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Nuclear Physics B 796 (2008) 1-24

www.elsevier.com/locate/nuclphysb

Geometric transitions and dynamical SUSY breaking

Mina Aganagica, Christopher Beema'*, Shamit Kachrub

a Department of Physics, University of California, Berkeley, CA 94720, USA b Department of Physics and SLAC, Stanford University, Stanford, CA 94305, USA

Received 26 November 2007; accepted 29 November 2007

Available online 5 December 2007

Abstract

We show that the physics of D-brane theories that exhibit dynamical SUSY breaking due to stringy instanton effects is well captured by geometric transitions, which recast the nonperturbative superpotential as a classical flux superpotential. This allows for simple engineering of Fayet, Polonyi, O'Raifeartaigh, and other canonical models of supersymmetry breaking in which an exponentially small scale of breaking can be understood either as coming from stringy instantons or as arising from the classical dynamics of fluxes. © 2007 Elsevier B.V. All rights reserved.

PACS: 11.25.Uv; 11.25.Tq; 11.25.Mj

Keywords: Dynamical SUSY breaking; Geometric transitions; Retrofitting; Stringy instanton; Polonyi; Fayet; O'Raifeartaigh

1. Introduction

It is of significant interest to find simple examples of dynamical supersymmetry breaking in string theory. One class of examples, where stringy D-instanton effects play a starring role, was described in [1]. These models exhibit "retrofitting" of the classic SUSY-breaking theories (Fayet, Polonyi and O'Raifeartaigh) [2], without incorporating any nontrivial gauge dynamics. Instead, stringy instantons [3] automatically implement the exponentially small scale of SUSY breaking in theories with only Abelian gauge fields. A related idea using disc instantons instead of D-instantons appears in [4]. These models are simpler in many ways than their existing field theory analogues [5].

* 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.2007.11.032

In this paper, we show that these results (and many generalizations) admit a clear and computationally powerful understanding using geometric transition techniques [6] (see also [7,8]). Such techniques are well known to translate quantum computations of superpotential interactions in nontrivial gauge theories to classical geometric computations of flux-induced superpotentials [9]. They are most powerful when the theories in question exhibit a mass gap. While the classic models we study do manifest light degrees of freedom (and hence do not admit a complete description in terms of geometry and fluxes), we find that a mixed description involving small numbers of D-branes in a flux background—which arises after a geometric transition from a system of branes at a singularity—nicely captures the relevant physics of supersymmetry breaking.1 In the original theory without flux, the SUSY-breaking effects are generated by D-instantons either in U(1) gauge factors or on unoccupied, but orientifolded, nodes of the quiver gauge theory (analogous to those studied in [1,15,16]). Both are in some sense "stringy" effects. Simple generalizations involve more familiar transitions on nodes with large N gauge groups.

The geometric transition techniques we apply have two advantages over the description using stringy instantons in a background without fluxes. First, they allow for a classical computation of the relevant superpotential instead of requiring a nontrivial instanton calculation. Second, they incorporate higher order corrections (due to multi-instanton effects in the original description) which had not been previously calculated.

The organization of this paper is as follows. In Section 2, we remind the reader of the relevant background about geometric transitions. In Section 3, we discuss the geometries we will use to formulate our DSB theories. In Sections 4-6, we give elementary examples that yield Fayet, Polonyi, and O'Raifeartaigh models that break SUSY at exponentially low scales. In Section 7, we present a single geometry that unifies the three models, reducing to them in various limits. In Section 8, we provide a more general, exact analysis of the existence of these kinds of SUSY-breaking effects. In Section 9, we give a few other examples of simple DSB theories (related to recent or well-known literature in the area). Finally, in Section 10, we extend the technology to orientifold models, in particular recovering models which are closely related to the specific examples of [1].

2. Background: Geometric transitions

Computing nonperturbative corrections in string theory, even to holomorphic quantities such as a superpotential, is in general very difficult. A surprising recent development [6,17] is that in some cases—namely for massive theories—these nonperturbative effects can be determined by perturbative means in a dual language.2

Consider, for example, N D5 branes in type IIB string theory wrapping an isolated, rigid P1 in a local Calabi-Yau manifold. In the presence of D5 branes, D1 brane instantons wrapping the P1 generate a superpotential for its Kahler modulus.3 The instanton effects are proportional to

1 For an application of geometric transitions to the study of supersymmetry breaking in the context of brane/antibrane systems, see [10-14].

2 For a two-dimensional example, see [18].

3 This is a slight misnomer, since t is a parameter, and not a dynamical field for a noncompact Calabi-Yau.

where t = fS2 (BNS + igsBRR). For general N, these D1 brane instantons are gauge theory instantons. More precisely, they are the fractional U(N) instantons of the low energy N = 1 U(N) gauge theory on the D5 brane. However, on the basis of zero-mode counting, one expects that stringy instanton effects are present even for a single D5 brane.

In the absence of D5 branes, the theory has N = 2 supersymmetry, and the Kahler moduli space is unlifted. In that case, the local Calabi-Yau with a rigid P1 is known to have another phase where the S2 has shrunk to zero size and has been replaced by a finite S3. The two branches meet at t = 0, where there is a singularity at which D3 branes wrapping the S3 become massless.

What happens to this phase transition in the presence of D5 branes? Classically, we can still connect the S2 to the S3 side by a geometric transition. The only difference is that to account for the D5 brane charge, we need there to be N units of RR flux through the S3,

Quantum mechanically the effect is more dramatic. In the presence of D5 branes, there is no sharp phase transition at all between the S2 and the S3 sides; the interpolation between them is completely smooth. As a consequence, the two sides of the transition provide dual descriptions of the same physics. Since the theory is massive now, the interpolation occurs by varying the coupling constants of the theory. The fact that the singularity where the S3 shrinks to zero size is eliminated is consistent with the fact that D3 branes wrapping an S3 with RR flux through it are infinitely massive. The most direct proof of the absence of a phase transition is in the context of M-theory on a G2 holonomy manifold [19-21]. This is related to the present transition by mirror symmetry and an M-theory lift. In M-theory, the transition is analogous to a perturbative flop transition of type IIA string theory at the conifold, except that in M-theory, the classical geometry gets corrected by M2 brane instantons instead of worldsheet instantons [19]. The argument that the two sides are connected smoothly is analogous to Witten's argument for the absence of a sharp phase transition in IIA [22]. In both cases, the presence of instantons is crucial for the singularities in the interior of the classical moduli space to be eliminated.

The fact that the two sides of the transition are connected smoothly implies that the superpotentials should be the same on both sides. The instanton-generated superpotential has a dual description on the S3 side as a perturbative superpotential generated by fluxes. The flux superpotential

is perturbative, given by

W = —S + Nds Fo,

where F0(S) is the prepotential of the Calabi-Yau, and

The first term in Eq. (2.1) comes from the running of the gauge coupling, t/gs, which implies that there is a nonzero HNS flux through a 3-chain on the S2 side. This 3-chain becomes the

noncompact 3-cycle dual to the S3 after the transition. Near the conifold point,

9sJo = S(tog( j3) - ^ +•••,

where the omitted terms are a model dependent power series in S, and A is a high scale at which t is defined. Integrating out S in favor of t, the superpotential W becomes

WW = - A3 exp^ - N—^j + ■■■

up to two- and higher-order instanton terms that depend on the power series in F0(S). The duality should persist even in the presence of other branes and fluxes, as long as the S2 that the branes wrap remains isolated, and the geometry near the branes is unaffected. As we will discuss in Section 10, this can also be extended to D5 branes wrapping P*'s in Calabi-Yau orientifolds.

3. The theories

To construct the models in question, we will consider type IIB on noncompact Calabi-Yau 3-folds which are Ar ADE type ALE spaces fibered over the complex plane C[x]. These are described as hypersurfaces in C4 as follows:

uv = [](z - zi(x)). (3.1)

This geometry is singular at points where u,v = 0 and zi (x) = zj (x) = z. At these points, there are P1 's of vanishing size which can be blown up by deforming the Kahler parameters of the Calabi-Yau. There are r 2-cycle classes, which we will denote

These correspond to the blow-ups of the singularities at zi = zi+1, i = 1,...,r. It is upon these P1 's that we wrap D5 branes to engineer our gauge theories.

The theory on the branes can be thought of as an N = 2 theory, corresponding to D5 branes wrapping 2-cycles of the ALE space, which is then deformed to an N = 1 theory by superpotentials for the adjoints. For the branes on S;2, this superpotential is denoted Wi($i). The adjoints describe the positions of the branes in the x-direction, and the superpotential arises because the ALE space is fibered nontrivially over the x -plane. The superpotential can be computed by integrating [23,24] W=/a

over a 3-chain C with one boundary as the wrapped S2. In this particular geometry, it takes an extra simple form (the details of the computation appear in Appendix A),

Wi(x) = J (zi(x) - zi+t(x)) dx. (3.2)

In addition to the adjoints, for each intersecting pair of 2-cycles S2, Sf+1 there is a bifundamental hypermultiplet at the intersection, consisting of chiral multiplets Qi,i+i and Qi+\^\, with

a superpotential interaction inherited from the N = 2 theory, Tr(6M+1 $i+iQi+u — Qu+1Qi+u$i).

Classically, the vacua of the theory correspond to the different ways of distributing branes on the minimal P1's in the geometry [25]. When one of the nodes is massive, the instantons corresponding to D1 branes wrapping the S2 can be summed up in the dual geometry after a geometric transition. As explained in [1], and as we will see in the next section, this can trigger supersym-metry breaking in the rest of the system.

As an aside, we note that the systems we are studying are a slight generalization of those described in [1,15]. Those geometries are related to the family of geometries studied here, but correspond to particular points in the parameter space where the adjoint masses have been taken to be large and the branes and/or O-planes have been taken to coincide in the x -plane. In addition, we allow the possibility of U(1) (or in some case [1,15] the instanton effects were associated with nodes that were only occupied by O-planes). Nevertheless, we will find the same qualitative physics as in [1] in this broader class of theories.

4. The Fayet model

We now turn to a specific geometry which will engineer the Fayet model at low energies. This is an A3 geometry, and (3.1) can be written explicitly as

uv = (z — mx)(z + mx)(z — mx)(z + m(x — 2a). (4.1)

After blowing up, we wrap M branes each on S2 at z1(x) = z2(x) and S| at z2(x) = z3(x), and one brane on at z3(x) = z4(x). The tree-level superpotential (3.2) is now given by

W = J2 Wi(®i) + Tr(Qn$2Q21 — Q21&1Q12) + Tr(Q23$3Q32 — Q32&2Q23), (4.2)

W1(01) = m@2, W2 (02) = —m$2, W3(03) = m(03 — a)2.

The branes on nodes one and two intersect, since both of the corresponding P1's are at x = 0. However, the third node, and the single brane on it, is isolated at x = a, and the theory living on it is massive (see Fig. 1). Correspondingly, the instanton effects due to D-instantons wrapping the third node can be summed up in a dual geometry where we trade S32 for a 3-cycle S3 with one unit of flux through it,

The geometry after the transition is described by the deformed equation uv = (z — mx)(z + mx)({z — mx)(z + m(x — 2a) — s}, where the size of the S3

i0 = S

Fig. 1. The A3 geometry, used for retrofitting the Fayet model, before the geometric transition. The red lines represent the P1 's, wrapped by D5 branes. The third node does not intersect the other two and is massive. The geometry after the transition sums up the corresponding instantons. For N = 1 branes on S|, the instantons are stringy. For N > 1, these are fractional instantons associated with gaugino condensation in the pure U(N) n = 1 gauge theory on that node. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

is given by S = s/m. It is fixed to be exponentially small by the flux superpotential, as we shall see shortly. The third brane is gone now, and so are the fields Q23, Q32 and $3. The effective superpotential can now be written to leading order in S as

Weff = W1($1) + W2(&2,S) + Tr (Q12 $2 Q 21 - Q21 $1 Q12) + Wflux(S). In this geometry, the exact flux superpotential is

Wflux = —S + s(log S - 1

without any polynomial corrections in S. It is crucial here that the superpotential for $2 has changed, due to the change in the geometry, to W2($2), where

W2(x) = f (z2(x) -Z3(x)) dx, while the superpotential for is unaffected. We have defined (z - Z3(x))(z -Z*(x)) = (z - z3(x^(z - Z4(x^ - s

with z3(x) being the branch which asymptotically looks like z3 (x) at large values of x .In other words,

W2(x) = f {-m(x> + a) -Vm2(x'- a)2 +s) ^

This superpotential sums up the instanton effects due to Euclidean branes wrapping node three.

Before the transition, the vacuum was at $2 = 0. At the end of the day, we expect it to be perturbed by exponentially small terms of order S, so the relevant part of the superpotential is

2 1 a - $2

W2($2) = -m Tr - 2 S Trlog 2 + •••, (4.4)

where we have omitted terms of order S2 and higher and dropped an irrelevant constant. We comment on the form of these corrections in Appendix B. The theories on nodes one and two are asymptotically free. If the fields S and &1j2 have very large masses, we can integrate them out and keep only the light degrees of freedom. Keeping only the leading instanton corrections, the relevant F-terms are

F&1 = 2m&i - Q12Q21,

F$2 = -2m&2 + Q21Q12 + ^-,

2(a - $2)

Fs = t/gs + log S/A3 - - Tr log (a - &2)/A. (4.5)

Setting these to zero, we obtain

S* = A3 exp^ - + 1

t = t - -Mgs log(a/A) 1

61,* = Q12Q21,

62,* = — Q21Q12 -S* + ---. (4.6)

2m 4ma

The omitted terms are higher order in Q21 Q12/ma and exp(-g^). The low energy, effective

superpotential is

Wf =- Tr(Q12 Q21Q12Q21) - —- Tr Q12Q21 + •••, m 4ma

where we have neglected corrections to the quartic coupling, and the higher order couplings of

Q's, all of which are exponentially suppressed. As shown in [1], in the presence of a generic FI

term for the off-diagonal U(1) under which Q12 and Q21 are charged,

D = Q12QI2 - Q21Q21 - r, the exponentially small mass for Q will trigger F-term supersymmetry breaking with an exponentially low scale; we can put Q12,* ^ Vr, and then

Fq21 ~ -;-S*.

Geometrically, turning the FI term corresponds to choosing the central charges of the branes on the two nodes to be misaligned. Combined with the fact that nodes one and two have become massive with an exponentially low mass, this provides an extremely simple mechanism of breaking supersymmetry at a low scale. The non-supersymmetric vacuum we found classically is reliable as long as the scale of supersymmetry breaking is far above the strong coupling scales of the U(M) x U(M) gauge theory. If we take N branes on the massive node instead of one, the story is the same, apart from the fact that the flux increases, and correspondingly the vacuum value of S changes to S* ~ A3 exp(-i/Ngs). In this case however, the instantons that trigger supersymmetry breaking are the fractional U(N) instantons.

5. The Polonyi model

In this section we construct the Polonyi model with an exponentially small linear superpotential term for a chiral superfield $. This will turn out to be somewhat more subtle, and the existence of the (meta)stable vacuum will depend sensitively on the Kahler potential. We describe specific cases where we know the relevant Kahler potential does yield a stable vacuum in Section 7.

Consider an A2 geometry given by

uv = (z - mx)(z - mx){z + m(x - 2a) (5.1)

which has one D5 brane wrapped on the S2 blown up at z\(x) = z2(x), and one D5 brane wrapped on the S2 blown up at z2(x) = z3 (x). This system has a tree-level superpotential

W = W\($i) + W2($2) + Q 12$2 Q21 - Q21$1Q12, (5.2)

W1($1) = 0, W2($2) = m($2 - a)2.

There is a classical moduli space of vacua parameterized by the expectation value of $1 and where Q12,* = Q21,* = 0, and $2,* = a.

At a generic point in the moduli space, away from $1 = a, the theory on the branes wrapping S22 is massive. Then, the instanton effects associated with D1 branes wrapping this node can be summed up by a geometric transition that replaces S22 by an S3 with one unit of flux through it. This deforms the Calabi-Yau geometry to

uv = (z - mx)({z - mx)(z + m(x - 2a) - s)

which now has an S3 of size

J a = S-

where S = s/m. With this deformation, the superpotential for node one is altered as well:

W1(x) = f(-m(a - x) + vm2(a - x)2 + ^dx.

The effective superpotential after the transition is simply

Weff = W 1($1,S) + Wflux(S), where the flux superpotential has the simple form:

Wflux(S) = — S + S(logS/A3 - 1).

Note that there is no supersymmetric vacuum, since F$1 = 0 always.

Suppose that at a point in the moduli space, say at $1 = 0, the Kahler potential takes the form

K =|$1|2 + c|$114 + ■

where the higher order terms are suppressed by a characteristic mass scale (which we set to one). Then, provided

|ca2| > 1, c < 0, the theory has a non-supersymmetric vacuum at 1

&1,* = — (5.3)

which breaks SUSY at an exponentially low scale. This can be seen as follows. Expanded about small &1, the superpotential W1 takes the form

W1&1) = -^ log(a - &1)/A +

where the subleading terms are suppressed by additional powers of S, but are otherwise regular at the origin of space. Integrating out S first, by solving its F-term constraint, we find

S* = A3 exp(-i/gs) + •••,

t = t - 2gs log(a/A)

and the subleading terms are of order &1/a, which will turn out to be small in the vacuum. For large t, S is generically very massive, so integrating it out is justified. The potential for now becomes

Veff(&1) = , , '^ ,2 , '"1 ,2 +•" . 1 + c\&1 \2 \a - &1\2

It is easy to see that, up to corrections of order 1/\a2c\ and S*/(ma2), this has a non-

supersymmetric vacuum at (5.3) where has a mass squared of order

This is positive, and the vacuum is (meta)stable, as long as c < 0. Note that we could have obtained the Polonyi model as a limit of the Fayet model in which we turn on a very large FI term for the off-diagonal U(1) of nodes one and two. In this case, the stability of the Fayet model for a generic (effectively canonical) Kahler potential guarantees that the Polonyi model obtained from it is stable. In fact, as we will review in Section 7, one can show this directly by computing the relevant correction to the Kahler potential, arising from loops of massive gauge bosons [1].

6. The O'Raifeartaigh model

To represent the third simple classic class of SUSY breaking models, we engineer an O'Raifeartaigh model. Consider the A3 fibration with

Z1(x) = mx, Z2(x) = mx, Z3(x) = mx, Z4(x) = -m(x - 2a). (6.1)

The defining equation of the noncompact Calabi-Yau is then

uv = (z - mx)(z - mx)(z - mx)(z + m(x - 2a) (6.2)

and we wrap a single D5 brane on each of S2 2 3. The adjoints $1 and $2 are massless, while $3 obtains a mass from its superpotential,

W3(x) = 1 (z3(x) - z4(x)) dx (6.3)

which gives

W3($3) = m($3 - a)2.

Of course, there are also quarks Q12,Q21 and Q23,Q32. They couple via superpotential couplings

Q12$1Q21 - Q12$2Q21 + Q23$2Q32 - Q23$3 Q32. (6.4)

Because $3 is locked at a, for generic values of $2, Q23 and Q32 are massive. Then node three is entirely massive, and we can perform a geometric transition.

The resulting theory has a new glueball superfield S, and effective superpotential

Weff = Q12$1Q21 - Q12$2Q21 - 1Slog(a - $2)/A

+ S(log(S/A3) - 1) + —S + •••. (6.5)

Integrating out the S field yields (at leading order)

S* = A3e-i/gs, (6.6)

t = t + 2 gs log (a/A). Plugging this into the superpotential yields

Weff = Q12$1 Q21 - Q12$2Q21 - 2S*$2/a + ••• . (6.7)

The omitted terms are suppressed by more powers of $2/a. We recognize (6.7) as the superpotential for an O'Raifeartaigh model, very similar to the one considered in [1]. We see that setting F$1 = F$2 = 0 is impossible, so one obtains F-term supersymmetry breaking, with a small scale set by A3 exp(-i/3gs).

The stability of the non-supersymmetric vacuum again depends on the form of (technically) irrelevant corrections to the Kahler potential. As in the case of the Polonyi model, corrections which yield a stable vacuum can be arranged by embedding the model in a slightly larger theory. We now turn to a general analysis of one such larger theory.

7. A master geometry

It is possible to construct one configuration of branes on an A4 geometry which, in appropriate limits, can be made to reduce to any of the three simple models discussed in the previous sections. The geometry is described by the defining equation

= (z - mx)(z - mx)(z + mx)(z - mx){z + m(x - 2a), (7.1)

Fig. 2. The master A4 geometry that gives rise to Fayet, Polonyi and O'Raifeartaigh models by turning on suitable FI terms. The stringy instantons associated with the massive fourth node generate the nonperturbative superpotential that triggers dynamical supersymmetry breaking in the rest of the theory.

where we wrap N branes on nodes one, two and three, and a single brane on node four

(see Fig. 2), leading to a superpotential given by

^master = J2 Wi(0i) + 12 Tr(QU+i$i+iQi+ij - Qi+1J^iQU+l). (7.2)

i = 1 i = 1

The superpotentials for the adjoints are given by

W1(01) = 0, W2(®2) = -m Tr(^|), W3(®3) = m Tr(032), W4(04) = -m(®4 - a)2.

In the interest of simplicity, we will set N = 1 in this section. The non-Abelian generalization is immediate, since all the nodes are asymptotically free (for large adjoint masses). As long as the scale of supersymmetry breaking driven by the geometric transition is high enough, we can ignore the non-Abelian gauge dynamics on the other nodes.

The master theory has a metastable, non-supersymmetric vacuum for generic, nonzero FI terms. We can recover all three of the models discussed above by introducing large Fayet-Iliopoulos terms for certain pairs of quarks, so we expect that these will have non-supersymmetric vacua as well. This approach to obtaining the canonical models is particularly useful in the case of Polonyi and O'Raifeartaigh models, for which we needed to assume a particular sign for the subleading correction to the Kahler potential. By obtaining the theories from the master theory, we can compute the leading corrections to the Kahler potential directly and show that they are of the type required to stabilize the SUSY-breaking vacua.

To see that the master theory has a metastable, non-supersymmetric vacuum, we can proceed as in the Fayet model. Node four is massive, and the corresponding nonperturbative superpotential can be computed in the geometry after a transition. The effective superpotential after performing the transition and integrating out the massive adjoints $2,3 is then

Weff = Q12 Q21&1 + (Q23Q32 + ■■■), 4ma

where we have omitted quartic and higher order terms in the Q's which do not affect the status of the vacuum. With generic FI terms setting

IQ12I2 -IQ21I2 = r2, | Q23I2 -IQ32I2 = r3,

this is easily seen to have an isolated vacuum which breaks supersymmetry.

We will now show that we can recover all of the three models studied so far in particular regimes of large FI terms.

7.1. O'Raifeartaigh

We can recover the O'Raifeartaigh construction by turning on a large FI term for Q23 and Q32—that is, for the U(1) under which these are the only charged quarks. This generates a D-term

do'R = IQ23I2 -IQ32I2 - ^3. (7.3)

Taking r3 > 0, this requires that Q23 acquire a large expectation value. Additionally, there is an F-term for Q32

fq32 = Q23 ($3 - $2) (7.4)

which, in light of the D-term constraint, will set $2 equal to $3. The superpotential then becomes just the O'Raifeartaigh superpotential of the previous section (with certain indices renamed),

WO'R = m($4 - a)2 + Q12Q21 ($2 - $1) + Q24Q42($4 - $2). (7.5)

By performing a geometric transition on the massive node, we recover the superpotential (6.5).

7.2. Fayet

Alternatively, we could have turned on a large FI term for Q12 and Q21, generating a D-term

DFayet = I Q12I2 -IQ21I2 - r2. (7.6)

In conjunction with the F-term for Q21, by the same process as in the O'Raifeartaigh model, $1 is set equal to $2. This time, the remaining superpotential is given by

WFayet = m$| - m$32 + m($4 - a)2 + Q23 Q32($3 - $2) + ••• (7.7)

which is precisely the superpotential associated with the Fayet geometry (4.1). Performing a geometric transition on S42, we recover the Fayet model as discussed in Section 4.

7.3. Polonyi

From the Fayet model above, before the geometric transition, we can turn on another D-term for the quarks Q23 and Q32 which, along with the F-term for Q32 sets $2 = $3. The superpotential becomes

W =-m$32 + m($4 - a)2 + Q34Q43($4 - $3)

which reproduces the Polonyi model of Section 5. Again, performing the geometric transition on S42 results in the actual Polonyi model.

7.4. The Kahler potential

The O'Raifeartaigh and Polonyi models have flat directions at tree level. As we discussed for, e.g., the Polonyi model, the existence of a stable SUSY-breaking vacuum depends on the sign of the leading quartic correction to the Kahler potential. When we obtain the model as a suitable limit of our master model as above, we can compute this correction and verify explicitly that the vacuum is stable. Let us go through this in some detail. In fact, for simplicity, we will focus on obtaining a stable Polonyi model as a limit of a Fayet model [1].

After the geometric transition in the Fayet model, the effective theory is characterized by a superpotential

W = — Q23Q32 + ••• (7.8)

and a D-term

D = IQ32I2 - | Q2312 - T3- (7.9)

Here r3 is the FI term for the U(1) under which only Q23 and Q32 carry a charge. We can expand this theory about the vev Q23 = . Renaming

X = Q32, the effective theory then has

W = —jn3X. (7.10)

To find the Kahler potential for X, we should integrate out the massive U(1) gauge multiplet. What happens to the potential contribution from the D-term of (7.9)? As explained in [26], in the theory with the U(1) gauge field, gauge invariance relates D-term and F-term vevs at any critical point of the scalar potential. When one integrates out the U(1) gauge field, there is a universal quartic correction to the Kahler potential which (using the relation) precisely reproduces the potential contribution from the D-term. For the theory in question, the quartic correction to the Kahler potential for X is just

AK = -gU(1- (XtX)2. (7.11)

MU (1)

Here MU(i) is the mass of the U(1) gauge boson, MU(\) ~ gU(\)^/r3. The result is a quartic correction to K

AK = - — (XfX)2. (7.12)

So in the notation of Section 5, _ 1

and the sign c < 0 results in a stable vacuum, as expected. Plugging in the F-term FX ~ ma Vr?, (7.12) gives X a mass

mX ~ —, ma

in agreement with what it was in the full Fayet model. Note that while one would obtain other quartic couplings in K after integrating out the U(1) gauge boson, they do not play any role. They involve powers of the heavy field Q34, and since Fq34 ^ FX, cross-couplings of the form Q34Q34X3X in K do not appreciably correct the estimate obtained above for the mass of X.

8. Generalization

We now present a very general argument for the existence of supersymmetry-breaking effects in a class of stringy quiver gauge theories which includes those just discussed. Suppose we have such an Ar quiver theory in which the last node is isolated and undergoes a transition. Note that this is the case in the master geometry considered in the previous section. In this case, the transition deforms the geometry to the following:

I r— 1

v =(n(z - zi(x))j((z - zr(x))(z - zr+1 (x)) - s)

in which case the superpotential for the branes on the second-to-last node becomes

Wr-1($r-1) = j dx (zr(x) - zr -1 (x)), where zr(x) is the solution to the equation

(z - zr(x))(z - zr+1(x)) = s (8.1)

which asymptotically approaches zr(x). We can rewrite the superpotential as a correction to the pre-transition superpotential as

Wr-1($r-1) = J dx (zr(x) - zr(x)) + Wr-1($r-1) and the F-term for $r -1 and the remaining adjoints are then given by F$r-1 = Wr-1($r-1) + (zr($) - zr($)) + Qr-1,rQr,r-1,

F$i = W[($i) + Qi-1,i Qi,i-1 - Qi,i+1 Qi+1,i (8.2)

which we can combine to obtain the constraint

Y,w'i($i) = zr($r-1) -lr($r-1). (8.3)

Note that the right-hand side here cannot vanish for any value of $r-1 since zr(x) can never solve (8.1), the solution to which defines zr(x).

If we now consider turning on generic FI terms for the U(1) gauge groups, the D-term constraints will require that, say, the Qi,i+1's acquire vevs while the Qi+1,i's get fixed at zero. The F-terms for the Qi+1,i 's will then in turn require

$i = $j

for all i, j. When the brane superpotentials for the first r - 1 nodes are of the form Wi($i) = Cim$f, i = 1,...,r - 1,

where ei = 0 ± 1, the left-hand side of (8.3) vanishes, while the right-hand side is strictly nonzero. It is exponentially small, as long as the last node was isolated,

Wr($r) = m($r — a)2,

before the transition. This generically triggers low-scale SUSY breaking.

In terms of the classic models discussed in this paper, one can immediately see that the SUSY breaking in the Fayet model and in the master geometry can be explained by the above analysis. In the case of the Polonyi and O'Raifeartaigh models, it is even simpler, since the left-hand side of (8.3) vanishes identically for those models. One could conduct a similar analysis for configurations with more complicated superpotentials and nongeneric F-terms on a case-by-case basis. What we see is that often the SUSY-breaking effects caused by the geometric transition can be understood at an exact level.

9. SUSY breaking by the rank condition

In this section, we present models which break supersymmetry due to the "rank condition". This class of models is very similar to those arising in studies of metastable vacua in SUSY QCD [27]. However, we work directly with the analogue of the magnetic dual variables, and the small scale of SUSY breaking is guaranteed by retrofitting [2]. Consider the A3 fibration with

zi(x) = mx, Z2 (x) = -mx, Z3(x) = -mx, z4(x) = —m(x — 2a). (9.1) Then the defining equation is

uv = (z — mx)(z + mx)(z + mx)(z + m(x — 2a). (9.2)

We choose to wrap Nf — Nc D5 branes on S2, Nf D5 branes on S|, and a single D5 on S2 (see Fig. 3). The tree level superpotential is

W = J2 Wi(<Pi) + J2 (Qi,i+i$i+iQi+i,i — Qi+i,i$iQi,i+i), (9.3)

i = 1 i = 1

W1(01) = m Tr№)2, W2(®2) = 0, W3(&3) = —m(<P3 — a)2.

Now, we replace the third (^(1)) node with an S3 with flux, and integrate out ^ trivially (we can take the mass to be very large). The result is a superpotential,

W = S (log (SM3) — 1) + —S — 2 S Tr log (a — — Qt2^2Q21 + •••, (9.4)

where the omitted terms are suppressed by additional powers of S. Integrating out S in a Taylor expansion about = 0 produces a theory with superpotential

W = S* Tr 02/a — Tr Q1202Q21 + •••, (9.5)

S* = A3exp(—t/gs),

hlt'ti

Fig. 3. The (magnetic) A3 geometry that retrofits the ISS model.

and t = t — Nf 1 ^ log(a/A). Computing F$2, we see that the contribution from the first term in (9.5) has rank Nf, while the contribution from the second term has maximal rank Nf — Nc < Nf. The two cannot cancel, and so SUSY is broken. However, due to the small coefficient of the Tr term, the breaking occurs at an exponentially small scale.

This model resembles the theories analyzed in [27] (for Nc + 1 < Nf < 2 Nc) and in Section 4 of [28]. One difference is that the origin of the small parameter is dynamically explained. The discussion of corrections due to gauging of the U(Nf) factor (which is a global group in [27]) is identical to that in [28] up to a change of notation, and we will not repeat it here. For large a, the higher order corrections to (9.5) (which are suppressed by powers of $2/a) should not destabilize the vacuum at the origin, described in [27,28].

We could also replace the U(1) at node 3 with a U(N) gauge group, still in the same geometry. Then, in (9.4), the coefficient of the Slog S term is changed to N. The only effect, after a geometric transition at node three, is the replacement e—t/gs ^ e—t/gsN in (9.6). This model, where the node which undergoes the geometric transition has non-Abelian gauge dynamics, is a literal example of the retrofitting constructions of [2]. The field appears in the gauge coupling function of the U(N) gauge group at node three because it controls the masses of the quarks Q23 and Q32 which are charged under U(N). At energies below the quark mass, the U(N) is a pure N = 1 gauge theory and produces a gaugino condensation contribution A3N in the superpotential. The standard result for matching the dynamical scale of the low-energy, pure U(N) theory to the scale AN,Nf of the higher energy theory with Nf quark flavors with mass matrix m is4

A3N a 3N—Nf _

AN = AN Nf f detm.

Identifying S with the gaugino condensate [6]

S ~ tr(W2) = A3

4 Here, we are assuming the adjoints are very massive, m ^ to, and are just matching the QCD theories with quark flavors.

and identifying the mass matrix m = a — @2, we predict

SN = ANf det(a — (9.8)

This is precisely what carefully integrating S out of (9.4) produces, with AN—f = A3N—Nf x

e—t/gs. So, in our model with N > 1, the small Tr(^2) term in (9.5) can really be thought of as arising from the presence of in the gauge coupling function for the U(N) factor.

10. Orientifold models

In the presence of orientifold 5-planes, we expect D1 brane instantons wrapping 2-cycles that map to themselves to contribute to the superpotential. The D1 brane instanton contributions should again be computable using a geometric transition that shrinks the S2 and replaces it with an S3. Geometric transitions with orientifolds have been studied, e.g., in [29,30].

After the transition, we generally get two different contributions to the superpotential. First, charge conservation for the D5/O5 brane that disappears after the transition requires a flux

through the S3 equal to the amount of brane charge, —

Wflux = — S + Nd5/o59s F0. gs

Second, there can be additional O5 planes that survive as the fixed points of the holomorphic involution after the transition. The O5 planes, just like D5 branes, generate a superpotential [31]

W05 = J V, £

where the integral is over a 3-chain with a boundary on the orientifold plane. The contributions to the superpotential due to O5 planes and RR flux of the orientifold planes are both computed by topological string RP2 diagrams. The contribution of physical brane charge comes from sphere diagrams.

In this way, geometric transitions can be used to sum up the instanton-generated superpotentials in orientifold models. In analogy to our discussion in the previous sections, this can be used to understand models of dynamical supersymmetry breaking. We will discuss the Fayet model in detail; other models can be seen to follow in naturally.

10.1. The Fayet model

Consider orientifolding the theory from Section 3 by combining worldsheet orientation reversal with an involution I of the Calabi-Yau manifold. For this to preserve the same supersymmetry as the D5 branes, the holomorphic involution I of the Calabi-Yau has to preserve the holomor-phic three-form V = d^dzdx = — dfdzdx.

An example of such an involution is one that takes

u ^ v, v ^ u.

A simple, Fayet-type model built on this orientifold is an A5 geometry that is roughly a doubling of that in Section 4,

uv = (z — mx)2(z + mx)2(z, — m(x — 2a) (z + m(x — 2a). We will blow this up according to the ordering

Z1(x) = mx, Z2(x) = —m(x — 2a), Z3(x) = mx, Z4(x) = —mx, Z5 (x) = m(x + 2a), Z6(x) = —mx.

It can be shown that the orientifold projection ends up mapping

S2 S2 ,

Si ^ S6—i,

fixing S|. Consider wrapping M branes on Sf for i = 1,2, and their mirror images, and 2N branes on S|. With a particular choice of orientifold projection, the gauge group on the branes is going to be

U(M) x U(M) x Sp(N).

Since the orientifold flips the sign of x, on the fixed node, S^. it converts to an adjoint of Sp(N). (Having chosen that the orientifold sends x to minus itself, the action on the rest of the coordinates is fixed by asking that it preserve the same SUSY as the D5 branes and that it remain a symmetry after blowing up.) In the model at hand, the tree-level superpotential is

W = J2 Wi($i) + Tr(Q12 $2Q21 — Q2101Q12) + Tr(Q23$3Q32 — Q32$2Q23), i = 1

W1($1) = m Tr($1 — a)2, W2($2) = — m Tr($2 — a)2, W3($3) = m Tr

Note that, even though the P1 is fixed by the orientifold action, it is not fixed pointwise. This means there is no O5+ plane charge on it. Instead, there are two noncompact orientifold 5-planes. This model is T-dual [32] to the O6 plane models of [15].

After the geometric transition that shrinks node three and replaces it with an S3,

S32 ^ S3, the geometry becomes

uv = (z — mx)(Z + mx)(Z — m(x — 2a) 2{z + m(x — 2a))2 ((z — mx)(Z + mx) — s),

with S = s/m. Since the orientation reversal acted freely on the S^, there are only N units of D5 flux through the S3,

J HRR = N,

which gives rise to a superpotential

Wflux = — S + N^ log -- — 1

2gs \ A3

The overall factor of 1 /2 comes from the fact that both the charge on the S2 and its size have been cut in half by the orientifold action. Above, t = fS2 k + igsBRR is the combination of Kahler

moduli that survives the orientifold projection. In addition, the two noncompact 05+ planes get pushed through the transition. Because the space still needs two blowups to be smooth, to give a precise description of the O5 planes would require using a geometry covered with 4 patches. At the end of the day, effectively, the O5 planes correspond to noncompact curves over the two points on the Riemann surface

(z — Z3(x)){z — Z4(x)) = ((z — mx)(z + mx) — s) = 0 located at x = 0 and the corresponding values of z, z±(0). They generate a superpotential

z—(0) Z+(0)

Wo5+ = J (Z3 —Z4)dx + j (Z3 —Z4)dx. One can show that the contribution of the O5 planes is W05+ = +S^ log | — 1

The fact that the RP2 contribution is proportional to that of the sphere is not an accident. It has been shown generally that the contribution of O5 planes in these classes of models is ±9sFs2 [30,33]. This means that the 05 planes and the fluxes add up to N + 1 units of an "effective" flux on the S3.

After the transition, the branes on node three have disappeared, and with them, the fields and Q23, Q32. In addition, the deformation of the geometry induces a deformation of the superpotential for node two,

W2(x) = f (z2(x) — h(x)) dx,

where one picks for Z3 the root that asymptotes to +mx. This deformed superpotential is then

W2(x) = J {—m(x — 2a) — V(mx)2 + s ) dx,

which, when expanded near the vacuum at x = a, gives

W2(&2) = — Tr m(&2 — a)2 — 1S Trlog(02/A) + •••. The full effective superpotential that sums up the instantons is thus

Weff = W1(01) + W2($2,S) + Tr (Q 12^2 Q21 — Q2101 Q12) + Wflux + W05.

Up to an overall shift of both $1,2 by a, this is the same model as in Section 3.

We expect a transition here even when N = 0, and there are no D5 branes on the S2. The transition for Sp(0) is analogous to the transition that occurs for the U(1) gauge theory of a single D-brane on the S2. In both cases, the smooth joining of the S2 and the S3 phases is due to instantons that correct the geometry. In the orientifold case at hand, it is important to note

that, while there is no flux through the S3, the D3 brane wrapping it is absent: the orientifold projection projects out [34] the N = 1 U(1) vector multiplet associated with the S3, and with it the D3 brane charged under it.

Picking the other orientifold projection, the Sp(N) gauge group gets replaced by SO(2N), with 03 becoming the corresponding adjoint. In this case, much of the story remains the same, except that the RP2 contribution becomes

Wo5— = —s( log J3 — 1

This means that the 05— planes and the fluxes add up to N — 1 units of an "effective" flux on the S3. This is negative or zero for N < 1. Naively, the negative effective flux breaks supersym-metry after the transition. This is clearly impossible. It has been argued in [30] that the correct interpretation of this is that in fact SO(2), SO(1) and S0(0) cases do not undergo geometric transitions. This has to correspond to the statement that, in these cases, there are no D1 brane instantons on node three, and so the classical picture is exact. This translates to the statement that in these cases, S should not be extremized, but rather set to zero identically in the effective superpotential,

Wef = Weff | S=0.

Note that with the SO projection on the space-filling branes, a D-instanton wrapping the same node enjoys an Sp projection. As discussed in [15,35], in this situation, direct zero-mode counting also suggests that the instanton should not correct the superpotential. There are more than two fermion zero modes coming from the Ramond sector of strings stretching from the instanton to itself. This is in accord with the results of [30]. In contrast, when one has an Sp projection on the space-filling branes, the instanton receives an SO projection, and the instanton with SO(1) worldvolume gauge group has the correct zero-mode count to contribute. The presence of instanton effects for this projection (and their absence without it), was also confirmed by direct studies of the renormalization group cascade ending in the appropriate geometry in [15].

Acknowledgements

We would like to thank B. Florea, B. Freivogel, T. Grimm, J. McGreevy, D. Poland, N. Saulina, E. Silverstein and C. Vafa for helpful discussions. The research of M.A. and C.B. was supported in part by the UC Berkeley Center for Theoretical Physics. The research of M.A. is also supported by a DOE OJI Award, the Alfred P. Sloan Fellowship, and NSF grant PHY-0457317. S.K. was supported in part by the Stanford Institute for Theoretical Physics, and by NSF grant PHY-0244728 and DOE contract DE-AC003-76SF00515. S.K. acknowledges the kind hospitality of the MIT Center for Theoretical Physics during the completion of this work.

Appendix A. Brane superpotentials

We can compute the superpotential W(@) as function of the wrapped 2-cycles £ by using

the superpotential [23,24] W=/ V

where C is a 3-chain with one boundary being £ and the other being a reference 2-cycle £0 in the same homology class. It is easy to show [23] that the critical points of the superpotential are holomorphic curves. We will evaluate it for the geometries at hand. We can write the holomorphic three-form of the noncompact Calabi-Yau in the usual way,

dv A dz A dx dv

£ =-~T~p-= — A dz A dx. (A.1)

Now for fixed values of x and z, the equation for the CY threefold becomes uv = const, which is the equation for a cylinder. By shifting the definition of u or v by a phase, we can insist that the constant is purely real, and then by writing u = x + iy, v = x — iy, the equation can be reformulated as two real equations in terms of the real (xR,yR) and imaginary (xj,yj) parts of x and y

xR + yR = C + xj + yj, xrxi = yRyi. (A.2)

The first of these can be solved for any given values of xj and yi to give an S1. The second equation restricts the possible values which we choose for xj and yj to a one-dimensional curve in the (xj,yj) plane, and so we have the topology of S1 x R, where the size of the S1 degenerates at the points where z = zi(x) for any i. By simultaneously shifting the phases of u and v according to

u ^ e'0 u,

—iO v ^ e iOv

the equation for the cylinder remains unchanged, and we simply rotate about the S1 factor. We can thus integrate £ around the circle and obtain

£ = dz A dx

up to an overall constant. Now the Px's on which we are wrapping the D5 branes are the product of the S1 just discussed and an interval in the z direction between values where the S1 fiber degenerates. Thus, for a given P1 class in which the vanishing S1 occurs for zi(x) and zj(x), we can integrate dz A dx over the interval in the z-plane and obtain

j £ = (zt(x) — zj(x)) dx.

S1 xiij

The superpotential for the D-branes then becomes a superpotential for the location of the branes on the t-plane. Defining an arbitrary reference point t-t., we then have

W(x) = J(zt(x) — zj(x)) dx. (A.3)

Of course, the contribution to the superpotential coming from the limit of integration at t. is just an arbitrary constant and is not physically relevant. Thus we write (A.3) instead as the indefinite integral

W(x) = J(zt(x) — zj(x)) dx. (A.4)

Appendix B. Multi-instanton contributions

In this appendix we demonstrate the computation of multi-instanton corrections to the superpotential using the Polonyi model of Section 5 as an example. All the information about these corrections is contained in the deformed superpotential for @,

- - a>2+^) dx (B„

along with the flux superpotential5

Wflux = g^S + S^log J - ^, (B.2)

where the scale J is determined by the one-loop contributions to the matrix model free energy. The models considered in this paper are particularly convenient since the purely quadratic superpotential for the massive adjoint at the transition node guarantees that the flux superpotential will be exact at one-loop order in the associated matrix model [17].

Extremizing the flux superpotential and expanding in powers of the instanton action

Sinst ~ exp(—t/Ngs),

we can determine multi-instanton contributions to a given superpotential term. Summing up the series contributing to a given term in will correspond to computing corrections to a fixed, explicit disc diagram, and so we might expect these series to exhibit some integrality properties.

We first expand the deformed superpotential W1(0) as a power series in the glueball super-field S,

W(<P) = f (m(x — a) — m(x — a)^1 + £ (—1^^ ^ y^j dx, (B.3)

where the expansion parameter y can also be expanded as a power series in x,

y = , S ,2 =^2(1 + £(n + 1)(—1)" (") J • (B.4)

m(x — a)2 ma2 y ^ \a J J

We can integrate (B.3) term by term to obtain an expansion of the effective superpotential in powers of @. However, it will be useful to represent this schematically

Wi (0) = ci Tr i> + c2 Tr i>2 + •••, ci = J2 c(n)Sn,

where the coefficients ci are themselves written as power series in S. Extremizing the superpotential with respect to S gives an equation for the values of S

log A = —1 — £ J2"c(n)Sn—1Tr (B.5)

n=1 i=1

5 In the case of the Polonyi model these two terms constitute the entire superpotential. In the more general case, however, there will be more fields with superpotential terms, but it will remain the case that only these two contributions play a role in determining instanton corrections.

which can be solved perturbatively in powers of Sinst. Reinserting the resulting values into the original superpotential then allows us to read off the instanton-corrected superpotential of the low energy theory up to any given number of instantons. Below we display the linear and quadratic terms at the three-instanton level.

Weff = f Tr 0 + m Tr 02

3 — _l 1 A6

f =--e gs ---— e

m =--t- e

5 A6 16 ma4

. 2l 9

16 m2a5 A9

32 m2a6

gs + ■■ ■ _ 31

Z gs + ■ ■

It may be interesting to see if there is some way to relate these to the exact formulae for multicovers derived in the resolution of the singularity in hypermultiplet moduli space when a 2-cycle shrinks in IIB string theory, given (up to mirror symmetry) in [36].

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