU(N) withN = Nf,1−M and dynamical
scale Λ. To get the matter content in the dual, we replace the microscopic flavors Q5k,
Q̃j5, X53, X15 by the dual flavors Q̃k5, Q5j , X35, X51 respectively. We also have the
mesons related to the fields in the electric theory by
M1k =
X15Q5K ; Ñj3 =
Q̃j5X53
M13 =
X15X53 ; Φ̃jk =
Q̃j5Q5k
(3.19)
There is a cubic superpotential coupling the mesons and the dual flavors
Wmes. = h (M1kQ̃k5X51 + M13X35X51 + Ñj3X35Q5j + Φ̃jkQ̃k5Q5j ) (3.20)
where h = Λ/Λ̂ with Λ̂ given by Λ
3Nc−Nf
elect Λ
3(Nf−Nc)−Nf = Λ̂Nf , where Λelect is the
dynamical scale of the electric theory. Writing the classical superpotential terms of the
new fields gives
Wclas. = −hµ0M13Y31 + λ′Q1iQ̃i3Y31 + µ′ Ñj3Q3j + µ′M1kQ̃k1
+ m1Q1iQ̃k1 + m3Q3jQ̃i3 − hµ 2TrΦ (3.21)
where µ0 = λΛ, µ
′ = λ′Λ, and µ 2 = −m5Λ̂. So the complete superpotential in the
Seiberg dual is
Wdual = −hµ0M13Y31 + λ′Q1iQ̃i3Y31 + µ′ Ñj3Q3j + µ′M1kQ̃k1
+ m1Q1iQ̃k1 + m3Q3jQ̃i3 − hµ 2TrΦ
+ h (M1kQ̃k5X51 + M13X35X51 + Ñj3X35Q5j + Φ̃jkQ̃k5Q5j ) (3.22)
This superpotential has a sector completely analogous to the ISS model, triggering
supersymmetry breaking by the rank condition. This suggests the following ansatz for
the point to become the one-loop vacuum
Q5k = Q̃
, (3.23)
with all other vevs set to zero. Following our technique as explained above, we expand
fields at linear order around this point. Focusing on Nf,1 = 2 and Nc = 1 for simplicity
(the general case can be easily recovered), we have
Q̃k5 =
µ+ δQ̃5,1
δQ̃5,2
; Q5k = (µ+ δQ5,1 ; δQ5,2) ; Φ =
δΦ0,0 δΦ0,1
δΦ1,0 δΦ1,1
Q̃k1 =
δQ̃1,1
δQ̃1,2
; Q1i = (δQ1,1 ; δQ1,2) ; Q̃i3 =
δQ̃3,1
δQ̃3,2
; Q3j = (δQ3,1 ; δQ3,2)
Ñj3 =
δÑ3,1
δÑ3,2
; M1k = (δM1,1 ; δM1,2) ; M13 = δM13 ; Y31 = δY31 ; X51 = δX51
X35 = δX35
(3.24)
Inserting this into equation (3.22) gives
Wdual = −hµ0 δM13δY31 + λ′ δQ1,1δQ̃3,1δY31 + λ′ δQ1,2δQ̃3,2δY31
+ µ′ δÑ3,1δQ3,1 + µ
′ δÑ3,2δQ3,2 + µ
′ δM1,1δQ̃1,1 + µ
′ δM1,2δQ̃1,2
+ m1δQ1,1δQ̃1,1 + m1δQ1,2δQ̃1,2 + m3δQ3,1δQ̃3,1 + m3δQ3,2δQ̃3,2
− hµ 2δΦ11 + h (µδM1,1δX51 + δM1,1δQ̃5,1δX51 + δM1,2δQ̃5,2δX51
+ δM13δX35δX51 + µδX35δÑ3,1 + δX35δÑ3,1δQ5,1 + δX35δÑ3,2δQ5,2
+ µδQ̃5,1δΦ00 + µδQ5,1δΦ00 + δQ5,1δQ̃5,1δΦ00 + µδΦ01δQ̃5,2
+ δQ5,1δΦ01δQ̃5,2 + µδΦ10δQ5,2 + δQ̃5,1δΦ10δQ5,2 + δQ̃5,2δΦ11δQ5,2).
We now need to identify the pseudomoduli, in other words the massless fluctuations at
tree level. We focus then just on the quadratic terms in the superpotential
Wmass = −hµ0 δM13δY31
+ µ′ δÑ3,1δQ3,1 +m3δQ3,1δQ̃3,1 + hµδX35δÑ3,1
+ µ′ δÑ3,2δQ3,2 +m3δQ3,2δQ̃3,2
+ µ′ δM1,1δQ̃1,1 + m1δQ1,1δQ̃1,1 + hµδM1,1δX51
+ µ′ δM1,2δQ̃1,2 + m1δQ1,2δQ̃1,2
+ hµδQ̃5,1δΦ00 + hµδQ5,1δΦ00
+ hµδΦ01δQ̃5,2 + µδΦ10δQ5,2. (3.25)
We have displayed the superpotential so that fields mixing at the quadratic level appear
in the same line. In order to identify the pseudomoduli we have to diagonalize4 these
fields. Note that the structure of the mass terms corresponds to the one in appendix C,
in particular around equation (C.9). From the analysis performed there we know that
upon diagonalization, fields mixing in groups of four (i.e., three mixing terms in the
superpotential, for example the δM1,1, δQ̃1,1, δQ1,1, δX51 mixing) get nonzero masses,
while fields mixing in groups of three (two mixing terms in the superpotential, for
example δM1,2, δQ̃1,2 and δQ1,2) give rise to two massive perturbations and a massless
one, a pseudomodulus. We then just need to study the fate of the pseudomoduli. From
the analysis in appendix C, the pseudomoduli coming from the mixing terms are
Y1 = m3δÑ3,2 − µ′δQ̃3,2 ,
Y2 = m1δM1,2 − µ′δQ1,2 ,
Y3 = hµ(δQ5,1 − δQ̃5,1) . (3.26)
In order to continue the analysis, one just needs to change basis to the diagonal fields
and notice that the one loop contributions to the pseudomoduli are described again by
the asymmetric model of appendix A.2, so they receive positive definite contributions.
The exact analytic expressions can be easily found with the help of some computer
algebra program, but we omit them here since they are quite unwieldy.
4 The general case
In the previous section we showed that several examples of quiver gauge theories on
DSB fractional branes have metastable vacua once additional flavors are included.
In this section we generalize the arguments for general DSB branes. We will show
how to add D7–branes in a specific manner so as to generate the appropriate cubic
flavor couplings and mass terms. Once this is achieved, we describe the structure of
the Seiberg dual theory. The results of our analysis show that, with the specified
configuration of D7–branes, the determination of metastability is greatly simplified
and only involves looking at the original superpotential. Thus, although we do not
prove that DSB branes on arbitrary singularities generate metastable vacua, we show
how one can determine the existence of metastability in a very simple and systematic
4As a technical remark, let us note that it is possible to set all the mass terms to be real by an
appropriate redefinition of the fields, so we are diagonalizing a real symmetric matrix.
manner. Using this analysis we show further examples of metastable vacua on systems
of DSB branes.
4.1 The general argument
4.1.1 Construction of the flavored theories
Consider a general quiver gauge theory arising from branes at singularities. As we have
argued previously, we focus on DSB branes, so that there is a gauge factor satisfying
Nf,0 < Nc, which can lead to supersymmetry breaking by the rank condition in its
Seiberg dual. To make the general analysis more concrete, let us consider a quiver
like that in Figure 6, which is characteristic enough, and let us assume that the gauge
factor to be dualized corresponds to node 2. In what follows we analyze the structure
of the fields and couplings in the Seiberg dual, and reduce the problem of studying the
meta-stability of the theory with flavors to analyzing the structure of the theory in the
absence of flavors.
PSfrag replacements
X21 Y21
Y32 Z32
X43 Y43
Figure 6: Quiver diagram used to illustrate general results. It does not correspond to any
geometry in particular.
The first step is the introduction of flavors in the theory. As discussed in [19], for any
bi-fundamental Xab of the D3-brane quiver gauge theory there exist a supersymmetric
D7-brane leading to flavors Qbi, Q̃ia in the fundamental (antifundamental) of the b
(ath) gauge factor. There is also a cubic coupling XabQbiQ̃ia. Let us now specify a
concrete set of D7-branes to introduce flavors in our quiver gauge theory. Consider a
superpotential coupling of the D3-brane quiver gauge theory, involving fields charged
under the node to be dualized. This corresponds to a loop in the quiver, involving node
2, for instance X32X21X14Y43 in Figure 6. For any bi-fundamental chiral multiplet in
this coupling, we introduce a set of Nf,1 of the corresponding D7-brane. This leads
to a set of flavors for the different gauge factors, in a way consistent with anomaly
cancellation, such as that shown in Figure 7. The description of this system of D7-
branes in terms of dimer diagrams is carried out in Appendix B. The cubic couplings
described above lead to the superpotential terms5
Wflavor = λ
′ (X32Q2bQb3 + X21Q1aQa2 + X14Q4dQd1 + Y43Q3cQc4 ) (4.1)
Finally, we introduce mass terms for all flavors of all involved gauge factors:
Wmass = m2Qa2Q2b + m3Qb3Q3c + m4Qc4Q4d + m1Qd1Q1a (4.2)
These mass terms break the flavor group into a diagonal subgroup.
PSfrag replacements
X21 Y21
Y32 Z32
X43 Y43
Q2b Qb3
Q4dQd1
Figure 7: Quiver diagram with flavors. White nodes denote flavor groups.
4.1.2 Seiberg duality and one-loop masses
We consider introducing a number of massive flavors such that node 2 is in the free
magnetic phase, and consider its Seiberg dual. The only relevant fields in this case are
those charged under gauge factor 2, as shown if Figure 8. The Seiberg dual gives us
Figure 9 where the M ’s are mesons with indices in the gauge groups, R’s and S’s are
5Here we assume the same coupling, but the conclusions hold for arbitrary non-zero couplings.
PSfrag replacements
X21 Y21
Y32 Z32Qa2
Figure 8: Relevant part of quiver before Seiberg duality.
PSfrag replacements
X̃12 Ỹ12
Ỹ23Z̃23
M1, . . . ,M6
Figure 9: Relevant part of the quiver after Seiberg duality on node 2.
mesons with only one index in the flavor group, and Xab is a meson with both indices
in the flavor groups. The original cubic superpotential and flavor mass superpotentials
become
Wflavor dual = λ
′ (S13bQb3 + R
a1Q1a + X14Q4dQd1 + Y43Q3cQc4 )
Wmass dual = m2Xab + m3Qb3Q3c + m4Qc4Q4d + m1Qd1Q1a (4.3)
In addition we have the extra meson superpotential
Wmesons = h (XabQ̃b2Q̃2a + R
a1X̃12Q̃2a + R
a1Ỹ12Q̃2a + S
3bQ̃b2X̃23 + S
3bQ̃b2Ỹ23
+ S33bQ̃b2Z̃23 + M
31X̃12X̃23 + M
31X̃12Ỹ23 + M
31X̃12Z̃23
+ M431Ỹ12X̃23 + M
31Ỹ12Ỹ23 + M
31Ỹ12Z̃23 ). (4.4)
The crucial point is that we always obtain terms of the kind underlined above, namely
a piece of the superpotential reading m2Xab + hXabQ̃b2Q̃2a. This leads to tree level
supersymmetry breaking by the rank condition, as announced. Moreover the superpo-
tential fits in the structure of the generalized asymmetric O’Raifeartaigh model studied
in appendix A.2, with Xab, Q̃b2, Q̃2a corresponding to X , φ1, φ2 respectively. The mul-
tiplets Q̃b2 and Q̃2a are split at tree level, and Xab is massive at 1-loop. From our
study of the generalized asymmetric case, any field which has a cubic coupling to the
supersymmetry breaking fields Q̃b2 or Q̃2a is one-loop massive as well. Using the gen-
eral structure of Wmesons, a little thought shows that all dual quarks with no flavor
index (e.g. X̃ , Ỹ ) and all mesons with one flavor index (e.g. R or S) couple to the
supersymmetry breaking fields.
Thus they all get one-loop masses (with positive squared mass). Finally, the flavors
of other gauge factors (e.g. Qb3) are massive at tree level from Wmass.
The bottom line is that the only fields which do not get mass from these interac-
tions are the mesons with no flavor index, and the bi-fundamentals which do not get
dualized (uncharged under node 2). All these fields are related to the theory in the
absence of extra flavors, so they can be already stabilized at tree-level from the original
superpotential. So, the criteria for a metastable vacua is that the original theory, in
the absence of flavors leads, after dualization of the node with Nf < Nc, to masses for
all these fields (or more mildly that they correspond to directions stabilized by mass
terms, or perhaps higher order superpotential terms).
For example, if we apply this criteria to the dP2 case studied previously, the original
superpotential for the fractional DSB brane is
W = −λX53Y31X15 (4.5)
so after dualization we get
W = −λM13Y31 (4.6)
which makes these fields massive. Hence this fractional brane, after adding the D7-
branes in the appropriate configuration, will generate a metastable vacua will all moduli
stabilized.
The argument is completely general, and leads to an enormous simplification in
the study of the theories. In the next section we describe several examples. A more
rigorous and elaborate proof is provided in the appendix where we take into account the
matricial structure, and show that all fields, except for Goldstone bosons, get positive
squared masses at tree-level or at one-loop.
4.2 Additional examples
4.2.1 The dP3 case
Let us consider the complex cone over dP3, and introduce fractional DSB branes of the
kind considered in [15]. The quiver is shown in Figure 10 and the superpotential is
W = X13X35X51 (4.7)
Node 1 has Nf < Nc so upon addition of massive flavors and dualization will lead
to supersymmetry breaking by the rank condition. Following the procedure of the
previous section, we add Nf,1 flavors coupling to the bi-fundamentals X13, X35 and
X51. Node 1 is in the free magnetic phase for P +1 ≤ Nf,1 < 32P +
. Dualizing node
1, the above superpotential becomes
W = X35M53 (4.8)
where M53 is the meson X51X13. So, following the results of the previous section, we
can conclude that this DSB fractional brane generates a metastable vacua with all
pseudomoduli lifted.
4.2.2 Phase 1 of PdP4
Let us consider the PdP4 theory, and introduce the DSB fractional brane of the kind
considered in [15]. The quiver is shown in Figure 11 . The superpotential is
W = −X25X51X12 (4.9)
U(P+1)
Figure 10: Quiver diagram for the dP3 theory with a DSB fractional brane.
U(M+P)
Figure 11: Quiver diagram for the dP4 theory with a DSB fractional branes.
Node 1 has Nf < Nc and will lead to supersymmetry breaking by the rank condition in
the dual. Following the procedure of the previous section, we add Nf,1 flavors coupling
to the bi-fundamentals X12, X25 and X51. Node 1 is in the free magnetic phase for
P + 2 ≤M +Nf,1 < 32(M + P ). Dualizing node 1, the above superpotential becomes
W = X25M52, where M53 is the meson X51X12. Again we conclude that this DSB
fractional brane generates a metastable vacua with all pseudomoduli lifted.
4.2.3 The Y p,q family
Consider D3-branes at the real cones over the Y p,q Sasaki-Einstein manifolds [36, 37,
38, 39], whose field theory were determined in [8]. The theory admits a fractional brane
[13] of DSB kind, which namely breaks supersymmetry and lead to runaway behavior
[15, 18]. The analysis of metastability upon addition of massive flavors for arbitrary
Y p,q’s is much more involved than previous examples. Already the description of the
field theory on the fractional brane is complicated. Even for the simpler cases of Y p,q
and Y p,p−1 the superpotential contains many terms. In this section we do not provide a
general proof of metastability, but rather consider the more modest aim of showing that
all directions related to the runaway behavior in the absence of flavors are stabilized by
the addition of flavors. We expect that this will guarantee full metastability, since the
fields not involved in our analysis parametrize directions orthogonal to the runaway at
infinity.
The dimer for Y p,q is shown in Figure 12 and consists of a column of n hexagons and
2m quadrilaterals which are just halved hexagons [18]. The labels (n,m) are related
to (p, q) by
n = 2q ; m = p− q (4.10)
• The Y p,1 case
The dimer for the theory on the DSB fractional brane in the Y p,1 case is shown
in Figure 13, a periodic array of a column of two full hexagons, followed by p− 1 cut
hexagons (the shaded quadrilateral has Nc = 0). As shown in [18], the top quadrilateral
which has Nf < Nc, and induces the ADS superpotential triggering the runaway. The
relevant part of the dimer is shown in Figure 14, where V1 and V2 are the fields that run
to infinity [18]. This node will lead to supersymmetry breaking by the rank condition
in the dual. It is in the free magnetic phase for M + 1 ≤ Nf,1 < pM + M2 . The piece
Figure 12: The generic dimer for Y p,q, from [18].
of the superpotential involving the V1 and V2 terms is
W = Y U2V2 − Y U1V1. (4.11)
In the dual theory, the dual superpotential makes the fields massive. Hence, the theory
has a metastable vacua where the runaway fields are stabilized.
Figure 13: The dimer for Y p,1.
(p−1)M
(p−2)M
(2p−1)M
(p−1)M
(p+1)M
Figure 14: Top part of the dimer for Y p,1. The hexagons are labeled by the ranks of
the respective gauge groups
• The Y p,p−1 case
The analysis for Y p,p−1 is similar but in this case it is the bottom quadrilateral
which has the highest rank and thus gives the ADS superpotential [18]. The relevant
part of the dimer is shown in Figure 15, and the runaway direction is described by the
fields V1 and V2. Upon addition of Nf,1 flavors, the relevant node in the in the free
magnetic phase for M + 1 ≤ Nf,1 < pM + M2 Considering the superpotential, it is
straightforward to show that the runaway fields become massive. Complementing this
with our analysis in previous section, we conclude that the theory has a metastable
vacua where the runaway fields are stabilized.
We have thus shown that we can obtain metastable vacua for fractional branes at
cones over the Y p,1 and Y p,p−1 geometries. Although there is no obvious generalization
for arbitrary Y p,q’s, our results strongly suggest that the existence of metastable vacua
extends to the complete family.
5 Conclusions and outlook
The present work introduces techniques and computations which suggest that the ex-
istence of metastable supersymmetry breaking vacua is a general property of quiver
gauge theories on DSB fractional branes, namely fractional branes associated to ob-
structed complex deformations. It is very satisfactory to verify the correlation between
a non-trivial dynamical property in gauge theories and a geometric property in their
(p−1)M
(p−1)M
(2p−1)M
(p−2)M
(2p−2)M
(2p−2)M
Figure 15: Bottom part of the dimer for Y p,p−1. The hexagons are labeled by the ranks
of the respective gauge groups
string theory realization. The existence of such correlation fits nicely with the remark-
able properties of gauge theories on D-branes at singularities, and the gauge/gravity
correspondence for fractional branes.
Beyond the fact that our arguments do not constitute a general proof, our analysis
has left a number of interesting open questions. In fact, as we have mentioned, all
theories on DSB fractional branes contain one or several fields which do not appear
in the superpotential. We expect the presence of these fields to have a direct physical
interpretation, which has not been uncovered hitherto. It would be interesting to find
a natural explanation for them.
Finally, a possible extension of our results concerns D-branes at orientifold singular-
ities, which can lead to supersymmetry breaking and runaway as in [27]. Interestingly,
in this case the field theory analysis is more challenging, since they would require
Seiberg dualities of gauge factors with matter in two-index tensors. It is very possible
that the string theory realization, and the geometry of the singularity provide a much
more powerful tool to study the system.
Overall, we expect other surprises and interesting relations to come up from further
study of D-branes at singularities.
Acknowledgments
We thank S. Franco for useful discussions. A.U. thanks M. González for encouragement
and support. This work has been supported by the European Commission under RTN
European Programs MRTN-CT-2004-503369, MRTN-CT-2004-005105, by the CICYT
(Spain), and by the Comunidad de Madrid under project HEPHACOS P-ESP-00346.
The research by I.G.-E. is supported by the Gobierno Vasco PhD fellowship program.
The research of F.S is supported by the Ministerio de Educación y Ciencia through an
FPU grant. I.G.-E. and F.S. thank the CERN Theory Division for hospitality during
the completion of this work.
A Technical details about the calculation via Feyn-
man diagrams
A.1 The basic amplitudes
In the main text we are interested in computing two point functions for the pseudo-
moduli at one loop, and in section 2.2 also tadpole diagrams. There are just a few kinds
of diagrams entering in the calculation, which we will present now for the two-point
function, see Figure 16. The (real) bosonic fields are denoted by φi and the (Weyl)
fermions by ψi. The pseudomodulus we are interested in is denoted by ϕ.
c) d)
a) b)
ϕ ϕ ϕ
ϕ ϕ ϕ
Figure 16: Feynman diagrams contributing to the one-loop two point function. The dashed
line denotes bosons and the solid one fermions.
Bosonic contributions
These come from two terms in the Lagrangian. First there is a diagram coming from
terms of the form (Figure 16b):
L = . . .+ λϕ2φ2 − 1
m2φ2, (A.1)
giving an amplitude (we will be using dimensional regularization)
iM = −2iλ
(4π)2
− γ + 1 + log 4π − logm2
. (A.2)
The other contribution comes from the diagram in Figure 16a:
L = . . .+ λϕφ1φ2 −
2, (A.3)
which contributes to the two point function with an amplitude:
iM = iλ
(4π)2
− γ + log 4π −
dx log∆
, (A.4)
where here and in the following we denote ∆ ≡ xm21 + (1− x)m22.
Fermionic contributions
The relevant vertices here are again of two possible kinds, one of which is nonrenor-
malizable. The cubic interaction comes from terms in the Lagrangian given by the
diagram in Figure 16c:
L = . . .+ ϕ(aψ1ψ2 + a∗ψ̄1ψ̄2) +
1 + ψ̄
2 + ψ̄
2). (A.5)
We are assuming real masses for the fermions here, in the configurations we study this
can always be achieved by an appropriate field redefinition. The contribution from
such vertices is given by:
−2im1m2
(4π)2
(a2 + (a2)∗)
− γ + log 4π − log∆
− 8i|a|
(4π)2
− γ + log 4π + 1
− log∆
. (A.6)
The other fermionic contribution, which one does not need as long as one is dealing
with renormalizable interactions only (but we will need in the main text when analyzing
the pseudomodulus θ), is given by terms in the Lagrangian of the form (Figure 16d):
L = . . .+ λϕ2(ψ2 + ψ̄2) + 1
m(ψ2 + ψ̄2), (A.7)
which contributes to the total amplitude with:
iM = 8λmi
(4π)2
− γ + 1 + log 4π − logm2
. (A.8)
A.2 The basic superpotentials
The previous amplitudes are the basic ingredients entering the computation, but in
general the number of diagrams contributing to the two point amplitudes is quite
big, so calculating all the contributions by hand can get quite involved in particular
examples6. Happily, one finds that complicated models (such as dP1 or dP2, studied in
the main text) reduce to performing the analysis for only two different superpotentials,
which we analyze in this section.
The symmetric case
We want to study in this section a superpotential of the form:
W = h(Xφ1φ2 + µφ1φ3 + µφ2φ4 − µ2X). (A.9)
6The authors wrote the computer program in http://cern.ch/inaki/pm.tar.gz which helped
greatly in the process of computing the given amplitudes for the relevant models.
http://cern.ch/inaki/pm.tar.gz
This model is a close cousin of the basic O’Raifeartaigh model. We are interested in
the one loop contribution to the two point function of X , which is massless at tree
level.
From the (F-term) bosonic potential one obtains the following terms entering the
one loop computation:
|hXφ2|2 + |h|2µ(Xφ2φ∗3 +X∗φ∗2φ3) + |h|2µ(Xφ1φ∗4 +X∗φ∗1φ4)
+ |h|2µ2(φ1φ2 + φ∗1φ∗2) +
|h|2µ2|φi|2 (A.10)
In order to do the computation it is useful to diagonalize the mass matrix by
introducing φ+ and φ− such that:
(φ+ + iφ−) φ2 =
(φ+ − iφ−) (A.11)
and φa, φb such that:
φ∗3 =
(φa + iφb) φ
(φa − iφb). (A.12)
With these redefinitions the bosonic scalar potential decouples into identical φ+ and
φ− sectors, giving two decoupled copies of:
V = |h|2|X|2|φ+|2 + |h|2µ2(|φ+|2 + |φa|2)
+|h|2µ(Xφ+φa +X∗φ∗+φ∗a)−
|h|2µ2
φ2+ + (φ
. (A.13)
Calculating the amplitude consists simply of constructing the (very few) two point
diagrams from the potential above and plugging the formulas above for each diagram
(the fermionic part is even simpler in this case). The final answer is that in this model
the one loop correction to the mass squared of X is given by:
δm2X =
|h4|µ2
(log 4− 1). (A.14)
The generalized asymmetric case
The next case is slightly more complicated, but will suffice to analyze completely all
the models we encounter. We will be interested in the one loop contribution to the
mass of the pseudomoduli Y in a theory with superpotential:
W = h(Xφ1φ2 + µφ1φ3 + µφ2φ4 − µ2X) + k(rY φ1φ5 + µφ5φ7), (A.15)
with k and r arbitrary complex numbers. The procedure is straightforward as above,
so we will just quote the result. We obtain an amplitude given by:
iM = −i
(4π)2
|h2rµ|2C
, (A.16)
where we have defined C(t) as:
C(t) = t
log 4− t
t− 1 log t
. (A.17)
Note that this is a positive definite function, meaning that the one loop correction
to the mass is always positive, and the pseudomoduli get stabilized for any (nonzero)
value of the parameters. Also note that the limit of vanishing t with |r|2t fixed (i.e.,
vanishing masses for φ5 and φ7, but nonvanishing coupling of Y to the supersymmetry
breaking sector) gives a nonvanishing contribution to the mass of Y .
B D7–branes in the Riemann surface
The gauge theory of D3-branes at toric singularities can be encoded in a dimer diagram
[40, 41, 42, 43, 44]. This corresponds to a bi-partite tiling of T 2, where faces corre-
spond to gauge groups, edges correspond to bi-fundamentals, and nodes correspond to
superpotential terms. As an example, the dimer diagram of D3–branes on the cone
over dP2 is shown in Figure 17. As shown in [43], D3–branes on a toric singularity are
mirror to D6–branes on intersecting 3-cycles in a geometry given by a fibration of a
Riemann surface Σ with punctures. This Riemann surface is just a thickening of the
web diagram of the toric singularity [45, 46, 47], with punctures associated to external
legs of the web diagram. The mirror D6-branes wrap non-trivial 1-cycles on this Rie-
mann surface, with their intersections giving rise to bi-fundamental chiral multiplets,
and superpotential terms arising from closed discs bounded by the D6-branes. In [19],
it was shown that D7–branes passing through the singular point can be described in
the mirror Riemann surface Σ by non-compact 1-cycles which come from infinity at one
puncture and go to infinity at another. Figure 18 shows the 1-cycles corresponding to
some D3- and D7-branes in the Riemann surface in the geometry mirror to the complex
cone over dP2. A D7-brane leads to flavors for the two D3-brane gauge factors whose
1-cycles are intersected by the D7-brane 1-cycle, and there is a cubic coupling among
the three fields (related to the disk bounded by the three 1-cycles in the Riemann
surface).
Figure 17: Dimer diagram for D3–branes at a dP2 singularity.
Figure 18: Riemann surface in the geometry mirror to the complex cone over dP2, shown
as a tiling of a T 2 with punctures (denoted by capital letters). The figure shows the non-
compact 1-cycles extending between punctures, corresponding to D7-branes, and a piece of
the 1-cycles that correspond to the mirror of the D3-branes.
U(M) U(M)
U(2M)
PSfrag replacements
Q1i Qi3
Figure 19: Quiver for the dP2 theory with M fractional branes and flavors.
As stated in Section 4, given a gauge theory of D3-branes at a toric singularity,
we introduce flavors for some of the gauge factors in a specific way. We pick a term
in the superpotential, and we introduce flavors for all the involved gauge factors, and
coupling to all the involved bifundamental multiplets. For example, the quiver with
flavors for the dP2 theory is shown in Figure 19.
On the Riemann surface, this procedure amounts to picking a node and introducing
D7-branes crossing all the edges ending on the node, see Figure 18. In this example
we obtain the superpotential terms
Wflavor = λ
′(Q1iQ̃i3Y31 +Q3jQ̃j5X53 +Q5kQ̃k1X15) (B.1)
In addition we introduce mass terms
Wmass = m1Q1iQ̃k1 +m2Q3jQ̃i3 +m5Q5kQ̃j5 (B.2)
This procedure is completely general and applies to all gauge theories for branes at
toric singularities7.
C Detailed proof of Section 4
Recall that in Section 4 we considered the illustrative example of the gauge theory
given by the quiver in Figure 20. Since node 2 is the one we wish to dualize, the only
relevant part of the diagram is shown in Figure 21. We show the Seiberg dual in Figure
22. The above choice of D7–branes, which we showed in appendix B can be applied
to arbitrary toric singularities, gives us the superpotential terms
Wflavor = λ
′ (X32Q2bQb3 + X21Q1aQa2 + X14Q4dQd1 + Y43Q3cQc4 )
Wmass = m2Qa2Q2b + m3Qb3Q3c + m4Qc4Q4d + m1Qd1Q1a (C.1)
Taking the Seiberg dual of node 2 gives
Wflavor dual = λ
′ (S13bQb3 + R
a1Q1a + X14Q4dQd1 + Y43Q3cQc4 )
Wmass dual = m2Xab + m3Qb3Q3c + m4Qc4Q4d + m1Qd1Q1a
Wmesons = h (XabQ̃b2Q̃2a
+ R1a1X̃12Q̃2a + R
a1Ỹ12Q̃2a
7This procedure does not apply if the superpotential (regarded as a loop in the quiver) passes twice
through the node which is eventually dualized in the derivation of the metastable vacua. However we
have found no example of this for any DSB fractional branes.
PSfrag replacements
X21 Y21
Y32 Z32
X43 Y43
Q2b Qb3
Q4dQd1
Figure 20: Quiver diagram with flavors. White nodes denote flavor groups
PSfrag replacements
X21 Y21
Y32 Z32Qa2
Figure 21: Relevant part of quiver before Seiberg duality.
PSfrag replacements
X̃12 Ỹ12
Ỹ23Z̃23
M1, . . . ,M6
Figure 22: Relevant part of the quiver after Seiberg duality on node 2.
+ S13bQ̃b2X̃23 + S
3bQ̃b2Ỹ23 + S
3bQ̃b2Z̃23
+ M131X̃12X̃23 + M
31X̃12Ỹ23 + M
31X̃12Z̃23
+ M431Ỹ12X̃23 + M
31Ỹ12Ỹ23 + M
31Ỹ12Z̃23 ) (C.2)
where we have not included the original superpotential. The crucial point is that
the underlined terms appear for any quiver gauge theory with flavors introduced as
described in appendix B. As described in the main text, supersymmetry is broken
by the rank condition due to the F-term of the dual meson associated to the massive
flavors. Our vacuum ansatz is (we take Nf = 2 and Nc = 1 for simplicity; this does
not affect our conclusions)
Q̃b2 =
; Q̃2a = (µ1Nc ; 0) (C.3)
with all other vevs set to zero. We parametrize the perturbations around this minimum
Q̃b2 =
µ+ φ1
; Q̃2a = (µ+ φ3 ; φ4) ; Xab =
X00 X01
X10 X11
(C.4)
and the underlined terms give
hXabQ̃b2Q̃2a − hµ2Xab = hX11 φ2 φ4 − hµ2X11 + hµ φ2X01 + hµ φ4X10
+ hµ φ1X00 + hµ φ3X00 + hφ1 φ3X00 + hφ2 φ3X01
+ hφ1 φ4X10 (C.5)
It is important to note that all the fields in (C.4) will have quadratic couplings only in
the underlined term (C.5). Thus, one can safely study this term, and the conclusions
are independent of the other terms in the superpotential. Diagonalizing (C.5) gives
hXabQ̃b2Q̃2a − hµ2Xab = hX11 φ2 φ4 − hµ2X11 + hµ φ2X01 + hµ φ4X10
2hµ φ+X00 +
φ2+X00 −
φ2−X00
(ξ+ − ξ−)φ2X01 +
(ξ+ + ξ−)φ4X10 (C.6)
where
(φ1 + φ3) ; ξ− =
(φ1 − φ3) (C.7)
This term is similar to the generalized asymmetric case studied in appendix A.2 with
X11 → X ; φ4 → φ1 ; φ2 → φ2 ; X10 → φ3 ; X01 → φ4 (C.8)
So here X11 is the linear term that breaks supersymmetry, and φ2, φ4 are the broken
supersymmetry fields. In (C.6), the only massless fields at tree-level are X11 and
ξ−. Comparing to the ISS case in Section 2.1 shows that Im ξ− is a Goldstone boson
and X11, Re ξ− get mass at tree-level. As for φ2 and φ4, setting ρ+ =
(φ2 + φ4) and
(φ2−φ4) gives us Re(ρ+) and Im (ρ−) massless and the rest massive. Following
the discussion in Section 2.1, Re(ρ+) and Im (ρ−) are just the Goldstone bosons of the
broken SU(Nf ) symmetry
8. We have thus shown that the dualized flavors (e.g. Q̃b2,
Q̃2a) and the meson with two flavor indices (e.g. Xab) get mass at tree-level or at 1-loop
unless they are Goldstone bosons. Now, we need to verify that this is the case for the
remaining fields.
PSfrag replacements
Q4dQd1
X̃12 Ỹ12
Ỹ23Z̃23
M1..M6
Figure 23: Quiver after Seiberg duality on node 2.
The Seiberg dual of the original quiver diagram is shown in Figure 23. The dual-
ized bi-fundamentals come in two classes. The first are the ones that initially (before
dualizing) had cubic flavor couplings, there will always be only two of those (e.g. X̃12,
X̃23). The second are those that did not initially have cubic couplings to flavors, there
is an arbitrary number of those (e.g. Ỹ12, Ỹ23, Z̃23). Figure 24 shows the relevant part
of the quiver for the first class. Recalling the superpotential terms (C.2), there are
several possible sources of tree-level masses. For instance, these can arise in Wflavor dual
and Wmass dual. Also, remembering our assignation of vevs in (C.3), tree-level masses
can also arise in Wmesons from cubic couplings involving the broken supersymmetry
fields (e.g. Q̃b2, Q̃2a). The first class of bi-fundamentals (e.g. X̃12, X̃23) only appear in
Wmesons coupled to their respective mesons (e.g. R
1, S1). In turn these mesons will ap-
8In the case where the flavor group is SU(2), these Goldstone bosons are associated to the gener-
ators tx and ty.
PSfrag replacements
M1, . . . ,M6
Figure 24: Relevant part of dual quiver for first class of bi-fundamentals.
pear in quadratic terms in Wflavor dual coupled to flavors (e.g. S
3bQb3 and R
a1Q1a), and
these flavors each appear in one term in Wmass. Thus there are two sets of three terms
which are coupled at tree-level and which always couple in the same way. Consider for
instance the term
λ′ S13bQb3 + m3Qb3Q3c + hS
3bQ̃b2X̃23 = λ
′ (S1 S2)
+m1(C1 C2)
+ h (S1 S2)
µ+ φ1
= λ′(S1B1 + S2B2) +m1(B1C1 +B2C2)
+ hµS1 X̃23 + hS1 φ1 X̃23 + hS2 φ2 X̃23
(C.9)
where Si, Bi, Ci and X̃23 are the perturbations around the minimum. Diagonalizing
(which can be done analytically for any values of the couplings), we get that all terms
except one get tree-level masses, the massless field being:
Y = m1S2 − λ′C2 (C.10)
This massless field has a cubic coupling to φ2 X̃23 and gets mass at 1-loop since φ2 is
a broken supersymmetry field, as described in appendix A.2.
Figure 25 shows the relevant part of the quiver for the second class of bi-fundamentals
(i.e. those that are dualized but do not have cubic flavor couplings).
These fields and their mesons only appear in one term, so will always couple in the
same way. Taking as an example
hR2a1Ỹ12Q̃2a =
Ỹ12 (µ+ φ3 ; φ4)
= µR1 Ỹ12 + R1 φ3 Ỹ12 +R2 φ4 Ỹ12 (C.11)
PSfrag replacements
M1, . . . ,M6
Figure 25: Relevant part of dual quiver for second class of bi-fundamentals.
This shows that R1 and Ỹ12 get tree-level masses and R2 gets a mass at 1-loop since
it couples to the broken supersymmetry field φ4. The only remaining fields are flavors
like Qc4, Q4d, which do not transform in a gauge group adjacent to the dualized node
(i.e. not adjacent in the quiver loop corresponding to the superpotential term used to
introduce flavors). These are directly massive from the tree-level Wmass term.
So, as stated, all fields except those that appear in the original superpotential (i.e.
mesons with gauge indices and bi-fundamentals which are not dualized) get masses
either at tree-level or at one-loop. So we only need to check the dualized original
superpotential to see if we have a metastable vacua.
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	Introduction
	The ISS model revisited
	The ISS metastable minimum
	The Goldstone bosons
	Meta-stable vacua in quiver gauge theories with DSB branes
	The complex cone over dP1
	Additional examples: The dP2 case
	The general case
	The general argument
	Construction of the flavored theories
	Seiberg duality and one-loop masses
	Additional examples
	The dP3 case
	Phase 1 of PdP4
	The Yp,q family
	Conclusions and outlook
	Technical details about the calculation via Feynman diagrams
	The basic amplitudes
	The basic superpotentials
	D7–branes in the Riemann surface
	Detailed proof of Section 4
