Available online at www.sclencedlrect.com

ScienceDirect

Nuclear Physics B 796 (2008) 44-65

www. elsevier. com/locate/nuclphysb

Geometric transitions and D-term SUSY breaking

Mina Aganagic, Christopher Beem *

Department of Physics, University of California, Berkeley, CA 94720, USA Received 26 November 2007; accepted 29 November 2007 Available online 5 December 2007

Abstract

We propose a new way of using geometric transitions to study metastable vacua in string theory and certain confining gauge theories. The gauge theories in question are N = 2 supersymmetric theories deformed to N = 1 by superpotential terms. We first geometrically engineer supersymmetry-breaking vacua by wrapping D5 branes on rigid 2-cycles in non-compact Calabi-Yau geometries, such that the central charges of the branes are misaligned. In a limit of slightly misaligned charges, this has a gauge theory description, where supersymmetry is broken by Fayet-Iliopoulos D-terms. Geometric transitions relate these configurations to dual Calabi-Yaus with fluxes, where Hrr, Hns and dJ are all non-vanishing. We argue that the dual geometry can be effectively used to study the resulting non-supersymmetric, confining vacua.

© 2007 Elsevier B.V. All rights reserved. PACS: 11.25.Uv; 11.25.Tq; 11.25.Mj

Keywords: Geometric transitions; SUSY breaking; D-term; Conifold; Geometric fluxes; Fayet-Illiopoulos

1. Introduction

In the interest of finding controllable, realistic string vacua, it is important to find simple and tractable mechanisms of breaking supersymmetry in string theory. A powerful method which has been put forward in [1] consists of geometrically engineering metastable vacua with D-branes wrapping cycles in a Calabi-Yau manifold, and using geometric transitions and topological string techniques to analyze them. In [1], metastable vacua were engineered by wrapping D5 branes and anti-D5 branes on rigid 2-cycles in a Calabi-Yau. The non-

* Corresponding author.

E-mail address: beem@berkeley.edu (C. Beem).

0550-3213/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2007.11.033

supersymmetric vacuum obtained in this fashion was argued to have a simple closed-string dual description, in which branes and anti-branes are replaced by fluxes. In [2], geometric transitions were used to study the physics of D-brane theories that break supersym-metry dynamically. Namely, the authors showed that the instanton-generated superpotential that triggers supersymmetry breaking can be computed by classical means in a dual geometry, where some of the branes are replaced by fluxes (for an alternative approach see [3]).

In this paper, we propose another way of using geometric transitions to study supersymmetry breaking. As in [1], we consider D5 branes wrapping rigid cycles in a non-compact Calabi-Yau X. If b2(X) > 1, supersymmetry can be broken by choosing the complexified Kahler moduli so as to misalign the central charges of the branes,

Since the 2-cycles wrapped by the D5 branes are rigid, any deformation of the branes costs energy, and the system is guaranteed to be metastable.1 In the extreme case of anti-aligned central charges, we recover the brane/anti-brane configurations of [1,5]. For slightly misaligned central charges, the system has a gauge theory description in terms of an N = 2 quiver theory deformed to N = 1 by superpotential terms [6,7], and with Fayet-Iliopoulos D-terms turned on [8-10].2 The D-terms trigger spontaneous supersymmetry breaking in the gauge theory.

We argue that the dynamics of this system is effectively captured by a dual Calabi-Yau with all branes replaced by fluxes. Turning on generic Kahler moduli on the open-string side has a simple interpretation in the dual low-energy effective theory as turning on a more generic set of FI parameters than hitherto considered in this context, but which are allowed by the N = 2 supersymmetry of the background. On shell, this breaks some or all of the N = 2 supersymmetry. Geometrically, this corresponds to not only turning on HNS and HRR fluxes on the Calabi-Yau, but also allowing for dJ = 0 [12,13].3 Moreover, we show that the Calabi-Yau geometries with these fluxes turned on have non-supersymmetric, metastable vacua, as expected by construction in the open-string theory.

The paper is organized as follows: In Section 2, we review the physics of D5 branes on a single conifold and the dual geometry after the transition, paying close attention to the effect of Fayet-Iliopoulos terms. In Section 3, we consider the case of an A2 geometry where misaligned central charges lead to supersymmetry breaking. We provide evidence that the dual geometry correctly captures the physics of the non-supersymmetric brane system. We show that the results are consistent with expectations from the gauge theory, to the extent that these are available. We also comment on the relation of this work to [2], and point out some possible future directions. In Appendix A, we lay out the general case for larger quiver theories. We show that in the limit of large separations between nodes, the theory has metastable, non-supersymmetric vacua in all cases where they are expected.

1 Supersymmetry-breaking by misaligning the central charges of D5 branes wrapped on rigid curves was also studied in [4] in the context of compact Calabi-Yau manifolds.

2 Some geometric aspects of supersymmetry breaking by F-terms in this context were recently discussed in [11].

3 For another example of supersymmetry breaking by turning on Hns , Hrr and dJ fluxes, see [14].

2. The conifold

In this section, we consider N D5 branes on the resolved conifold. We will first review the open-string theory on the branes, and then discuss the dual closed-string description. Our discussion will be more general than the canonical treatment in that we will consider the case where the D5-branes possess an arbitrary central charge.

2.1. The D-brane theory

To begin with, let us recall the well-known physics of N D5 branes wrapping the S2 tip of the resolved conifold. This geometry can be represented as a hypersurface in C4 [u, v, z, t],

uv = z(z — mt).

The geometry has a singularity at the origin of C4 which can be repaired by blowing up a rigid P1. This gives the P1 a complexified Kahler class

Z = J (J + iBNs) = j + ibNs. (2.1)

The theory on the D5 branes at vanishing j reduces in the field theory limit to a d = 4, N = 1, U(N) gauge theory with an adjoint-valued chiral superfield of mass m. The bare gauge coupling is given by

4n bNS

<?ym gs

for positive bNS. In string theory, the tension of the branes generates an energy density related to the four-dimensional gauge coupling by

2N bNS

V* = — = N b—. (2.3)

gYM 2ngs

Turning on a small, nonzero j can be viewed as a deformation of this theory by a Fayet-Iliopoulos parameter for the U(1) center of the gauge group [8,9]. This deforms the Lagrangian by

AL = 42% Tr D, (2.4)

where D is the auxiliary field in the N = 1 vector multiplet and

% = ~r~ , (2.5)

where the factor of gs comes from the disk amplitude. This deformation (2.4) breaks the N = 1 supersymmetry which was linearly realized at j = 0. In particular, turning on j, increases the energy of the vacuum: integrating out D from the theory by completing the square in the auxiliary field Lagrangian,

Ld = —|^Tr D2 + Tr D,

raises the vacuum energy to

bNS ( 1 j2 \

V* = N-N^ 1 + - j- . (2.6)

2ngs\ 2 bNSz

Supersymmetry is not broken, however. At nonzero j, a different N = 1 supersymmetry is preserved4—one that was realized nonlinearly at vanishing j [15,16]. Which subgroup of the background N = 2 supersymmetry is preserved by the branes is determined by Z in (2.1), the BPS central charge in the extended supersymmetry algebra.5 For any Z, the open-string theory on the branes has an alternative description which is manifestly N = 1 supersymmetric, with vanishing FI term and bare gauge coupling related to the magnitude of the central charge [7],

1 ^S +j 2 (2.7)

gYM 4ngs

Geometrically, this is just the quantum volume of the resolving P1. As such, the central charge also determines the exact tension of a single D5 brane at nonzero j,

VbNs + j2

V* = N^--. (2.8)

For small Kahler parameter, j < bNS,

this agrees with the vacuum energy in the field theory limit (2.6).

For any j, the theory is massive; it is expected to exhibit confinement and gaugino condensation at low energies, leaving an effective U(1) gauge theory in terms of the center of the original U(N) gauge group. We will show next that the strongly coupled theory has a simple description for any value of Z in terms of a large N dual geometry with fluxes.

2.2. The geometric transition at general Z

We will now discuss the large N dual geometry for general values of the central charge Z. Special cases (either vanishing j or vanishing bNS) have been considered in the literature, but the present, expanded discussion is, to our knowledge, new.6 We will see that the dual geometry exactly reproduces the expected D5 brane physics. From the perspective of the low-energy effective action, the consideration of general central charge corresponds to turning on a more general set of N = 2 FI terms than previously considered in this context. Geometrically, this will lead us to consider generalized Calabi-Yau manifolds, for which dJ is non-vanishing in addition to having HNS and HRR fluxes turned on. This will provide a local description of the physics for each set of branes in the more general supersymmetry-breaking cases of Section 3 and Appendix A.

4 This is true even in the field theory limit, despite the presence of the constant FI term. Namely, a second, nonlinearly realized supersymmetry is present in the gauge theory as long as there is only a constant energy density [15]. We thank A. Strominger for explaining this to us.

5 Strictly speaking, the central charge of N branes is NZ. In this paper, we will always take the number of branes N to be positive, so that we interpolate between the branes and anti-branes by varying Z.

6 See related discussion in [7].

To begin, let us recall the large N dual description of the D5 brane theory at vanishing j. This is given in terms of closed-string theory on the deformed conifold geometry,

uv = z(z — mt) + s. (2.9)

This is related to the open-string geometry by a geometric transition which shrinks the P1 and replaces it with an S3 of nonzero size,

where A is the 3-cycle corresponding to the new S3, and the period of the holomorphic three-form over A is related to the parameters of the geometry by S = s/m. The D5 branes have disappeared and been replaced by N units of Ramond-Ramond flux through the S3,

HRR = N. (2.10)

There are Ramond-Ramond and Neveu-Schwarz fluxes through the dual, non-compact B-cycle as well,

a = j (Hrr + iHns/gs) = bRR + ibns/gs, (2.11)

which corresponds to the complexified gauge coupling of the open-string theory,

a = — +

2n ' gYM-

The B -cycle is cut off at the scale, at which a is measured.7 The dependence of a on the IR cutoff in the geometry corresponds to its renormalization group running in the open-string theory.

If it were not for the fluxes, the theory would have N = 2 supersymmetry, with S being the lowest component of an N = 2 U(1) vector multiplet. That theory is completely described by specifying the prepotential, F0(S), which can be determined by a classical geometry computation,

Q = d F0. dS 0

The presence of nonzero fluxes introduces electric and magnetic Fayet-Iliopoulos terms in the low-energy theory for the U(1) vector multiplet and its magnetic dual [17-20]. The effect of the fluxes (2.10), (2.11) can also be described in the language of N = 1 superspace as turning on a superpotential for the N = 1 chiral superfield with S as its scalar component,

RR + iHNS'

W(S) = j Q A (HRR + iHNS/g.

7 For simplicity, the IIB axion is set to zero in this paper.

For the background in question, this takes the form

W(S) = aS — N —Fo. (2.12)

In terms of the parameters of the D-brane theory, S is identified with the vev of the gaugino condensate. One way to see this is by comparing the superpotentials on the two sides of the duality. The aS superpotential on the closed string side corresponds to the classical superpotential term 4 Tr Wa Wa on the gauge theory side.

What does the FI term deformation of the D-brane theory correspond to in the closed-string theory? To begin with, let us address this question from the perspective of the low-energy effective action. We know that the U(1) gauge field after the transition coincides [6,21] with the U(1) gauge field that is left over after the SU(N) factor of the gauge group confines. This suggests that we should simply identify Fayet-Iliopoulos D-terms on the two sides. More precisely, the Lagrangian of the theory after the transition can be written in terms of N = 1 superfields,

5 = S + 4l0f + 00F, (2.13)

Wa = -ika + OaD + 2 (<7^0)^v, (2.14)

as an N = 2 action deformed to N = 1 by the superpotential (2.12),

L = ±Im( f d20d20 S F + i d201 ^ W a Wa + 2 f 4n \J dS J 2 dS2 a J

The Fayet-Iliopoulos deformation (2.4) should produce an additional term in this Lagrangian, AL =—j-D, (2.16)

2V2ngs

corresponding to an FI term as in (2.5). Note that on the D-brane side, the center of mass U(1) corresponds to 1/N times the identity matrix in U(N), so that the normalization of (2.16) precisely matches (2.4).

We now show that the deformation (2.16) leads to exactly the physics that we expect on the basis of large N duality. After turning on the FI term, the effective potential of the theory becomes

y = 4nGSS (|9S W I2 + j/gs I2) + const (2.17)

1 d2 Gss = — (T — T), T = —t To-2i 9S2 0

We have also shifted the potential by an (arbitrary) constant, which we choose to be the tension of the branes at vanishing j,

const = N -

for convenience. We can then rewrite (2.17) as

V = 1 (|a — Nt |2 + Ij/gs |2) + const = / Ia — Nt |2 + const, (2.18) 2n(T — t) 2n(T — t)

a = bRR +--JbNS + j2

gsy NS

and the constant has shifted. As expected from the D-brane picture, the effective potential of the theory with the FI term turned on and with gauge coupling (2.2) is the same as that of the theory without the FI term and with gauge coupling (2.7).

In this simple example, the prepotential is known to be given exactly by

mm=2 S2(>og( -3

The vacuum of the theory is determined by the minimum of (2.18), which occurs at

<5- NT = 0, (2.19)

or, in terms of the expectation value of the gaugino bilinear, at

S*= mA2exp(2nia/N). (2.20)

Finally, we note that the energy in the vacuum (2.20) is larger than that in the j = 0 vacuum by the constant that enters (2.18), giving

%s + j 2

V* = N-, (2.21)

which is precisely the tension of the brane after turning on j. This is a strong indication that we have identified parameters correctly on the two sides of the duality. It is easy to see that in the vacuum, neither the F-term

F = dS W,

nor the D-term vanishes. Nevertheless, as will now show, this new vacuum preserves half of the N = 2 supersymmetry of the theory we started with, though not the one manifest in the action as written. Defining the SU(2)R doublet of fermions

- = (t

the relevant part of the supersymmetry transformations of the N = 2 theory are

= xij€j

where X is a matrix of F- and D-terms, shifted by an imaginary part due to the presence of a "magnetic" FI term (see, for example, [1] and references therein)

X (-Yl -f2 +N y „) (2.22)

V2V. Y3 Y1 - 1Y2 + N) y ;

where the N = 2 auxiliary fields are identified with the auxiliary F-term of S in (2.13) and the D-term of the gauge field in (2.14) according to

(Y1 + iY2) = 2iF, Y3 =V2D.

Note that the triplet Y = (Y1,Y2,Y3) transform like a vector of the SU(2)R symmetry of the N = 2 theory. In the vacuum (2.19)

iN (bNS - ybNs + j2 j '

y2(bNs + j 2 A j bNS -yjbNs + j 2 /

The supersymmetry manifest in (2.15) corresponds to e1, and it is clearly broken in the vacuum for non-vanishing j, since neither the F- nor the D-term vanish. However, the determinant of X vanishes, and so there is a zero eigenvector corresponding to a preserved supersymmetry.

So far, we have identified turning on j with turning on an FI term in the low-energy effective action. It is natural to ask what this corresponds to geometrically in the Calabi-Yau manifold. In [17,18] (following [22,23]) it was shown that turning on a subset of the FI terms of the low-energy N = 2 theory arising from IIB compactified on a Calabi-Yau manifold corresponds to turning on HNS and HRR fluxes in the geometry. This is what we used in (2.12). The question of what corresponds to introducing the full set of FI terms allowed by N = 2 supersymmetry was studied, for example, in [12,13]. To make the SU(2)R symmetry of the theory manifest, we can write the triplet of the N = 2 FI terms as

E=_L (-e1 - iE2 E3

V2 V E3 El - iE2 where (E1 ,E2,E3) transform as a vector under SU(2)R and enter the action as 1

— Re(Tr XE).

These are given in terms of ten-dimensional quantities by8

E1 = j HNS/gs, E2 = j Hrr, E3 = j dJ/gs.

Note that this agrees precisely with what we have just derived using large N duality. Just as the bare gauge coupling fS2 BNS /gs = bNS /gs gets mapped to fB HNS /gs after the transition due to running of the coupling, SU(2)R covariance of the theory demands that turning on fS2 J/gs = j/gs before the transition get mapped to turning on fB dJ/gs after the transition. Moreover, we saw in this section that the latter coupling gets identified as a Fayet-Iliopoulos D-term for the U(1) gauge field on the gravity side. This exactly matches the result of [12,13], since E3 is the Fayet-Iliopoulos D-term parameter. It is encouraging to note that [12,13] reach this conclusion via arguments completely orthogonal to our own. Finally, we observe that an SO(2) c SU(2)R rotation can be used to set the Fayet-Iliopoulos D-term E3 = j/gs to zero, at the expense of

replacing E1 = bNS/gs by E1 = ^bNS + j2/gs, and this directly reproduces (2.18). 3. An A2 fibration and the geometric engineering of a metastable vacuum

By wrapping D5 branes on rigid P1's in more general geometries with b2(X) > 1, we can engineer vacua which are guaranteed to be massive and break supersymmetry by choosing the

8 This follows from Eq. (3.53) of [13] up to an SU(2)r rotation and specializing to a local Calabi-Yau. More precisely, to derive this statement one needs to look at the transformations of the N = 2 gauginos, not the gravitino as in [13], but these are closely related. See, for example, [23].

central charges of the branes to be misaligned. Since the D-brane theories experience confinement and gaugino condensation at low energies, we expect to be able to study the dynamics of these vacua in the dual geometries where the branes are replaced by fluxes.

In this section, we will consider the simple example of an A2 singularity fibered over the complex plane C[i]. This is described as a hypersurface in C4,

uv = z(z — mt)(z — m(t — a). (3.1)

This geometry has two singular points at u,v,z = 0 and t = 0,a. The singularities are isolated, and blowing them up replaces each with a rigid P1. The two Px's are independent in homology, and the local geometry near each of them is the same as that studied in the previous section.

Consider now wrapping Ni D5 branes on the P1 at t = 0 and N2 branes on the P1 at t = a. If the central charges of the branes,

7A = J J + iBNS = ji + ibNS,i

Zi = J + iBNS = ji + ibNS,i (3.2)

are aligned (e.g., if the Kahler parameters ji both vanish), the theory on the branes has N = 1 supersymmetry. At sufficiently low energies, it reduces to a U(N1) x U(N2) gauge theory with a bifundamental hypermultiplet Q, Q, a pair of adjoint-valued chiral fields and a superpotential given by

W = - Tr $2 — ~m Tr $22 — a Tr QQ+ Tr(Q^1 Q — QQ$2)- (3.3)

For a small relative phase of the central charges, e.g., if the theory at vanishing Kahler parameters is deformed by ji by

ji/bNSi« 1, (3.4)

we expect this to have a pure gauge theory description at low energies in terms of the supersym-metric theory with Fayet-Iliopoulos terms for the two U(1)'s. Misaligning the central charges such that

Z1 = C12Z2, (3.5)

for any positive, real constant c12, should break all the supersymmetries of the background. Nevertheless, for large enough m and a, the vacuum should be stable. Since the theory is massive, we expect it to exhibit confinement at very low energies, with broken supersymmetry. Nevertheless, as we will now argue, the dynamics of the theory can be studied effectively for any ji in the dual geometry, where the branes have been replaced by fluxes.

3.1. Large N dual geometry

The Calabi-Yau (3.1) has a geometric transition which replaces the two P1's by two S3's, S2 ^ S?, i = 1, 2.

The complex structure of the geometry after the transition is encoded in its description as a hypersurface,

= z(z — mt)(z — m(t — a) + ct + d, (3.6)

where c, d are related to the periods, S1,2, of the 3-cycles, A1,2, corresponding to the two S3's, d

f Q, — F0 = f Q.

J , dSi 0 J

As before, Bi are the non-compact 3-cycles dual to Ai, and F0 is the prepotential of the N = 2 theory. The prepotential in this geometry is again given by an exact formula,

^=1^ a-) -3)+2 <'»g( A-) -1)—^^ A;>

The theory with Ni D5 branes on the Pi1 before the transition is dual to a theory with Ni units of RR flux through S3 after the transition:

hrr = Ni.

There are additional fluxes turned on through the non-compact, dual B -cycles,

ai = j (Hrr + iHNS/gs) = bRR,i + ibNS,i/gs, Bi

corresponding to running gauge couplings, and

dJ/gs = ji/gs

corresponding to Fayet-Iliopoulos terms. The fluxes generate a superpotential,

W = J Q A (Hrr + iHNS/gs) X

^ d W = £ ai Si — Ni— F0,

and Fayet-Iliopoulos D-terms,

? 2^7iizgs 1,

where Di are auxiliary fields in the two N = 1 vector multiplets.

Large N duality predicts that for misaligned central charges (3.5), the fluxes should break all supersymmetries, and moreover, that the non-supersymmetric vacuum should be metastable. We will now show that this indeed the case. The tree-level effective potential of the theory is

y = 4nGik & W dkW + jijk/g2) + const,

1 d2 r

G = — (T — T)ik, Tik =-J^0,

lk 2i( 'k dSidSk

and the Kahler metric is determined by the off-shell N = 2 supersymmetry of the background. We have shifted the zero of the potential energy by the tension of the branes at vanishing ji,

V^ at bNS,i

const = Ni-

i = 1,2

In the case at hand,

' 2ngs

T11 = logf' t22 =

2ni \A2mJ 2ni \A^(-m)

whereas r12 is a constant9 independent of the S7, _ 1 / a 12 2ni \ Ao

It is straightforward to see that the critical points of the potential correspond to solutions of

Re(ai) + Re(rik)Nk = 0,

GjiGjk(Im(ai) Im(ak) + jtjk/g,2) = (Nj f.

The first equation fixes the phase of Si's, and the second their magnitude. Consider the case where the two nodes are widely separated, namely, where the sizes Si of the two S3's are much smaller than the separation a between them. In this limit, the equations of motion can be easily solved to obtain

N / a \ N2 cos 012

sN,1 = (A0m)NM —) exp(2nia 1) + ■■■,

N / a \N1cos 012

sNl = {-A2m)N2( — ) exp(2nic5 2) + •••, (3.7)

where 0ij is the relative phase between the central charges Zi and Zj. We can see that in the limit where the Zi are aligned, this reduces to the simple case without FI terms where the effective gauge coupling has been replaced with the parameter ai. The case of anti-aligned central charges was studied in [1,5]. The weak coupling limit of two widely separated nodes, in which our approximation is justified, corresponds to

SUt. « a< Ao. (3.8)

S7,*'s should be identified with the vev's of gaugino condensates on the branes, and are the order parameters of the theory. This is the case even in the presence of FI terms, as explained in the previous section. For small FI terms, this relies only on the off-shell N = 1 supersymmetry of the theories on both sides of the duality and a comparison of superpotentials. In [1] it was conjectured that this also holds in the brane/anti-brane case, where the central charges are anti-aligned and supersymmetry is maximally broken. It is natural, then, that the above limit should correspond to the theory being weakly coupled at the scale of the superpotential (3.3). In the same limit, the vacuum energy is given by

v/bNS,1 + j12 VbNS,2 + j2 1 / a \

V* = N11—-- + N2-—-- + —T N1N2 log — (1 - cos 012) + •

2ngs 2ngs 4n2 \AoJ

9 For convenience, we will take T12 to be purely imaginary.

Fig. 1. The A2 geometry from in the text, drawn in the T-dual NS5 brane picture. The D5 branes map to D4 branes and appear as red lines. The NS5 branes are drawn as blue lines/points. At 912 = 0 the system is supersymmetric. For any other value of 9i2, supersymmetry is broken. Varying 9i2 continuously from zero to n produces a geometry which is related to the original A2 geometry by a flop. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Note that in the limit of aligned central charges, the potential energy is simply the brane tension. This is in fact true exactly, and is related to the fact (which we will demonstrate later on) that in this case supersymmetry is preserved. For any other value of the angle, there is an additional attraction. In the extreme case, when we increase 912 from zero to n, we end up with a brane/anti-brane system on the flopped geometry. We can view this as varying one of the Zi's until the BNS field through that cycle goes to minus itself. This is a flop, and by comparing to [5], it follows that the solution we found above for 912 = n precisely corresponds to a brane/anti-brane system in the flopped geometry.

To see that supersymmetry is broken by the vacuum at non-vanishing 912, we write the action (2.15) in an N = 2 invariant way in terms of N = 2 chiral multiplets Ai consisting of N = 1 chiral multiplets Si, and Wf,

Ai = (Si,wa)

A = S + + + 2 )F,V +

The appropriate N = 2 Lagrangian is given by

L = 41n Im(J d 49 d 4x Fo(A,^ + ^ Re(xfE'ab)

where xab is defined as in (2.22). Then, the relevant supersymmetry variations of the fermions are given by

= xabeb, (3.10)

and at the extrema of the effective potential,

X. = -i- (N + G'k Im(ak) -G'kjk/gs \ (311)

' V2\ -Gikjk/gs Ni - Gik Im(«kV' )

The equations of motion imply that the determinants of both X1i2 vanish. For each node, then, xi has one zero eigenvalue. It can be shown (see the general discussion of Appendix A), that (3.10) and (3.11) imply that a global supersymmetry is preserved if and only if the central charges

(3.2) are aligned, i.e., if a positive real constant c12 exists such that Z1 = C12Z2.

This is exactly as expected from the open-string picture, and provides a nice test of the large N duality conjecture for general central charges.

Now we will show that the vacuum is indeed metastable. Consider the masses of bosonic fluctuations about the non-supersymmetric vacuum. As was found to be the case in [5], the Hessian of the scalar potential can be block diagonalized. After changing variables to bring the kinetic terms into their canonical form, the eigenvalues become

M? „ =

(a2 + b2 + 2abv) (a + b)2(a - b)2 + 4abv(a + b)(b + a)

2(1 - v)2 ,

2 (a2 + b2 + 2abv cos0) ± V(a + b)2(a - b)2 + 4abv(a + b cos 0)(b + a cos 0) M^ = 2(1 - v)2

(3.12)

where we have adopted the notation of [1] in defining

N1 , N2

2tfGn|S1|' 2^G22|S2|'

v = G12

G11G22 '

with all quantities evaluated in the vacuum. In addition, we have introduced a new angle, which is defined by the equation

Y1 • Y2 = N1N2cos 0,

where Y' are related to X' as in (2.22) for each i. In the limit (3.8),

0 = 012 + O(v),

so we can treat this as being the same as the phase which appears in (3.7), (3.9). Note that in the limit

cos 0 ^ 1, (3.13)

we recover the results for a supersymmetric system, with the masses of bosons becoming pair-wise degenerate. The other extreme of anti-aligned charges can be shown to correspond to

cos 0 ^ -1,

where the results of [5] should be recovered. Indeed by plugging in and rearranging terms we recover the mass formulas from p. 25 of [5]. These then provide exact values for the tree-level masses of the component fields in the supersymmetry-breaking vacuum. For any 0, the masses of the bosons are all positive as long as

v < 1.

This, in turn, is ensured as long as the metric on moduli space is positive definite in the vacuum. So indeed, the system is metastable, as expected.

To get a measure of supersymmetry breaking, let us now compare the masses of the bosons and the fermions in this vacuum. The fermion masses arise from the superspace interaction which appears as

¿ lm( / d 40 d4 x 2 j )j )(Xkcd0c0d )),

where Fijk = di dj dcF0. For the geometry in question, the prepotential is exact at one-loop order, and the third derivatives vanish except when all derivatives are with respect to the same field. We can then write the fermion mass matrices for a given node (and non-canonical kinetic terms) as

i _ 1 i

Mab = ig^2S €acXcd€db-

Performing a change of basis to give the fermion kinetic terms a canonical form, we can diago-nalize the resulting mass matrix and obtain the mass eigenvalues. There are two zero modes,

Mkh2 = 0,

corresponding to two broken supersymmetries. In addition there are two massive fermions, which we label by 2i,

„ (a + b) ±V(a - b)2 + 2abv(1 + cos0)

M-é, . =-. (3.14)

2(1 - v)

Note that in the supersymmetric limit (3.13), the masses of él and f2 match those of and 02,4, which have become pairwise degenerate. For small misalignment, and large separation of the two nodes, the mass splittings of bosons and fermions are easily seen to go like

Ml - Mé o2

2-2 ~ v022,

M2 + M2 12

where v goes to zero in the limit of large separation, and 012 measures the misalignment of the central charges.

3.2. Gauge theory limit

In the gauge theory limit (3.4), the vacuum energy (3.9) reduces to

v. = Z*i ^ (1 + 2 j-) + ¿ NN2'»g( ¿)( j - bNS2 )2 + ■■■.

i NS,i (3.15)

The first terms are classical contributions, as we saw in Section 2. The last term comes from a one-loop diagram in string theory, with strings stretched between the two stacks of branes running around the loop.

To begin with, consider the Abelian case,10

U(1) x U(1),

10 Although the rank of the gauge group is not large in this case, the geometric transition is still expected to provide a smooth interpolation between the open- and closed-string geometries. For a recent review, see [2], and references therein. It is natural to expect that for small deformations by FI terms that break supersymmetry, the two sides still provide dual descriptions of the same physics.

when the gauge theory has no strong dynamics at low energies. We should be able to reproduce (3.15) directly in the field theory by computing the one-loop vacuum amplitude in a theory with FI terms turned on. We can write the classical F- and D-term potential of the gauge theory as

Vtree = |F*1 I2 + |F*2 I2 + IFqI2 + IFqI2 + 1 ^,1(1?|2 -|q|2 - )2

+ 2 gYM,2(IqI2 -Iq I2 -^h)2,

F^1 = m$1 - qq, F$2 = m$2 - qq, Fq =q(a + $2 - $1), Fq= q(a + $2 - $1)

and $i, q,q are the lowest components of the corresponding chiral superfields. The gauge theory quantities are related to those of the string theory construction by

1 = bNS,i ^ = ji

The identification between the field theory FI term and the string theory parameter is expected to hold only for small ji/bNS,i. For nonzero f1,2, supersymmetry appears to be broken since the two D-term contributions cannot be simultaneously set to zero with the F-terms. In fact, we know that if the central charges are aligned, this is just a relic of writing the theory in the wrong superspace.

For large m, a, this potential has a critical point at the origin of field space. At this point, all the F-terms vanish, and there is pure D-term supersymmetry breaking. The spectrum of scalar adjoint and gauge boson masses is still supersymmetric at tree-level, since the only contribution to the masses in the Lagrangian is the FI-dependent piece for the bifundamentals. This means that the only relevant contribution to the one-loop corrected potential is from the bifundamental fields. The scalar components develop a tree-level mass which is simply given by11

m2q = a2 + r, m2 = a2 - r (3.16)

while the fermion masses retain their supersymmetric value,

2 2 2 mtq = mfi= a .

We have defined the constant

r = V2 (^Ym,2 - ^Ym,0 = V2 (j - jj^). (3.17)

The one-loop correction to the vacuum energy density is given by

^ = (Em4log A - Em4f log A

\ b A0 f Ao

where mb,f are the boson and the fermion masses, and A0 is the UV cutoff of the theory. The limit in which we expect a good large N dual is when the charged fields are very massive, r ^ a2,

11 One can easily check that for small r, the masses agree with what we expect from string theory. The bifundamental matter is the same as for the 0-4 system, with small B-fields turned on along the D4 branes. See, e.g., [24,25].

and at low energies the theory is a pure gauge theory. Expanding to the leading order in r/a2, the one-loop potential is then given by

y = ^ + ^r2log A'

We have omitted the A0 independent terms which correspond to the finite renormalization of the couplings in the Lagrangian and are ambiguous. We see that this exactly agrees with effective potential (3.9), (3.15) as computed in the dual geometry, after the transition.

In the general, U(Ni) x U(N2) case, we have a strongly coupled gauge theory at low energies. Nevertheless, since in the (Ni, Ni) sector supersymmetry is preserved, the one-loop contribution of that sector to the vacuum energy density should vanish beyond the classical contribution. Thus, we expect that only the bifundamental fields contribute to the vacuum energy at this level. The one-loop computation then goes through as in the Abelian case, up to the N1N2 factor from multiplicity, once again reproducing the answer (3.15) from large N dual geometry.

3.3. Relation to the work of [2]

We close with a comment on the relation to the work of [2], to put the present work in context. The A2 model at hand is the same as the geometry used to engineer the Fayet model in [2]. More precisely, the authors there engineered a "retrofitted" Fayet model. The parameter a that sets the mass of the bifundamentals was generated by stringy or fractional gauge theory instantons, and thus was much smaller than the scale set by the FI terms, which were taken to be generic. That resulted in F-term supersymmetry breaking which was dynamical.

In the present context, we still have a Fayet-type model, but we find ourselves in a different regime of parameters of the field theory, where r/a2 < 1, with r defined in terms of the FI parameters as in (3.17). Outside of this regime, the vacuum at the origin of field space, with Q and Q vanishing, becomes tachyonic even in the field theory, as can be seen from (3.16). Once this becomes the case, the large N dual presented here is unlikely to be a good description of the physics. For example, for N1 = N2 = N and r/a2 > 1, it was found in [2] that the theory has a non-supersymmetric vacuum where all the charged bifundamental fields are massive and the gauge symmetry is broken to U(N). This may still have a description in terms of some dual geometry with fluxes, but not the one at hand. This may be worth investigating.

Thus, unlike the models of [2], those considered here break supersymmetry spontaneously but not dynamically. It would be nice to find a way to retrofit the current models and to generate low scale supersymmetry breaking in this context. This would require finding a natural way of obtaining small FI terms. The mechanism of [2] does not apply here, since the terms in question are D-terms and not F-terms. This may be possible in the context of warped compactifications12 and compact Calabi-Yau manifolds, perhaps along the lines of [29].

Acknowledgements

We would like to thank T. Grimm, S. Hellerman, S. Kachru, L. Mazzucato, Y. Nakayama, D. Poland, J. Seo, A. Strominger, and C. Vafa for valuable discussions. This research was sup-

12 The effects of warping in the context of [1] have been studied in [26]. In [27], the authors constructed supergravity solutions that were subsequently interpreted in [28], to correspond to D5 branes on the conifold with Fayet-Iliopoulos terms turned on, i.e., the theory we studied in Section 2. We thank Y. Nakayama for pointing out to us the latter work.

ported in part by the UC Berkeley Center for Theoretical Physics and NSF grant PHY-0457317. The research of M.A. is also supported by a DOE OJI Award, the Alfred P. Sloan Fellowship.

Appendix A. Fayet-Iliopoulos terms for ADE singularities

The large N duality we studied in the previous sections should generalize to other ADE fibered geometries. In this appendix we will demonstrate that the large N dual geometries for these more general spaces have some of the same qualitative features. Consider the ADE type ALE spaces

which are fibered over the complex t -plane, allowing the coefficients parameterizing the deformations to be t dependent. The requisite deformations of the singularities are canonical (see [7] and references therein). In fibering this over the t-plane, the z7 become polynomials zi(t). At a generic point in the t-plane, the ALE space is smooth, with singularities resolved by blowing up r independent 2-cycle classes

where r is the rank of the corresponding Lie algebra. This corresponds to turning on Kahler moduli

The 2-cycles Sf intersect according to the ADE Dynkin diagram of the singularity. Consider now wrapping N7 D5 branes on the ith 2-cycle class. The theory on the branes is an N = 2 quiver theory with gauge group

with a bifundamental hypermultiplet Q7j, Qj7 for each pair of nodes connected by a link in the Dynkin diagram. The fibration breaks the supersymmetry to N = 1 by turning on superpotentials W7(@7) for the adjoint chiral multiplets @7,

which compute the holomorphic volumes of the 2-cycles at fixed t. The superpotentials Wi(t) can be thought of as parameterizing the choice of complex structure of the ALE space at each point in the t -plane. The full tree-level superpotential of the theory is given by

A, : x2 + y2 + zk+1 = 0, Dr : x2 + y2z + zr-1 = 0,

E6: x2 + y3 + z4 = 0, Et: x2 + y3 + yz3 = 0, E&: x2 + y3 + z5 = 0

Sf, 7 = 1,...,r,

Y\U(Ni),

W = J2TrW7(07) + J2Tr(Q7jQj7$7 - Qij&jQjï)

where the latter sum runs over nodes that are linked.

For vanishing ji, the structure of the vacua of the theory was computed in [7]. For each positive root ej of the lie algebra,

ei = ^ n\ei i

for positive integers nlI, one gets a rigid P1 at points in the t -plane t = ai,p,

Wi (a I,p) = J2 niW[(ahp) = 0. (A.1)

Here I labels the positive root and p runs over all the solutions to (A.1) for that root. The choice of vacuum breaks the gauge group down to

Y\U(Mip) i,p

Ni = £ Mi,pni. i

Turning on generic Fayet-Iliopoulos terms for the U(1) centers of the gauge group factors, ji

AC = -Tr Di,

i 2V2ngs

2niF0(S) = 5? S2H wê^Ù - 2) + "j*»-'^ >°g(

breaks supersymmetry while retaining (meta)stability of the vacuum as long as ji is much smaller than the mass of all the bifundamentals in the vacuum.

The ALE fibrations have geometric transitions in which each P1 is replaced by a minimal S3. The leading order prepotential F0 for all these singularities was computed in [30], and is given by

S 2 f- ( Sb \ 3 \ 1

+ •••, (A.2)

where the sum is over all critical points b = (i,p),

and i(b) = i denotes the root i to which the critical point labeled by b corresponds. We are neglecting cubic and higher order terms in the Si,p, which are related to higher loop corrections in the open string theory. Above, ei • ej is the inner product of two positive, though not necessarily simple, roots. Geometrically, the inner product is the same as minus the intersection number of the corresponding 2-cycles classes in the ALE space. In addition, there are fluxes turned on in the dual geometry which are determined by holography:

J Hrr = Ma,

j ^HRR + — HNS^ = bRR,I(a) + —bNS,I(a), j dJ = ji(a)

The theory on this geometry without fluxes is an N = 2, U(\)k gauge theory, where k is the number of S3's. The effect of the fluxes on the closed-string theory in this background was determined in [12,13]. The result is a set of electric and magnetic N = 2 Fayet-Iliopoulos terms, which enter the N = 2 superspace Lagrangian,

L = Im(J d 40 Fo(Aa)) + Re(Ya • Ea ).

(bNs ja

Ea = ~^,bRR,a, — V gs gs

and where the auxiliary fields Ya are shifted by the magnetic FI term,

Ma = (0, Ma, 0). The auxiliary field Lagrangian then has the form

Laux = -1 Gab Re(Y)a • Re(Y)b + -1 Re(tab) Re(Y)a • Mb + -1 Re(Y)a • Ea 8n 4n 4n

and integrating out the auxiliary fields sets them equal to their expectation values,

Ya = -Gab(Eb + Re(Tbc)Mc) + iMa or, to be more precise,

-GabYb = (bNS/gs, bRR + TabMb, ja/gs ).

We can make contact with the more familiar form of this action and its scalar potential by reducing to N = 1 superspace. There, the auxiliary fields Ya get identified with auxiliary fields of the vector and chiral multiplets corresponding to the ath S3, and the fluxes give rise to the usual flux superpotential

W = J2 a a Sa - MadSa Fo(S).

In addition, there are Fayet-Iliopoulos terms for the U(1)'s, and the total scalar potential is given by

y = -1nGab(da W dbW + ja jb/g2)

aa = bRR,a + ibNS,a/gs.

There are vacua at the field values which satisfy

day - FabcGCeGbd ((ae - iemMm)(a d - idnM") + jejdg) = 0.

At one-loop order, the prepotential has non-vanishing third derivatives only when all derivatives are with respect to the same field. The vacuum condition can be simplified to this order, and upon

considering the equation as two real equations for the real and imaginary part, the conditions become

(bRR,a - Re(Tab))Mb = 0,

GaCGad(bNS,cbNS,d + jcjd) = (Mags)2.

The first of these can be solved easily for the phases of the Sa. Moreover, we see that it is equivalent to the condition that the real part of the auxiliary fields Y2a vanish for all a in the N = 2 superspace Lagrangian,

Gab Re(y|) = 0. In light of that result, the second condition can be written as

Ya • Ya = 0. (A.3)

Since the supersymmetry transformations are Se^a = XaC + •••

X _ i-Ya - i Re(Y2)a + Ma Xa = Y3 Y? - i Re(Y2)a + Ma

(A.3) is precisely the condition that there exists some supersymmetry transformation on each node which is locally preserved by the vacuum. Of course, for supersymmetry to be conserved globally, these supersymmetry transformations must match for all nodes. The condition for this to be the case is

a b a b

YL = YL = YL

Ma Mb, Ma Mbl which, along with the requirement that the metric on moduli space be positive definite in the vacuum, requires that

Za = cabZb

for a positive, real constant cab. This conforms to our intuition from the open-string picture that preserving supersymmetry should require that the complex combination of the FI terms and gauge couplings should have the same phase on each node.

We can also see that the vacuum we just found is metastable, as we expect based on large N duality. Consider the Hessian of the potential,

-ndadcy = 8n2S1Scg2 (GiaGacGcj (bNSbNS + ]i]}) - GacMaMcg2),

4ndad-c V = 8 S 2 (G'aGacGcj (bNsbNs + J<Jj) + GacMaMcg2s), 8n 2SaS-g2

and similarly for complex conjugates. The eigenvalues of the Hessian are manifestly positive in the limit where Gab vanishes for a = b, which corresponds to widely separated nodes, and where the matrix ddV is diagonal. Moreover, the determinant of the Hessian is strictly positive for any Gab, so the one-loop Hessian remains positive definite for any Gab.

Finally, we can compute the value of the vacuum energy in the limit where the branes are far separated. The relevant limit in this more general case is

Sa,* ^ abc < Ao. The vacuum energy is then given by <2 , ;2

7bNs,b + jb 1

EtV NS,b + Jb 1 j, „ abc

M^^^--J2 eibb) • ej(c)M M log ^(1 - cos 0bc)

which reduces to the one-loop value in the gauge-theory limit, as in the A2 case.

References

[1] M. Aganagic, C. Beem, J. Seo, C. Vafa, Geometrically induced metastability and holography, hep-th/0610249.

[2] M. Aganagic, C. Beem, S. Kachru, Geometric transitions and dynamical SUSY breaking, arXiv: 0709.4277 [hep-th].

[3] O. Aharony, S. Kachru, Stringy instantons and cascading quivers, JHEP 0709 (2007) 060, arXiv: 0707.3126 [hep-th].

[4] D.E. Diaconescu, A. Garcia-Raboso, K. Sinha, A D-brane landscape on Calabi-Yau manifolds, JHEP 0606 (2006) 058, hep-th/0602138.

[5] M. Aganagic, C. Beem, B. Freivogel, Geometric metastability, quivers and holography, arXiv: 0708.0596 [hep-th].

[6] F. Cachazo, K.A. Intriligator, C. Vafa, A large N duality via a geometric transition, Nucl. Phys. B 603 (2001) 3, hep-th/0103067.

[7] F. Cachazo, S. Katz, C. Vafa, Geometric transitions and N = 1 quiver theories, hep-th/0108120.

[8] M.R. Douglas, G.W. Moore, D-branes, Quivers, and ALE instantons, hep-th/9603167.

[9] D.E. Diaconescu, M.R. Douglas, J. Gomis, Fractional branes and wrapped branes, JHEP 9802 (1998) 013, hep-th/9712230.

[10] M.R. Douglas, Enhanced gauge symmetry in M(atrix) theory, JHEP 9707 (1997) 004, hep-th/9612126.

[11] M. Buican, D. Malyshev, H. Verlinde, On the geometry of metastable supersymmetry breaking, arXiv: 0710.5519 [hep-th].

[12] A. Lawrence, J. McGreevy, Local string models of soft supersymmetry breaking, JHEP 0406 (2004) 007, hep-th/0401034.

[13] M. Grana, J. Louis, D. Waldram, Hitchin functionals in N = 2 supergravity, JHEP 0601 (2006) 008, hep-th/0505264.

[14] P.G. Camara, M. Grana, No-scale supersymmetry breaking vacua and soft terms with torsion, arXiv: 0710.4577 [hep-th].

[15] J.A. Harvey, G.W. Moore, On the algebras of BPS states, Commun. Math. Phys. 197 (1998) 489, hep-th/9609017.

[16] M. Aganagic, C. Popescu, J.H. Schwarz, Gauge-invariant and gauge-fixed D-brane actions, Nucl. Phys. B 495 (1997) 99, hep-th/9612080.

[17] T.R. Taylor, C. Vafa, RR flux on Calabi-Yau and partial supersymmetry breaking, Phys. Lett. B 474 (2000) 130, hep-th/9912152.

[18] C. Vafa, Superstrings and topological strings at large N, J. Math. Phys. 42 (2001) 2798, hep-th/0008142.

[19] I. Antoniadis, H. Partouche, T.R. Taylor, Spontaneous breaking of N = 2 global supersymmetry, Phys. Lett. B 372 (1996) 83, hep-th/9512006.

[20] E.A. Ivanov, B.M. Zupnik, Modified N = 2 supersymmetry and Fayet-Iliopoulos terms, Phys. At. Nucl. 62 (1999) 1043, Yad. Fiz. 62 (1999) 1110, hep-th/9710236.

[21] F. Cachazo, C. Vafa, N = 1 and N = 2 geometry from fluxes, hep-th/0206017.

[22] J. Polchinski, A. Strominger, New vacua for type II string theory, Phys. Lett. B 388 (1996) 736, hep-th/9510227.

[23] J. Michelson, Compactifications of type IIB strings to four dimensions with non-trivial classical potential, Nucl. Phys. B 495 (1997) 127, hep-th/9610151.

[24] N. Seiberg, E. Witten, String theory and noncommutative geometry, JHEP 9909 (1999) 032, hep-th/9908142.

[25] M. Berkooz, M.R. Douglas, R.G. Leigh, Branes intersecting at angles, Nucl. Phys. B 480 (1996) 265, hep-th/9606139.

[26] M.R. Douglas, J. Shelton, G. Torroba, Warping and supersymmetry breaking, arXiv: 0704.4001 [hep-th].

[27] A. Butti, M. Grana, R. Minasian, M. Petrini, A. Zaffaroni, The baryonic branch of Klebanov-Strassler solution: A supersymmetric family of SU(3) structure backgrounds, JHEP 0503 (2005) 069, hep-th/0412187.

[28] A. Dymarsky, I.R. Klebanov, N. Seiberg, On the moduli space of the cascading SU(M + p) x SU(p) gauge theory, JHEP 0601 (2006) 155, hep-th/0511254.

[29] Y. Nakayama, M. Yamazaki, T.T. Yanagida, Moduli stabilization in stringy ISS models, arXiv: 0710.0001 [hep-th].

[30] F. Cachazo, B. Fiol, K.A. Intriligator, S. Katz, C. Vafa, A geometric unification of dualities, Nucl. Phys. B 628 (2002) 3, hep-th/0110028.