Scholarly article on topic 'Geometric metastability, quivers and holography'

Geometric metastability, quivers and holography Academic research paper on "Physical sciences"

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{"Geometric transitions" / Metastability / "Large N duality" / Brane/antibrane / Holography / Supersymmetry-breaking}

Abstract of research paper on Physical sciences, author of scientific article — Mina Aganagic, Christopher Beem, Ben Freivogel

Abstract We use large N duality to study brane/antibrane configurations on a class of Calabi–Yau manifolds. With only branes present, the Calabi–Yau manifolds in question give rise to N = 2 ADE quiver theories deformed by superpotential terms. We show that the large N duality conjecture of [M. Aganagic, C. Beem, J. Seo, C. Vafa, Geometrically induced metastability and holography, hep-th/0610249] reproduces correctly the known qualitative features of the brane/antibrane physics. In the supersymmetric case, the gauge theories have Seiberg dualities, which are represented as flops in the geometry. Moreover, the holographic dual geometry encodes the whole RG flow of the gauge theory. In the non-supersymmetric case, the large N duality predicts that the brane/antibrane theories also enjoy such dualities, and allows one to pick out the good description at a given energy scale.

Academic research paper on topic "Geometric metastability, quivers and holography"

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Nuclear Physics B 795 (2008) 291-333

www.elsevier.com/locate/nuclphysb

Geometric metastability, quivers and holography

Mina Aganagic, Christopher Beem *, Ben Freivogel

University of California, Berkeley, CA 94720, USA Received 27 September 2007; accepted 21 November 2007 Available online 26 December 2007

Abstract

We use large N duality to study brane/antibrane configurations on a class of Calabi-Yau manifolds. With only branes present, the Calabi-Yau manifolds in question give rise to N = 2 ADE quiver theories deformed by superpotential terms. We show that the large N duality conjecture of [M. Aganagic, C. Beem, J. Seo, C. Vafa, Geometrically induced metastability and holography, hep-th/0610249] reproduces correctly the known qualitative features of the brane/antibrane physics. In the supersymmetric case, the gauge theories have Seiberg dualities, which are represented as flops in the geometry. Moreover, the holographic dual geometry encodes the whole RG flow of the gauge theory. In the non-supersymmetric case, the large N duality predicts that the brane/antibrane theories also enjoy such dualities, and allows one to pick out the good description at a given energy scale. © 2007 Elsevier B.V. All rights reserved.

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

Keywords: Geometric transitions; Metastability; Large N duality; Brane/antibrane; Holography; Supersymmetry-breaking

1. Introduction

Geometric transitions have proven to be a powerful means of studying the dynamics of supersymmetric D-branes. String theory relates these transitions to large N dualities, where before the transition, at small't Hooft coupling, one has D-branes wrapping cycles in the geometry, and after the transition, at large 't Hooft coupling, the system is represented by a different geometry, with branes replaced by fluxes. The AdS/CFT correspondence can be thought of in this way.

* 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.2007.11.031

Geometric transitions are particularly powerful when the D-branes in question wrap cycles in a Calabi-Yau manifold. Then, the topological string can be used to study the dual geometry exactly to all orders in the't Hooft coupling. In [1] it was conjectured that topological strings and large N dualities can also be used to study non-supersymmetric, metastable configurations of branes in Calabi-Yau manifolds, that confine at low energies. This conjecture was considered in greater detail in [2,3]. String theory realizations of metastable, supersymmetry breaking vacua have appeared in [4-13]. The gauge theoretic mechanism of [14] has further been explored in string theory in [15-22].

In this paper we study D5 brane/anti-D5 brane systems in IIB on non-compact, Calabi-Yau manifolds that are ADE type ALE space fibrations over a plane. These generalize the case of the A\ ALE space studied in detail in [1-3]. The ALE space is fibered over the complex plane in such a way that at isolated points, the 2-cycles inherited from the ALE space have minimal area. These minimal 2-cycles are associated to positive roots of the corresponding ADE Lie algebra. Wrapping these with branes and antibranes is equivalent to considering only branes, but allowing both positive and negative roots to appear, corresponding to two different orientations of the S2's. The system can be metastable since the branes wrap isolated minimal 2-cycles, and the cost in energy for the branes to move, due to the tensions of the branes, can overwhelm the Coulomb/gravitational attraction between them.

The geometries in question have geometric transitions in which the sizes of the minimal S2's go to zero, and the singularities are resolved instead by finite sized S3's. The conjecture of [1] is that at large N, the S2's disappear along with the branes and antibranes and are replaced by S3's with positive and negative fluxes, the sign depending on the charge of the replaced branes. As in the supersymmetric case (see [23-25]), the dual gravity theory has N = 2 supersymmetry softly broken to N = 1 by the fluxes. The only difference is that now some of the fluxes are negative. On-shell, the positive and the negative fluxes preserve different halves of the original supersymmetry, and with both present, the N = 2 supersymmetry is completely broken in the vacuum (see [26] for discussion of a similar supersymmetry breaking mechanism and its phe-nomenological features in the context of heterotic M-theory). The topological string computes not only the superpotential, but also the Kahler potential.1 We show that the Calabi-Yau's with fluxes obtained in this way are indeed metastable, as expected by holography. In particular, for widely separated branes, the supersymmetry breaking can be made arbitrarily weak.2 In fact, we can use the gravity dual to learn about the physics of branes and antibranes. We find that at one-loop, the interaction between the branes depends on the topological data of the Calabi-Yau in a simple way. Namely, for every brane/antibrane pair, so for every positive root e+ and negative root e-, we find that the branes and the antibranes attract if the inner product

e+ ■ eis positive. They repel if it is negative, and do not interact at all if it is zero. In the Ak type ALE spaces, this result is already known from the direct open string computation [27,28], so this is a simple but nice test of the conjecture for these geometries. Moreover, we show that certain aspects

1 While the superpotential is exact, the Kahler potential is not. Corrections to the Kahler potential coming from warping, present when the Calabi-Yau is compact, have been investigated in [13].

2 The natural measure of supersymmetry breaking in this case is the mass splitting between the bosons and their

superpartners. For a compact Calabi-Yau, the scale of supersymmetry breaking is set by the mass of the gravitino, which is of the order of the cosmological constant. In our case, gravity is not dynamical, and the mass splittings of the dynamical fields are tunable [1].

of these systems are universal. We find that generically, just like in [2], metastability is lost when the't Hooft coupling becomes sufficiently large. Moreover, once stability is lost, the system appears to roll down toward a vacuum in which domain walls interpolating between different values of the fluxes become light. We also present some special cases where the non-supersymmetric brane/antibrane systems are exactly stable. In these cases, there are no supersymmetric vacua to which the system can decay.

When all the branes are D5 branes and supersymmetry is preserved, the low energy theory geometrically realizes [24,25] a 4d N = 2 supersymmetric quiver gauge theory with a superpotential for the world-volume adjoints which breaks N = 2 to N = 1. These theories are known to have Seiberg-like dualities [29] in which the dual theories flow to the same IR fixed point, and where different descriptions are more weakly coupled, and hence preferred, at different energy scales. The Seiberg dualities are realized in the geometry in a beautiful way [25]. The ADE fibered Calabi-Yau geometries used to engineer the gauge theories have intrinsic ambiguities in how one resolves the singularities by blowing up S2's. The different possible resolutions are related by flops that shrink some 2-cycles, and blow up others. The flops act non-trivially on the brane charges, and hence on the ranks of the gauge groups. The flop of a 2-cycle S2 corresponds to a Weyl reflection about the corresponding root of the Lie algebra. On the simple roots ei, this acts by

Brane charge conservation then implies that the net brane charges transform satisfying

Moreover, from the dual gravity solution one can reconstruct the whole RG flow of the gauge theory. The sizes of the wrapped 2-cycles encode the gauge couplings, and one can read off how these vary over the geometry, and correspondingly, what is the weakly coupled description at a given scale. Near the S3 's, close to where the branes were prior to the transition, corresponds to long distances in the gauge theory. There, the S2's have shrunken, corresponding to the fact that in the deep IR the gauge theories confine. As one goes to higher energies, the gauge couplings may simply become weaker, and the corresponding S2's larger, in which case the same theory will describe physics at all energy scales. Sometimes, however, some of the gauge couplings grow stronger, and the areas of the S2's eventually become negative. Then, to keep the couplings positive, the geometry must undergo flop transitions.3 This rearranges the brane charges and corresponds to replacing the original description at low energies by a different one at high energies. Moreover, the flops of the S2's were found to coincide exactly with Seiberg dualities of the supersymmetric gauge theories.

In the non-supersymmetric case, there is generally no limit in which these brane constructions reduce to field theories with a finite number of degrees of freedom. Thus there are no gauge theory predictions to guide us. However, the string theory still has intrinsic ambiguities in how the singularities are resolved. This is exactly the same as in the supersymmetric case, except that now not all Ni in (1.1) need be positive. Moreover, we can use holography to follow the varying sizes of 2-cycles over the geometry, and find that indeed in some cases they can undergo

3 It is important, and one can verify this, that this happens in a completely smooth way in the geometry, as the gauge coupling going to infinity corresponds to zero Kâhler volume of the 2-cycle, while the physical size of the 2-cycle is finite everywhere away from the S3 's.

Sf ^ S2 = S2 - (ei ■ el0)S2

flops in going from the IR to the UV. When this happens, descriptions in terms of different brane/antibrane configurations are more natural at different energy scales, and one can smoothly interpolate between them. This is to be contrasted with, say, the A1 case, where regardless of whether one considers just branes or branes and antibranes, it is only one description that is ever really weakly coupled, and the fact that another exists is purely formal.

The paper is organized as follows. In Section 2 we introduce the metastable D5 brane/anti-D5 brane configurations, focusing on Ak singularities, and review the conjecture of [1] applied to this setting. In Section 3 we study in detail the A2 case with a quadratic superpotential. In Section 4 we consider general ADE type geometries. In Section 5 we discuss Seiberg-like dualities of these theories. In Section 6 we study a very simple, exactly solvable case. In Appendices A and B, we present the matrix model computation of the prepotential for A2 ALE space fibration, as well as the direct computation from the geometry. To our knowledge, these computations have not been done before, and the agreement provides a direct check of the Dijkgraaf-Vafa conjecture for these geometries. Moreover, our methods extend easily to the other An cases. In Appendix C, we collect some formulas useful in studying the metastability of our solutions in Section 3.

2. Quiver branes and antibranes

Consider a Calabi-Yau which is an Ak type ALE space, k+1

X2 + y2 +n(z - Zi(t)) = 0, (2.1)

fibered over the t plane. Here, zi(t) are polynomials in t. Viewed as a family of ALE spaces parameterized by t, there are k vanishing 2-cycles,

Sf, i = 1,...,k (2.2)

that deform the singularities of (2.1). In the fiber over each point t in the base, the 2-cycle in the class Si2 has holomorphic area given by

«2'° = zi(t) - zi+1(t), (2.3)

where w2,0 is the reduction of the holomorphic three-form £ on the fiber. The only singularities are at points where x = y = 0 and

Zi(t) = Zj(t), i = j (2.4)

for some i and j . At these points, the area of one of the 2-cycles inherited from the ALE space goes to zero.

These singularities can be smoothed out by blowing up the 2-cycles, i.e., by changing the Kahler structure of the Calabi-Yau to give them all non-vanishing area.4 The homology classes of the vanishing cycles (2.4) then correspond to positive roots of the Ak Lie algebra (see e.g. [24]).5 In this case, the k simple, positive roots ei correspond to the generators of the second

4 As we will review later, the blowup is not unique, as not all the Kahler areas of the cycles in (2.2) need to be positive for the space to be smooth. Instead, there are different possible blowups which differ by flops.

5 The negative roots correspond to 2-cycles of the opposite orientation.

homology group. These are the classes of the S2 mentioned above which resolve the singularities where zi (t) = zi+i (t). We denote the complexified Kahler areas of the simple roots by

k + iBNS,

where k is the Kahler form. In most of our applications, will take the real part of ri to vanish. The string theory background is nonsingular as long as the imaginary parts do not also vanish. They are positive, per definition, since we have taken the Sf to correspond to positive roots. In classical geometry, the ri are independent of t. Quantum mechanically, in the presence of branes, one finds that they are not.

There are also positive, non-simple roots eJ = J2\=j ei, for I > j where zl+i(t) = Zj(t). The 2-cycle that resolves the singularity is given by

s2 = E S2 i=j

in homology. Its complexified Kahler area is given as a sum of Kahler areas of simple roots

rj = ri. i=j

The total area A(t) of a 2-cycle Sj2 at a fixed t receives contributions from both Kahler and holomorphic areas:

Aj(t) = ^\rj |2 +\W'j (t)\2. (2.5)

The functions Wj capture the holomorphic volumes of 2-cycles, and are related to the geometry byI

Wj(t) = Y, Wi(t),

Wi(t) = j (zi(t) - Zi+i() dt. (2.6)

These will reappear as superpotentials in matrix models which govern the open and closed topo-logical string theory on these geometries.

For each positive root J there may be more than one solution to (2.4). We will label these with an additional index p when denoting the corresponding 2-cycles, Sj p. For each solution there is an isolated, minimal area S2, but they are all in the same homology class, labeled by the root. They have minimal area because (2.5) is minimized at those points in the t plane where Wj (t) vanishes. These, in turn, correspond to solutions of (2.4). We will consider wrapping branes in the homology class

E MJ,pS2>

with I running over all positive roots, and p over the corresponding critical points. We get branes or antibranes on Sf depending on whether the charge MIpP is positive or negative.6 We will study what happens when we wrap branes on some of the minimal S2's and antibranes on others.

The brane/antibrane system is not supersymmetric. If we had branes wrapping all of the S2's, they would have each preserved the same half of the original N = 2 supersymmetry. However, with some of the branes replaced by antibranes, some stacks preserve the opposite half of the original supersymmetry, and so globally, supersymmetry is completely broken. The system can still be metastable. As in flat space, there can be attractive Coulomb/gravitational forces between the branes and the antibranes. For them to annihilate, however, they have to leave the minimal 2-cycles that they wrap. In doing so, the area of the wrapped 2-cycle increases, as can be seen from (2.5), and this costs energy due to the tension of the branes. At sufficiently weak coupling, the Coulomb and gravitational interactions should be negligible compared to the tension forces— the former are a one-loop effect in the open string theory, while the latter are present already at tree-level—so the system should indeed be metastable. For this to be possible, it is crucial that the parameters of the background, i.e. the Kahler moduli ri and the complex structure moduli that enter into the Wi (t), are all non-normalizable, and so can be tuned at will.

While this theory is hard to study directly in the open string language, it was conjectured in [1] to have a holographic dual which gives an excellent description when the number of branes is large.

2.1. Supersymmetric large N duality

Here we review the case where only branes are wrapped on the minimal S2's, and so super-symmetry is preserved. Denoting the net brane charge in the class Sf by Ni, this geometrically engineers an N = 2 supersymmetric ]"[k=l U(Ni) quiver gauge theory in four dimensions, deformed to N = 1 by the presence of a superpotential. The corresponding quiver diagram is the same as the Dynkin diagram of the Ak Lie algebra. The k nodes correspond to the k gauge groups, and the links between them to bifundamental hypermultiplets coming from the lowest lying string modes at the intersections of the S2's in the ALE space. The superpotential for the adjoint valued chiral field , which breaks the supersymmetry to N = 1, is

Wi(&i), i = 1,..., k,

where Wi(t) is given in (2.6). The chiral field describes the position of the branes on the t plane. As shown in [24], the gauge theory has many supersymmetric vacua, corresponding to all possible ways of distributing the branes on the S2's,

J^NiS2 = £ MpjS2,

i=1 I,p

where I labels the positive roots and p the critical points associated with a given root. This breaks the gauge symmetry as

nU(Ni) ^n U(Mpj). (2.7)

6 We could have instead declared all the Mipp to be positive, and summed instead over positive and negative roots.

At low energies the branes are isolated and the theory is a pure N = 1 gauge theory with gauge group (2.7). The SU(Mi,p) subgroups of the U(Mi,p) gauge groups experience confinement and gaugino condensation.

This theory has a holographic, large N dual where branes are replaced by fluxes. The large N duality is a geometric transition which replaces (2.1) with a dual geometry

x2 + y2 + n (Z - Zi {t)) = fr-i(t)zk-1 + f2r-I(t)zk-2 + ■■■ + fkr-l(t), (2.8)

where fn(t) are polynomials of degree n, with r being the highest of the degrees of zi(t). The geometric transition replaces each of the Sj p's by a three-sphere, which will be denoted Aip, with Mip units of Ramond-Ramond flux through it,

hrr + thns = m, p p.

In addition, there is flux through the non-compact dual cycles Bi, p,

HRR + tHns = -a,,

where t is the IIB axion-dilaton t = a + g-. These cycles arise by fibering Sj p over the t plane, with the 2-cycles vanishing at the branch cuts where the S3's open up. The nonzero H flux through the B -type cycles means that

S2 Sl,p

BRR + TBNS

varies over the t plane. In the gauge theory, this combination determines the complexified gauge coupling. Since

ffL = 1 [bns, ^^ =f BRR + aBNS,

g2 8s J 2n J

one naturally identifies a{ with the gauge coupling of the U(N{), N = 2 theory at a high scale 0i 4ni

a = —---T ■

For each positive root i, we then define al as

ai = ^2, ai

7 For the large N dual to be an honest Calabi-Yau, as opposed to a generalized one, we will work with fg k = 0.

Turning on fluxes gives rise to an effective superpotential [30]

Weff = f (HRR + tHNS) a i2. CY

Using the special geometry relations

J n = Sip, J n = ds,pFo, Aip Bip

the effective superpotential can be written as

Weff = aiSip + Mi,pdsiiP Fo. (2.9)

Here, SI,P gets identified with the value of the gaugino bilinear of the U(MI,P) gauge group factor on the open string side. The effective superpotential (2.9) can be computed directly in the gauge theory. Alternatively, it can be shown [31,32] that the relevant computation reduces to computing planar diagrams in a gauged matrix model given by the zero-dimensional path integral

—-1- f Ud0idQu+i dQi+i,i expf --Tr W (<P, Q)

nLvol U(Ni)J f=\ , , V gs

TrW($, Q) = J2TrW(0i) + Tr(Qi+ij0iQu+i - Qu+i$i+iQi+u). i = 1

The critical points of the matrix model superpotential correspond to the supersymmetric vacua of the gauge theory. The prepotential F0(SI,P) that enters the superpotential (2.9) is the planar free energy of the matrix model [31,33-35], expanded about a critical point where the gauge group is broken as in (2.7). More precisely, we have

2niFo(S) = F0nP(S) + £Fo,{ha} ]"[ Shaa

{ha} a

where F0{ha} ]~[a(Mags)ha is the contribution to the planar free energy coming from diagrams with ha boundaries carrying the index of the U(Ma) factor of the unbroken gauge group. Here a represents a pair of indices,

a = (I, p),

and we have denoted Sa = Mags. The "nonperturbative" contribution, F^p(S), to the matrix model amplitude comes from the volume of the gauge group (2.7) that is unbroken in the vacuum at hand [33,35], and is the prepotential of the leading order conifold singularity corresponding to the shrinking S3, which is universal. We will explain how to compute the matrix integrals in Appendix A. The supersymmetric vacua of the theory are then given by the critical points of the superpotential Weff,

dsa Weff = 0.

2.2. Non-supersymmetric large N duality

Now consider replacing some of the branes with antibranes while keeping the background fixed. The charge of the branes, as measured at infinity, is computed by the RR flux through the S3 that surrounds the branes. In the large N dual geometry, the S3 surrounding the wrapped S2p is just the cycle Ai,p. Replacing the branes with antibranes on some of the S2's then has the effect of changing the signs of the corresponding Mi,p's. Moreover, supersymmetry is now broken, so the vacua of the theory will appear as critical points of the physical potential

y = GSaSb dSa WeffdSWff + Vo. (2.10)

The superpotential Weff is still given by (2.9), and G is the Kahler metric of the N = 2 theory,

Gab,= Im(T)ab

Tab = dSadSb Fo

and a, b stand for pairs of indices (i, p). In the absence of gravity, we are free to add a constant, V0, to the potential,8 which we will take to be

Vo = £ Mip. (2.11)

A priori, V0 can be either positive or negative, depending on the charges. However, will see that in all the vacua where the theory is weakly coupled, the leading contribution to the effective potential at the critical point will turn out to be just the tensions of all the branes, which is strictly positive.

3. A simple example

We now specialize to an A2 quiver theory with quadratic superpotential. The geometry which engineers this theory is given by (2.1), with

zi(t) = -m\(t - a\), Z2(t) = 0, Z3(t) = m2(t - a2). There are three singular critical points (2.4) (assuming generic mi) corresponding to

t = ai, i = 1,2, 3,

where a3 = (m\a\ + m2a2)/(m\ + m2). Blowing up to recover a smooth Calabi-Yau, the singular points are replaced by three positive area S2's,

222 , S2, S3

with one homological relation among them,

S32 = S2 + S2. (3.1)

8 This simply adds a constant to the Lagrangian, having nothing to do with supersymmetry, or its breaking.

S2 2 then correspond to the two simple roots of the A2 Lie algebra, e1i2, and is the one nonsimple positive root, e1 + e2. Now consider wrapping branes on the three minimal 2-cycles so that the total wrapped cycle C is given by

C = M S + M2S2 + M3 S32.

If some, but not all, of the Mj are negative, supersymmetry is broken. As was explained in the previous section, as long as the branes are widely separated, this system should be perturbatively stable.

Non-perturbatively, we expect the branes to be able to tunnel to a lower energy state. The minimum energy configuration that this system can achieve depends on the net brane charges in the homology classes S2 and S2, given by N1 = Mi + M3 and N2 = M2 + M3. When N1 and N2 have the same sign, the system can tunnel to a supersymmetric vacuum with new charges

where all the M'j share the same sign, and the net charges N\ = M[ + M'3 and N2 = + M'3 are unchanged. All the supersymmetric vacua are degenerate in energy, but for the metastable, non-supersymmetric vacua, the decay rates will depend on the mj . Alternatively, if one of the Ni,2 is positive and the other is negative, the lowest energy configuration is necessarily not supersymmetric. In this way we get a stable, non-supersymmetric state which has nowhere to which it can decay.

In the remainder of this section, we will study these systems using the large N dual geometry with fluxes.

3.1. The large N dual

The large N dual geometry in this case is given by

x2 + y2 + z(z - mx(t - ax))(z + m2(t - a2)) = cz + dt + e. (3.2)

The three S2's at the critical points have been replaced by three S3's, A7, whose sizes are related to the coefficients c, d, e above. There are also three non-compact, dual 3-cycles B i . The geometry of the Calabi-Yau is closely related to the geometry of the Riemann surface obtained by setting x = y = 0 in (2.8). The Riemann surface can be viewed as a triple cover of the t plane, by writing (3.2) as

where zi (t) correspond to the zi (t) which are deformed in going from (2.1) to (3.2). In particular, the holomorphic three-form £2 of the Calabi-Yau manifold descends to a 1-form on the Riemann surface, as can be seen by writing

£ = a?,0 A dt

and integrating a2,0 over the S2 fibers, as in (2.3). The A and B cycles then project to 1-cycles on the Riemann surface. The three sheets are glued together over branch cuts which open up at t = ai. We have

Mj ^ Ml

0 = (z - zi(t)){z - z2(t)){z - z3(t))

for cyclic permutations of distinct I, J and K. This allows one to compute the prepotential F0 by direct integration (see Appendix B). Alternatively, by the conjecture of [33], the same prepotential can be computed from the corresponding matrix model. The gauge fixing of the matrix model is somewhat involved, and we have relegated it to Appendix A, but the end result is very simple. The field content consists of:

(a) Three sets of adjoints Фц of U(Mi), which describe the fluctuations of the branes around the three S2's.

(b) A pair of bifundamental matter fields Q12, Q21, coming from the 12 strings.

(c) Anticommuting bosonic ghosts, B13, C31 and B32, C23, representing the 23 and 31 strings.

Note that physical bifundamental matter from S2's with positive intersection corresponds to commuting bosonic bifundamentals in the matrix model, whereas W bosons between S2's with negative intersection in the physical theory correspond to bosonic ghosts, similarly to what happened in [35].

The effective superpotential for these fields is

Weff = 1 m i Tr $2i + 1 m2 Tr ^ + 2 m.3 Tr ^

+ ai2Tr Q12Q21 + <23 Tr B32C23 + <231 Tr B13C31

+ Tr(B32^22 C23 — C23 ^33 B32) + Tr(B13&33C31 — C31&11B13 )

+ Tr(Q 21^11 Q12 — Q 12^22 Q 21) where aij = ai — aj. From this we can read off the propagators

{ФцФц) =

(Ql2 Q 21) =

B23C32) =--,

(B31C13 ) =--,

as well as the vertices.

Keeping only those contractions of color indices that correspond to planar diagrams, and carefully keeping track of the signs associated with fermion loops, we find:

2niFo(Si) = 1 S2( log

2_1 КшлЛ2

- l) + 2 ^V т2Л2

+ 2 S32llo4 т3Л2

- ^S) S1S2 + ЧЛ0) S1S3 + ^Л3) S2S3

+ -2J (S2S2 + S22S1 + S32S1 + S32S2 - S2S3 - S22S3 - 6S1S2S3) + .

3 Ш1Ш2 2

12, m3 = m1 + m2.

The terms quadratic in the Si's correspond to one-loop terms in the matrix model, the cubic terms to two-loop terms, and so on. The fact that the matrix model result agrees with the direct computation from the geometry is a nice direct check of the Dijkgraaf-Vafa conjecture for quiver theories. The large N limit of quiver matrix models was previously studied using large N saddle point techniques in [24,25,36,37].

Consider now the critical points of the potential (2.10),

dSlV = 0.

The full potential is very complicated, but at weak't Hooft coupling (we will show this is consistent a posteriori) it should be sufficient to keep only the leading terms in the expansion of F0 in powers of S/A3. These correspond to keeping only the one-loop terms in the matrix model. In this approximation, the physical vacua of the potential (2.10) correspond to solutions of

ai + J2 xijMJ + J2 TJJMj = 0. (3.4)

Mj>0 MJ<0

To be more precise, there are more solutions with other sign choices for ±MJ, but only this choice leads to Im(r) being positive definite. Since Im(r) is also the metric on the moduli space, only this solution is physical.

Depending on how we choose to distribute the branes, there are two distinct classes of non-supersymmetric vacua which can be constructed in this way. We will discuss both of them presently.

3.2. Mi < 0, M2,3 > 0

In this case, the critical points of the potential correspond to

|M2 I \ -|M3|

o|M1| /72—\\M1\(a12\1 27 a31 \ 1 31 , _ . ,

S[ 11 = (^0m0 ( A) (a) exp(-2^i«1),

,-\ |M1| / \ -|M3|

S2M2| = (^0m2)|M2|(a^) (exp(-2W),

/ \ -|M2| /-\-|M1|

S3 |M31 = Km3)|M3|( exp(-2.i«3).

The Si are identified with the gaugino condensates of the low energy, U(M1) x U(M2) x U(M3) gauge theory. The gaugino condensates are the order parameters of the low energy physics and as such should not depend on the cutoff A0. Let us then introduce three new confinement scales, Ai, defined as

Si = A3.

In fact, only two of these are independent. As a consequence of homology relation (3.1), the gauge couplings satisfy a1 + a2 = a3, which implies that

'ATy a^31 M2 = ^ A^31 M3'

where A is given in (3.3). Requiring that the scales Ai do not depend on the cutoff scale, we can read off how the gauge couplings run with Ao,

g-1(Ao) = — log

'-(Ao) = —log

— log(

— log

\ |M-|

— log

1 (Ao — log —

— |M3|

— |M3|

As was noticed in [1], this kind of running of the gauge couplings and relation between strong coupling scales is very similar to what occurs in the supersymmetric gauge theory (as studied in [25]) obtained by wrapping Mi branes of the same kind on the three S2's. The only difference is that branes and antibranes lead to complex conjugate running, as if the spectrum of the theory remained the same, apart from the chirality of the fermions on the brane and the antibrane getting flipped. This is natural, as the branes and the antibranes have opposite GSO projections, so indeed a different chirality fermion is kept. In addition, the open string RR sectors with one boundary on branes and the other on antibranes has opposite chirality kept as well, and this is reflected in the above formulas.

To this order, the value of the potential at the critical point is

M | 1 y* = E Hr — ||m-| log

j g2 2n

+ —|Mi||M31 log 2n

The first terms are just due to the tensions of the branes. The remaining terms are due to the Coulomb and gravitational interactions of the branes, which come from the one-loop interaction in the open string theory. There is no force between the M2 branes wrapping and the M3 branes on S|, since M2,3 are both positive, so the open strings stretching between them should be supersymmetric. On the other hand, the M1 antibranes on S"2 should interact with the M2,3 branes as the Coulomb and gravitational interactions should no longer cancel. This is exactly what one sees above. The M1 antibranes on S"2 attract the M3 branes on Sf, while they repel the branes on Sf. We will see in the next section that more generally, branes and antibranes wrapping 2-cycles with negative intersection numbers (in the ALE space) attract, and those wrapping 2-cycles with positive intersection numbers repel. Since9

ei ■ e2 = 1, ei ■ e3 = —1,

this is exactly what we see here. 3.3. Mu > 0, M3 < 0

With only the non-simple root wrapped by antibranes, the critical points of the potential now correspond to

IM2I \ -IM3I

c|Mi| 2 \|Mi|/ a12\| 2Y a3^ | 3| / o • n S{ 1 = (A2mi)' ^ exp(—2mai),

/ \ |M1| /-\ —|M3|

sfU (A0m2)|M2|(ag) exp(—2ni«2),

9 The second relation is due to the self intersection numbers of s2 and S? being —2.

S3M3| = (A0m3 )|M3I^

3 0 A0

a3A-|M1|

I — I exp(-2nia3).

In this case, the Kahler parameters a1,2 run as

g^A) = - log

g22(A0) = - log

Ac\ |M2|_ a12J

i (A0 log —

\ |M2|

2 ) - log

31 1 A)

A3 \3|M3|

This follows the same pattern as seen in [1] and in the previous subsection. The branes and antibranes give complex conjugate running, as do the strings stretching between them. The value of potential at the critical point is, to this order,

^ M | 1 (

v* = E ~MT + 'M1HM31 log^

|M2||M3| log

Again, the first terms are universal, coming from the brane tensions. The remaining terms are the one-loop interaction terms. There is no force between the M1 branes wrapping S2 and the M2 branes on S|, since now both M1,2 have the same sign. The M3 antibranes on S| attract both M1 branes on S2 and the M2 branes on S|, since, in the ALE space

e1 ■ e3 = e2 ■ e3 = -1.

In the next subsection, we will show that both of these brane/antibrane systems are perturba-tively stable for large separations.

3.4. Metastability

The system of branes and antibranes engineered above should be perturbatively stable when the branes are weakly interacting—in particular, at weak't Hooft coupling. The open/closed string duality implies that the dual closed string vacuum should be metastable as well. In this subsection, will show that this indeed is the case. Moreover, following [2], will show that per-turbative stability is lost as we increase the't Hooft coupling. While some details of this section will be specific to the A2 case discussed above, the general aspects of the analysis will be valid for any of the ADE fibrations discussed in the next section.

To begin with, we note that the equations of motion, derived from the potential (2.10), are

d*V FkefGaeGbf(aa + Mcrac)(a b + Mdrbd) = 0,

and moreover, the elements of the Hessian are

dpdqV = GiaGbjiFabpq (a + Mkxki)(aj + Mrxr))

+ 2GiaGbcGdjiFabpi Fcdq (ai + MkTki)(a j + MrTrj),

dqdq V = -GiaGbcGdjiFabpi Fcdq{ai + Mk^){a j + Mrtrj)

- GicGdaGbjiFabpiFcdq(ai + Mktki)(aj + MrTj), (3.7)

where we have denoted dcTab = Fabc, and similarly for higher derivatives of t .

In the limit where all the't Hooft couplings g2Ni are very small, the sizes of the dual 3-cycles Sa = A3a are small compared to the separations between them, so we can keep only the leading terms in the expansion of F0 in powers of S, i.e., the one-loop terms in the matrix model. At one-loop, the third and fourth derivatives of the prepotential are nonzero only if all of the derivatives are with respect to the same variable. Expanding about the physical solution to this order,

aa + J2 Tab Mb + J2 Tab Mb = 0. (3.8)

Mb>0 Mb<0

The non-vanishing elements of the Hessian are

1 |M!'Mj | i; didj V = — SS G j, i, j opposite type,

1 |M!'Mj | ii

did j V = —- --^-G'J, i, j same type, (3.9)

j 4n2 SiSj , n

where the 'type' of an index refers to whether it corresponds to branes or antibranes.

To get a measure of supersymmetry breaking, consider the fermion bilinear couplings. Before turning on fluxes, the theory has N = 2 supersymmetry, and the choice of superpotential (2.6) breaks this explicitly to N = 1. For each 3-cycle, we get a chiral multiplet (Si ,fi) and a vector multiplet (Ai,Xi) where fi,'ki are a pair of Weyl fermions. It is easy to work out [1] that the coefficients of the non-vanishing fermion bilinears are

b = 2 Gcd(ad + -eTde) Fabc,

mkakb = 2 Gcd (ad + MeTde)Fabc, and evaluating this in the vacuum we find

_ 1 Ma + |Ma |

mfafb = 4^ s °ab,

1 Ma -— |

mX*Xb = --^-dab.

Bose-Fermi degeneracy is restored in the limit where we take (Gij) /GuGjj « 1, i, j opposite type.

In this limit we get a decoupled system of branes and antibranes except that for nodes wrapped with branes, the Sa get paired up with f's, and for nodes wrapped with antibranes they pair with A.'s, corresponding to a different half of N = 2 supersymmetry being preserved in the two cases.10 This is the limit of extremely weak't Hooft coupling, and the sizes of the cuts are the

10 The kinetic terms of both bosons and fermions are computed with the same metric Gab.

smallest scale in the problem by far Ai aij A

A«j A< 1- (3.10)

A A0 A0

In this limit the Hessian is manifestly positive definite. In fact the Hessian is positive definite as long as the one-loop approximation is valid. To see this note that the determinant of the Hessian is, up to a constant, given by

/ 1 n - -• 2

Det(92y) ^ De^n

. Det G

\ i = 1

(3.11)

It is never zero while the metric remains positive definite, so a negative eigenvalue can never appear. Thus, one can conclude that as long as all the moduli are in the regime where the't Hooft couplings are small enough for the one-loop approximation to be valid, the system will remain stable to small perturbations.

Let us now find how the solutions are affected by the inclusion of higher order corrections. At two loops, an exact analysis of stability becomes difficult in practice. However, in various limits one can recover systems which can be understood quite well. For simplicity, we will assume that the ai are all pure imaginary, and all the parameters -ij and A0 are purely real. Then there are solutions where the Si are real. In Appendix B, we show that in this case, upon including the two-loop terms, the determinant of the Hessian becomes

|MCIV* iFbbbb&b

(Det G-bfiu 4De/5cb + G^^f )

V ' V}iFccJ V iFbbbiFcccIMc\)

x Detf&cb - Gcb iFbbbb&b ) (3.12)

Vcb cb iFbbbiFcccIMc\) y '

Sk =-r-Fk-b(-\M-\\Mb\ + M-Mb) (3.13)

2IMk IFkkk 1

and Scb is the Kronecker delta. The first two terms in (3.12) never vanish, since the metric has to remain positive definite, so we need only analyze the last two determinants. We can plug in the one-loop values for the various derivatives of the prepotential, and in doing so obtain

( S- xb \

Det S-b ± 2nG-b—r-- = 0 (3.14)

\-b A3 IM-1J

with either choice of sign. Above, we have rewritten Eq. (3.13) as

SS- = — x-. (3.15)

This is a convenient rewriting because S/A3 is the parameter controlling the loop expansion, and x- is simply a number which depends on the N' but no other parameters.

Consider the case where, for some i, a given S3 grows much larger than the other two. We can think of this as increasing the effective't Hooft coupling for that node, or more precisely, increasing

( AMi \ 3 ( 1 \

, . = exp---

A ) \ I Mi IgffA)

Recall that the two-loop equations of motion for real Si, are given by

g"2ff(A) = —\Mi | log

+ GikSk

(3.16)

g-e2ff(^) = g-2(^0) -—1^12 + L13) + |-2|L12 -—3^13,

g2-e2ff(^) = g2-2(^0) - | —21 (L12 + L23) + |M1|L12 - —3^3, g-ff^) = g3-2(^0) -|—3|(L13 + L23) - |—1 |L13 -|—2|L23.

Here we have adopted the notation Lij = log and the Sk are as defined in (3.13). Note that in each case, two of the equations can be solved straight off. It is the remaining equations which provide interesting behavior and can result in a loss of stability. Correspondingly, the vanishing of the Hessian determinant in (3.12) is then equivalent to the vanishing of its ii entry (where we have assumed a vacuum at real S):

1 ± Gi

A3 \Mi \

We will see that we can approximate

(3.17)

Gii = —log

+ Li - Li

where we have defined

Li = Lij + Lik, i = j = k, so this provides the following conditions:

±1 = LrXi Si

(3.18)

\Mi\ A3'

The above equation, taken with positive sign, is equivalent to the condition for stability being lost by setting the determinant of the gradient matrix of the equations to zero. The equation with minus sign comes from losing stability in imaginary direction. Correspondingly, the equation of motion for the one node with growing't Hooft coupling becomes

Si Si g—ef =— M \ log A + L<XlA.

(3.19)

One of Eqs. (3.18) must be solved in conjunction with (3.19) if stability is to be lost.

The sign of xi can vary depending on the specifics of the charges. In all the cases, as the effective't Hooft coupling increases, solutions move to larger values of Si. For sufficiently large values, in the absence of some special tuning of the charges, (3.18) will be satisfied for one of the two signs. The only question then is whether the Si can get large enough, or whether a critical value above which the equation of motion can no longer be solved is reached before an instability sets in. In the equation above, if xi is negative, then there will be no such critical value, and Si can continue to grow unbounded. Correspondingly, a large enough value of the't Hooft coupling can always be reached where (3.18) is satisfied with negative sign. Alternatively, if the coefficient xi is positive, there will be a critical value for Li at which the right-hand side of the equation

takes a minimum value. This occurs at (Si.tJA3) = XL, which is precisely (3.18) with positive

' XiLi

left-hand side. So, for any value of xi an instability develops at finite effective't Hooft coupling corresponding to

Si,* \Mi |

|xi\Li'

or more precisely, at

\Mi\gffA) = log-1( M ' \xiLi

This critical value of the effective't Hooft coupling can be achieved by increasing the number of branes on that node, or, in case of nodes one and two, by letting the corresponding bare't Hooft coupling increase. This is true as long as supersymmetry is broken and the corresponding two-loop correction is non-vanishing, i.e. as long as xi = 0. It is reasonable to suspect that in the degenerate case, where charges conspire to set xi to zero even with broken supersymmetry, the instability would set in at three loops.

It is natural to ask the fate of the system once metastability is lost. It should be the case [2] that it rolls to another a critical point corresponding to shrinking the one compact B-type cycle, B1 + B2 - B3. To describe this point in the moduli space, introduce a new basis of periods in which this shrinking B-cycle becomes one of the A periods:

H = M1 + M3, j> H = M2 + M3, j> H = 0,

A1 A2 A3

J H = a1, j H = a2, J H = M3,

B1 B2 B3

where H = HRR + tHns. In particular, there is no flux through the new cycle A3. In fact, by setting M1 = M2 = —M3 = M, there is no flux through any of the A' cycles.11 For Si = fA, Q sufficiently large that we can ignore the light D3 branes wrapping this cycle,

1 S1 — 2n l0g A3, Ti=j —const,

it is easy to see that the system has an effective potential that would attract it to the point where the Si = 0 and the cycles shrink:

Veff — Vo + ^

log \ \

where ci — fB, H. By incorporating the light D3 branes wrapping the flux-less, shrinking cycles, the system would undergo a geometric transition to a non-Kahler manifold [38]. There, the cycle shrinks and a new 2-cycle opens up, corresponding to condensing a D3 brane hypermultiplet. However, this 2-cycle becomes the boundary of a compact 3-cycle B1 which get punctured in the transition that shrinks the A' cycles. A manifold where such a 2-cycle has nonzero volume is automatically non-Kahler, but it is supersymmetric. As will review shortly, the shrinking cycle A3

11 In the more general case, the system should be attracted to a point where only A3 shrinks.

is also the cycle wrapped by the D5 brane domain walls that mediate the non-perturbative decay of the metastable flux vacua. The loss of metastability seems to be correlated with existence of a point in the moduli space where the domain walls become light and presumably fluxes can annihilate classically (this also happened in the A1 model studied in [2]). In particular, in the last section of this paper, will provide two examples of a system where the corresponding points in the complex structure moduli space are absent, but which are exactly stable perturbatively even though they are non-supersymmetric (one of them will be stable non-perturbatively as well). It must be added, as discussed in [2], that it is far from clear whether the light domain walls can be ignored, and so whether the system truly rolls down to a supersymmetric vacuum. A more detailed analysis of the physics at this critical point is beyond the scope of this paper.

It was suggested in [2] that the loss of stability might be related to the difference in the value of V* between the starting vacuum and a vacuum to which it might tunnel becoming small, and thus the point where Coulomb attraction starts to dominate in a subset of branes. In the more complicated geometries at hand, it seems that such a simple statement does not carry over. This can be seen by noting that, for certain configurations of brane charges in our case, an instability can be induced without having any effect on the AV* between vacua connected by tunneling events. We are led to conclude that the loss of stability is a strong coupling effect in the non-supersymmetric system, which has no simple explanation in terms of our open string intuition. This should have perhaps been clear, in that the point to which the system apparently rolls has no straightforward explanation in terms of brane annihilation.

3.5. Decay rates

We now study the decays of the brane/antibrane systems of the previous section. This closely parallels the analysis of [1]. We have shown that when the branes and antibranes are sufficiently well-separated, the system is perturbatively stable. Non-perturbatively, the system can tunnel to lower energy vacua, if they are available. In this case, the available vacua are constrained by charge conservation—any two vacua with the same net charges

N1 = M1 + M3, N2 = M2 + M3

are connected by finite energy barriers. The false vacuum decay proceeds by the nucleation of a bubble of lower energy vacuum.

The decay process is easy to understand in the closed string language. The vacua are labeled by the fluxes through the three S3's

j HRR = Mi, I = 1, 2, 3.

Since RR three-form fluxes jump in going from the false vacuum to the true vacuum, the domain walls that interpolate between the vacua are D5 branes. Over a D5 brane wrapping a compact 3-cycle C in the Calabi-Yau, the fluxes jump by an amount

AMI = #(C n AI).

In the present case, it is easy to see that there is only one compact 3-cycle C that intersects the A-cycles,

C = B1 + B2 - B3.

So, across a D5 brane wrapping C, the fluxes through A1,2 decrease by one unit, and the flux through A3 increases by one unit. Note that this is consistent with charge conservation for the branes. In fact, the domain walls in the open and the closed picture are essentially the same. In the open string language, the domain wall is also a D5 brane, but in this case it wraps a 3-chain obtained by pushing C through the geometric transition. The 3-chain has boundaries on the minimal S2's, and facilitates the homology relation (3.1) between the 2-cycles. The decay rate r is given in terms of the action Smst of the relevant instanton

r ~ exp(-Sjnst).

Since the Calabi-Yau we have been considering is non-compact, we can neglect gravity, and the instanton action is given by

27n S4n

c. __D

Sinst. —

2 (AV,)3

where SD is the tension of the domain wall, and AV, is the change in the vacuum energy across the domain wall. While this formula was derived in [39] in a scalar field theory, it is governed by energetics, and does not depend on the details of the theory as long as the semi-classical approximation is applicable.

In the present case, the tension of the domain wall is bounded below by

Sd — — f Q, (3.20)

gs J C

since the fC Q computes the lower bound on the volume of any 3-cycle in this class, and the classical geometry is valid to the leading order in 1 /N, the order to which we are working. The tension of the domain wall is thus the same as the tension of a domain wall interpolating between the supersymmetric vacua, and to leading order (open-string tree-level) this is given by the difference between the tree-level superpotentials (2.6)

Q - W3(a3) - Wi(ai) - W2(a2) — -¿3,

where is defined in (3.3). This is just the "holomorphic area" of the triangle in Fig. 1. The area is large as long as all the brane separations are large, and as long as this is so, it is independent of the fluxes on the two sides of the domain wall.

At the same time, the difference in the potential energy between the initial and the final states is given by the classical brane tensions,

AV — V - Vf — £ (\Mj | - \M'j\)/g2.

The fate of the vacuum depends on the net charges. If Ni,2 are both positive, then the true vacuum is supersymmetric. Moreover, there is a landscape of degenerate such vacua, corresponding to all possible ways of distributing branes consistent with charge conservation such that M'} are all positive. Starting with, say, (M\,M2,M3) — (N\ + k, N2 + k, -k), where k > 0, this can decay to (N1 ,N2, 0) since

AV — V - Vf — 2-^,

Fig. 1. The figure corresponds to the A2 singularity in the z-t plane with quadratic "superpotential". There are three conifold singularities at Zi = Zj which can be blown up by three S2's, spanning two homology classes. Wrapping —1 anti-D5 branes on S12 and M2,3 D5 branes on S22,3 , we can engineer a metastable vacuum. The orientations of the branes are indicated by arrows.

corresponding to k branes on S| getting annihilated, where r3 is the Kahler area of S|. The decay is highly suppressed as long as string coupling gs is weak and the separation between the branes is large. The action of the domain wall is k times that of (3.20),12 so

Sinst —

27n k j^j

27n j g3 j3

lg31 -kjAj12.

(3.21)

32 gs jr3 j3 32 g4

The instanton action (3.21) depends on the cutoff scale A0 due to the running of the gauge coupling g3 (A0). The dependence on A0 implies [12] that (3.21) should be interpreted as the rate of decay corresponding to fluxes decaying in the portion of the Calabi-Yau bounded by A0.

If instead we take say N1 > 0 > N2, then the lowest energy state corresponds to N1 branes on node 1, N2 antibranes on node 2, with node 3 unoccupied. This is the case at least for those values of parameters corresponding to the system being weakly coupled. In this regime, this particular configuration gives an example of an exactly stable, non-supersymmetric vacuum in string theory—there is no other vacuum with the same charges that has lower energy. Moreover, as will discuss in Section 6, for some special values of the parameters m1,2 the system is exactly solvable, and can be shown to be exactly stable even when the branes and the antibranes are close to each other.

4. Generalizations

Consider now other ADE fibrations over the complex plane. As in (2.1) we start with the deformations of 2-complex dimensional ALE singularities:

x2 + / + zk+1 = 0, x2 + y2z + zr-1 = 0,

12 All quantities being measured in string units.

Fig. 2. Some of the two-loop Feynman graphs of the matrix model path integral, which are computing the prepotential Fo. The path integral is expanded about a vacuum corresponding to distributing branes on the three nodes. Here, the boundaries on node one are colored red, on node two are green and on node three are blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

£5: x2 + y

£7: x2 + y:

£s: x2 + y:

„4 .

and fiber these over the complex t plane, allowing the coefficients parameterizing the deformations to be t dependent. The requisite deformations of the singularities are canonical (see [24] and references therein). For example, the deformation of the Dr singularity is

x2 + y2z + z-1|

' r r \ r

v№ - z2) -n z2 + 2n

\i = 1 i = 1 ' i = 1

In fibering this over the t plane, the zi become polynomials zi(t) in t.13 After deformation, at a generic point in the t plane, the ALE space is smooth, with singularities resolved by a set of r independent 2-cycle classes

i = 1,.

where r is the rank of the corresponding Lie algebra. The 2-cycle classes intersect according to the ADE Dynkin diagram of the singularity: The deformations can be characterized by "superpotentials",

W'At) = v

which compute the holomorphic volumes of the 2-cycles at fixed t. For each positive root ej, which can be expanded in terms of simple roots ei as

= E »Ie

13 This is the so-called "non-monodromic" fibration. The case where the Zi are instead multi-valued functions of t corresponds to the "monodromic" fibration [24].

Fig. 3. Dynkin diagrams of the ADE Lie algebras. Every node corresponds to a simple root and to a 2-cycle class of self intersection -2 in the ALE space. The nodes that are linked correspond to 2-cycles which intersect with intersection number +1.

for some positive integers nj, one gets a zero-sized, primitive 2-cycle at points in the t-plane where

W'j (t) = J2 nijW[(t) = 0. (4.1)

Blowing up the singularities supplies a minimal area to the 2-cycles at solutions of (4.1), t = aj,p,

where i labels the positive root and p runs over all the solutions to (4.1) for that root.

As shown in [24] and references therein, the normal bundles to the minimal, holomorphic S2's obtained in this way are always O(-1) © O(-1), and correspondingly the S2's are isolated.14 This implies that when branes or antibranes are wrapped on the S2's, there is an energy cost to moving them off. Moreover, the parameters that enter into defining the Wi, as well as the Kahler classes of the S2's, are all non-dynamical in the Calabi-Yau. As a consequence, if we wrap branes and antibranes on minimal S2's, the non-supersymmetric system obtained is metastable, at least in the regime of parameters where the S2's are well separated.

The ALE fibrations have geometric transitions in which each minimal S2 is replaced by a minimal S3. A key point here is that none of the 2-cycles have compact, dual 4-cycles, so the transitions are all locally conifold transitions. The one-loop prepotential F0 for all these singularities was computed in [25], and is given by

2niF0 (S) = S2flo/ Sb ^ 3

2^S^yw'/(ab)A2J - 2, + IV eJC)SbSc log( Afy, (4.2)

14 In [24] the authors also considered the monodromic ADE fibrations, where the 2-cycles of the ALE space undergo monodromies around paths in the t plane. In this case, the novelty is that the S2's can appear with normal bundles O © O(-2) or O(-1) © O(3). Wrapping branes and antibranes on these cycles is not going to give rise to new metastable vacua, since there will be massless deformations moving the branes off of the S2's. It would be interesting to check this explicitly in the large N dual.

where the sum is over all critical points b = (J, p), and J(b) = J denotes the root J to which the critical point labeled by b corresponds. We are neglecting cubic and higher order terms in the SJ,p, which are related to higher loop corrections in the open string theory. Above, WJ(t) is the superpotential corresponding to the root eJ, and eJ ■ 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.

Consider wrapping Mb branes or antibranes on the minimal S2's labeled by b = (J, p). We'll take all the roots to be positive, so we get branes or antibranes depending on whether Mb is positive or negative. The effective superpotential for the dual, closed-string theory is given by (2.9). From this and the corresponding effective potential (2.10), we compute the expectation values for Sb in the metastable vacuum to be

The first term in the potential is just the contribution of the tensions of all the branes and antibranes. The second term comes from the Coulomb and gravitational interactions between branes, which is a one-loop effect in the open string theory. As expected, at this order only the brane/antibrane interactions affect the potential energy. The open strings stretching between a pair of (anti)branes, are supersymmetric, and the (anti)branes do not interact. The interactions between branes and antibranes depend on

which is minus the intersection number—in the ALE space—of the 2-cycle classes wrapped by the branes. The branes and antibranes attract if the 2-cycles they wrap have negative intersection, while they repel if the intersection number is positive, and do not interact at all if the 2-cycles do not intersect.

For example, consider the Ak quiver case, and a set of branes and antibranes wrapping the 2-cycles obtained by blowing up the singularities at

Zi(t) = Zj(t), Zm(t) = Zn(t)

where i < j and m < n. The branes do not interact unless i or j coincide with either m or n. The branes attract if i = m or j = n, in which case the intersection is either -1 or -2, depending on whether one or both of the above conditions are satisfied. This is precisely the case when the branes and antibranes can at least partially annihilate. If j = m or i = n, then the 2-cycles have intersection +1, and the branes repel. In this case, the presence of branes and antibranes

Mb< 0,

Mb> 0.

The value of the effective potential at the critical point is given by

ei • ej

should break supersymmetry, but there is a topological obstruction to the branes annihilating, even partially. In fact, in the Ak type ALE spaces, this result is known from the direct, open string computation [27,28]. The fact that the direct computation agrees with the results presented here is a nice test of the conjecture of [1].

5. A non-supersymmetric Seiberg duality

In the supersymmetric case, with all Mi positive, the engineered quiver gauge theories have Seiberg-like dualities. In string theory, as explained in [25], the duality comes from an intrinsic ambiguity in how we resolve the ADE singularities to formulate the brane theory.15 The different resolutions are related by flops of the S2's under which the charges of the branes, and hence the ranks of the gauge groups, transform in nontrivial ways. The RG flows, which are manifest in the large N dual description, force some of the S2's to shrink and others to grow, making one description preferred over the others at a given energy scale. In this section, we argue that Seiberg dualities of this sort persist even when some of the branes are changed to antibranes and supersymmetry is broken.

5.1. Flops as Seiberg dualities

For a fixed set of brane charges, one can associate different Calabi-Yau geometries. There is not a unique way to blow up the singularity where an S2 shrinks, and the different blowups are related by flops that shrink some 2-cycles and grow others. Instead of giving a 2-cycle class Si2 a positive Kahler volume

ri = I BNS

we can give it a negative volume, instead. This can be thought of as replacing the 2-cycle class by one of the opposite orientation

51 ^ S2 = -S2.

The flop of a simple root Si2 acts as on the other roots as a Weyl reflection which permutes the positive roots

52 ^ S2 = S2 - (ej • ei)Sf. (5.1)

The net brane charges change in the process, but in a way consistent with charge conservation

I>S?=E

^NtSf = V NiSf. (5.2)

We can follow how the number of branes wrapping the minimal 2-cycles change in this process. If i is the simple root that gets flopped,16 then Mi,p goes to Mip = -Mip and for other roots

15 The idea that Seiberg dualities have a geometric interpretation in string theory goes back a long while, see for example [40-44]. The fact that these dualities arise dynamically in string theory has for the first time been manifested in [25,45].

16 Flopping non-simple roots can be thought of in terms of a sequence of simple node flops, as this generates the full Weyl group.

labeled by j = i

Mj,p = Mw(j)p (5.3)

where w(j) is the image of j under the Weyl group action.

The size of the wrapped S2 is proportional to the inverse gauge coupling for the theory on the wrapping branes,

g~2(t) a-/ Bns , (5.4)

S2 Si,t

so the flop (5.1) transforms the gauge couplings according to

g-2 ^ g-2 = g-2 - (ej ' ei )g~2- (5.5)

Generally, there is one preferred description for which the gauge couplings are all positive. In the geometry, we have the freedom to choose the sizes of the 2-cycles Si2,t at some fixed high scale, but the rest of their profile is determined by the one-loop running of the couplings (3.5) throughout the geometry and by the brane charges. The most invariant way of doing this is to specify the scales Ai at which the couplings (5.4) become strong. We can then follow, using holography, the way the B-fields vary over the geometry as one goes from near where the S3's are minimal, which corresponds to low energies in the brane theory, to longer distances, far from where the branes were located, which corresponds to going to higher energies. The S2's have finite size and shrink or grow depending on whether the gauge coupling is increasing or decreasing. We will see that as we vary the strong coupling scales of the theory, we can smoothly interpolate between the two dual descriptions. Here it is crucial that the gauge coupling going through zero is a smooth process in the geometry: while the Kahler volume of the 2-cycle vanishes as one goes through a flop, the physical volume, given by (2.5), remains finite. Moreover, we can read off from the geometry which description is the more appropriate one at a given scale.

5.2. The A2 example

For illustration, we return to the example of the A2 quiver studied in Section 3. To begin with, for a given set of charges Mi, we take the couplings g-2 of the theory to be weak at the scale A set by the "superpotential". This is the characteristic scale of the open-string ALE geometry. Then Si/A3 is small in the vacuum, and the weak coupling expansion is valid. From (3.5), we can deduce the one-loop running of the couplings with energy scale / = t

J/-2/) = (2|M1i + |M3| - |M2|),

/j/g-2/) = (2|M2| + |M3|- |Mx|). (5.6)

Suppose now, for example

2|M1| + |M3| < |M2|, (5.7)

so then at high enough energies, g-2/) will become negative, meaning that the size of S2t has become negative. To keep the size of all the S2's positive, at large enough t, the geometry

undergoes a flop of S12 that sends

s2 ^ ^2 = -S2,

S2 ^ S2 = S2 + S2, (5.8) and correspondingly,

N1 = N2 - N1, N2 = N2, (5.9)

M = -M1, M2 = M3, M3 = M2. (5.10)

Recall the supersymmetric case first. The supersymmetric case with M1 = 0 was studied in detail in [25]. It corresponds to a vacuum of a low energy U(Ni) x U(N2), N = 2 theory where the superpotential breaks the gauge group to U(M2) x U(M3). The formulas (5.6) are in fact the same as in the supersymmetric case, when all the Mi are positive—the beta functions simply depend on the absolute values of the charges. If (5.7) is satisfied, the U(Ni) factor is not asymptotically free, and the coupling grows strong at high energies. There, the theory is better described in terms of its Seiberg dual, the asymptotically free U(N 1) x U(N2) theory, broken to U(M2) x U(M3) by the superpotential.17 The vacua at hand, which are visible semi-classically in the U(N1) x U(N2) theory, are harder to observe in the U(N 1) x U(N2) theory, which is strongly coupled at the scale of the superpotential. But, the duality predicts that they are there. In particular, we can smoothly vary the strong coupling scale AN1 of the original theory from (i) AN1 < A < i, where the description at scale l is better in terms of the original U(Ni) x U(N2) theory, to (ii) A < i < AN1, where the description is better in terms of the dual U(N 1) x U(N2) theory.

For the dual description of a theory to exist, it is necessary, but not sufficient (as emphasized in [18]), that the brane charges at infinity of the Calabi-Yau be the same in both descriptions. In addition, the gauge couplings must run in a consistent way. In this supersymmetric A2 quiver, this is essentially true automatically, but let us review it anyway with the non-supersymmetric case in mind. On the one hand, (5.5) implies that the under the flop, the couplings transform as

g1 (l) ^ g1 (l) = -<?1 (l),

_2 _2 _2 _2

g2 (l) ^ g2 (l) = g1 (l) + g2 (l). (5.11)

On the other hand, from (3.5) we know how the couplings g; corresponding to charges Mi run with scale l. The nontrivial fact is that the these two are consistent—the flop simply exchanges M2 = M3 and M3 = M2, and this is consistent with (5.11).

Now consider the non-supersymmetric case. Let us still take M1 = 0, but now with M2 > 0 > M3, such that (5.7) is satisfied. It is still the case that if we go to high enough energies, i.e. large enough l, the gauge coupling g1-2 will become negative, and the corresponding S12 will undergo a flop. We can change the basis of 2-cycles as in (5.5) and (3.8) that the couplings are all positive, and then the charges transform according to (5.10). Moreover, just as in the supersymmetric theory, after the flop the gauge couplings run exactly as they should given the new charges Mi,

17 The superpotential of the dual theory is not the same as in the original. As explained in [25], we can think of the flop as permuting the Z; (t), in this case exchanging z'j (t) with z^t), which affects the superpotential as ^ (#1) ^ -^1(^1), and W2(#2) ^'w1(#2) + W2(#2).

which are again obtained by exchanging node two and three. Moreover, by varying the scale AN1

where g1 becomes strong, we can smoothly go over from one description to the other, just as in the supersymmetric case. For example, in the A2 case we have a non-supersymmetric duality relating a ^(|^1|) x U(N2) theory, where the rank N1 = M3 is negative and N2 = M2 + M3 positive, which is a better description at low energies, to a U(N 1) x U(N2) theory with positive ranks N 1 = N2 - N1 = M2 and N2 = N2 = M2 + M3, which is a better description at high energies.

More generally, one can see that this will be the case in any of the ADE examples of the previous section. This is true regardless of whether all Mjp are positive and supersymmetry is unbroken, or they have different signs and supersymmetry is broken. In the case where super-symmetry is broken, we have no gauge theory predictions to guide us, but it is still natural to conjecture the corresponding non-supersymmetric dualities based on holography. Whenever the charges are such that in going from low to high energies a root ends up being dualized

S2 . _ S 2

Si,p ^ Si,p,

there should be a non-supersymmetric duality relating a brane/antibrane system which is a better description at low energies to the one that is a better description at high energies, with charges transforming as in (5.2) and (5.3). The theories are dual in the sense that they flow to the same theory in the IR, and moreover, there is no sharp phase transition in going from one description to the other. This can be seen from the fact that by varying the strong coupling scales of the theory, one can smoothly interpolate between one description and the other being preferred at a given energy scale /. We do not expect these to correspond to gauge theory dualities (in the sense of theories with a finite number of degrees of freedom and a separation of scales), but we do expect them to be string theory dualities.

5.3. Dualizing an occupied root

When an occupied node gets dualized, negative ranks M < 0 will appear. This is true even in the supersymmetric case. It is natural to wonder whether this is related to the appearance of non-supersymmetric vacua in a supersymmetric gauge theory. Conversely, starting with a non-supersymmetric vacuum at high energies, one may find that the good description at low energies involves all the charges being positive. We propose that when an occupied node gets dualized, there is essentially only one description which is ever really weakly coupled. In particular, "negative rank" gauge groups can appear formally but never at weak coupling. Moreover, while the supersymmetric gauge theories can have non-supersymmetric vacua, the phenomenon at hand is unrelated to that. This is in tune with the interpretation given in [25].

Consider the A2 theory in the supersymmetric case, M1i2,3 > 0, with both gauge groups U(N h2) being asymptotically free. The U(NV1) x U(N2) theory gives a good description at low energies, for

aN1 « A

where A is the characteristic scale of the ALE space, and An is the strong coupling scale of the U(N 1) theory. Now consider adiabatically increasing the strong coupling scale until

ann1 > A.

Then the U(Ni) x U(N2) description appears to be better at low energies, with N's related as in (5.9). Namely, from (5.11) we can read off the that the strong coupling scales match up as

AN1 = AfJl, so at least formally this corresponds to a more weakly coupled, IR free U(N\) theory. However, after dualizing node 1, its charge becomes negative

M1 = -M1.

How is the negative rank M1 < 0 consistent with the theory having a supersymmetric vacuum?

The dual theory clearly cannot be a weakly coupled theory. A weakly coupled theory of branes and antibranes breaks supersymmetry, whereas the solution at hand is supersymmetric. Instead, as we increase AN1 and follow what happens to the supersymmetric solution, the scale AM1 associated with gaugino condensation on node 1 increases as well, A < AM1 ~ AN1, and we find that at all energy scales below AM1 we have a strongly coupled theory, without a simple gauge theoretic description. The holographic dual theory of course does have a weakly coupled vacuum with charges M1 < 0, M2,3 > 0, which breaks supersymmetry. However, the gauge couplings in this vacuum run at high energies in a different way than in the supersymmetric U(N1) x U(N2) gauge theory. As emphasized in [18], this means we cannot interpret this non-supersymmetric vacuum as a metastable state of the supersymmetric gauge theory.

We could alternatively start with a weakly coupled, non-supersymmetric A2 theory with M1 < 0, M2,3 > 0. If (5.7) is not satisfied, the theory is asymptotically free. Increasing the strong coupling scale AN1 of this theory until AN1 ~ A, the theory becomes strongly coupled, and one is tempted to dualize it to a theory with M' > 0 at lower energies. However, from the vacuum solutions in Section 3, we can read off that, just as in the supersymmetric case, this implies that the scale AM1 of the gaugino condensate of node 1 becomes larger than the scale A, and no weakly coupled description exists. What is new in the non-supersymmetric case is that, as we have seen in Section 3, increasing the strong coupling scale AM1 to near A causes the system to lose stability.

Nevertheless, we can formally extend the conjectured Seiberg dualities to all the supersym-metric and non-supersymmetric vacua even when the node that gets dualized is occupied, except that the dual description is, in one way or another, always strongly coupled.

6. A very simple case

Let us now go back to the A2 case studied in Section 3 and suppose that two of the masses are equal and opposite m1 = -m2 = -m, so18

Z1(t) = 0, Z2(t) = -mt, Z3(t) = -m(t - a). (6.1)

It is easy to see from (2.4) that there are now only two critical points at t = 0 and t = a, which get replaced by S2 and S|. The third intersection point, which corresponds to the simple root S|, is absent here, and so is the minimal area 2-cycle corresponding to it. We study this as a special case since now the prepotential F0 can be given in closed form, so the theory can be solved exactly. This follows easily either by direct computation from the geometry, or from the corresponding matrix model (see Appendix A). The large N dual geometry corresponds to the two S2's being replaced by two S3 's:

X2 + y2 + z(z + mt)(z + m(t - a) = 51 (z + ma) + 53(z + m(t - a).

18 More precisely, relative to the notation of that section, we have performed a flop here that exchanges z1 and z2.

Fig. 4. There are only two minimal S s in the A2 geometry with m1 = -m2. The figure on the left corresponds to the first blowup discussed in the text, with two minimal S2's of intersection number +1 in the ALE space wrapped by M1 anti-D5 branes and M3 D5 branes. The figure on the right is the flop of this.

The exact prepotential is given by

2niFo(S) = 1 S2( log

2~1V"°\mA2

2} ■ 2~3V"°VmA2

+ 1 S32 (log

+ Si S3 log

We can now consider wrapping, say, M1 antibranes on S2 and M3 branes on S|. We get an exact vacuum solution at

^1M1I = (A2m) |Mi| exp(-2ni«i),

... . -MH

S3|M3| = Am) |M3| exp(-2^3),

where the potential between the branes is given by

|M1| |M3| 1 /' '

= M + M + — |M1||M31 log

Using an analysis identical to that in [1], it follows that the solution is always stable, at least in perturbation theory. Borrowing results from [1], the masses of the four bosons corresponding to fluctuations of S1j3 are given by

, >2 (a2 + b2 + 2abcv) ± ^(a2 + b2 + 2abcv)2 - 4a2b2(1 - v)2

{m±(c) =-2(1—v)2-

and the masses of the corresponding fermions are

|m^11 =

1 - v'

|m^2 1 =

where c takes values c = ±1, and

2nA3 Im r11

2nA3 Im t33

The parameter controlling the strength of supersymmetry breaking v is defined by

(Im T13)2

Im TnImt33

That v controls the supersymmetry breaking can be seen here from the fact that at v = 0, the masses of the four real bosons become degenerate in pairs, and match up with the fermion masses [1]. The masses of bosons are strictly positive since the metric on the moduli space Im t is positive definite, which implies

1 > v > 0, An1,2 ^ a

where AN12 is the scale at which the gauge coupling g-2 becomes strong.19

The fact that the system is stable perturbatively is at first sight surprising, since from the open string description one would expect that for sufficiently small a an instability develops, ultimately related to the tachyon that appears when the brane separation is below the string scale. In particular, we expect the instability to occur when the coupling on the branes becomes strong enough that the Coulomb attraction overcomes the tension effects from the branes. However, it is easy to see that there is no stable solution for small a. As we decrease a, the solution reaches the boundary of the moduli space,

A0exp( --

V g2,3lM3,1i,

where Im t is positive definite, before the instability can develop.20 Namely, if we view A0 as a cutoff on how much energy one has available, then for a stable solution to exist at fixed coupling, the branes have to be separated by more than ~ A0, and said minimum separation increases as one moves towards stronger coupling. The couplings, however, do run with energy, becoming weaker at higher A0, and because of that the lower bound on a actually decreases with energy. Alternatively, as we will discuss in the next subsection, there is a lower bound on how small lal can get, set by the strong coupling scales AN12 of the brane theory. When this bound is violated, the dual gravity solution disappears.

The fact that the system is perturbatively stable should be related to the fact that in this case there is no compact B cycle. Namely, in Section 3 we have seen that when perturbative stability is lost, the system rolls down to a new minimum corresponding to shrinking a compact B-cycle without flux through it. In this case, such a compact B -cycle is absent, so the system has no vacuum it can roll away to, and correspondingly it remains perturbatively stable.

The theory has another vacuum with the same charges, which can have lower energy. This vacuum is not a purely closed string vacuum, but it involves branes. Consider, for example, the case with M1 = -M3 = -M. In this case, the brane/antibrane system should be exactly stable for large enough separation a. However, when a becomes small enough, it should be energetically favorable to decay to a system with simply M branes on S|, which is allowed by charge conservation. This should be the case whenever

A(S22) < A(S2) + a(S|),

19 —2 Ani

19 From the solution, one can read off, e.g., g1 = —(2|M11 + IM31) log( aq1 ).

20 Since Im(T) is a symmetric real matrix of rank two, a necessary condition for the eigenvalues to be positive is that the diagonal entries are positive. The equation we are writing corresponds to the positivity of the diagonal entries of Im(T) evaluated at the critical point. For weak gauge coupling, this is also the sufficient condition.

where the areas on the right-hand side refer to those of the minimal S2's at the critical points of W[(t) = Z1(t) - Z2(t) and W3(t) = Z1(t) - Z3(t),

A(S2) = |n|, A(S|) = |n| + |r2|.

In the class of S|, there is no holomorphic 2-cycle, as W2(t) = z2(t) - z3(t) = -ma never vanishes, so

A(S2) =J\r22+m\2. (6.6)

Clearly, when a is sufficiently small, the configuration with M branes on S22 should correspond to the ground state of the system. If instead M1,3 are generic, we end up with a vacuum with intersecting branes, studied recently in [21]. Here one has additional massless matter coming from open strings at intersection of the branes, and correspondingly there is no gaugino condensation and no closed string dual. As a result, the methods based on holography we use here have nothing to say about this vacuum.

6.1. A stable non-supersymmetric vacuum

Consider now the flop of the simple A2 singularity of the previous subsection, where Z1 and Z2 get exchanged,

Z1(t) = -mt, Z2(t) = 0, Z3(t) = -m(t - a), and where

S2 ^ S2 = -S2.

We now wrap M1 < 0 antibranes on S»'2 and M2 > 0 branes on S|. In this case, one would expect the system to have a stable, non-supersymmetric vacuum for any separation between the branes. This is the case because the system has nowhere to which it can decay. Suppose we wrap one antibrane on S2 and one brane on S|. If a cycle C exists such that

c = -Sf+S| (6.7)

then the brane/antibrane system can decay to a brane on C. In the present case, such a C does not exist. The reason for that is the following. On the one hand, all the curves in this geometry come from the ALE space fibration, and moreover all the S2's in the ALE space have self intersection number -2. On the other hand, because the intersection number of S2 and S| is +1, Eq. (6.7) would imply that the self intersection of C is -6. So, the requisite C cannot exist. The vacuum is, in fact, both perturbatively and non-perturbatively stable; we will see that the holographic dual theory has no perturbative instabilities for any separation between the branes. Because the z's have been exchanged and the geometry is now different; we get a new prepotential FF0 and effective superpotential

Weff = J2 aiSi + Mid~s.F0(S), (6.8) i=1,2 i

tonus)=1 S2(>og( (-m^)-1)+1 Si(i»g( -!)- log( A0>

Alternatively, we should be able to work with the old geometry and prepotential (6.2), but adjust the charges and the couplings consistently with the flop. The charges and the couplings of the two configurations are related by

M1 = -M1, M2 = M3, (6.9)

where M1,3 are now both positive, and

g-2 = -gl-2, ^2-2 = g--2, (6.10) 2

where g- is now negative. The effective superpotential is

Weff =J2 °iSt + MidsiF0(S), (6.11)

i = 1,3

in terms of the old prepotential (6.2). Indeed, the two are related by F0(S1,S3) = FF0(S1, S2) and a simple change of variables

S1 = -S1, SS2 = S3,

leaves the superpotential invariant. The critical points of the potential associated to (6.11) with these charges are

= |M1l ) mM,\( a\^

S 1 = — j exp(-2^i'«1),

SiM11 =( — )M^ ^yM2'

- - /-\|M1|

SM21 = (A0^|M27 a J exp(-2^i«2) with effective potential at the critical point

|M11 |M2| 1 ~ ~ (

V, = M + ^ - — |M1||M2| log

The masses of the bosons in this vacuum are again given by (6.3), (6.5) with the obvious substitution of variables. Just as in the previous subsection, the masses are positive in any of these vacua. Moreover, because there are no two-loop corrections to the prepotential, as we have seen in Section 3, the vacuum is stable as long as the metric remains positive definite. In the previous section, we expected an instability for small enough a, and found that the perturbatively stable non-supersymmetric solution escapes to the boundary of the moduli space (defined as the region where Im t is positive definite) when this becomes the case. In this case, we do not expect any instability for any a, as there is nothing for the vacuum to decay to. Indeed, we find that Im t is now positive definite for any a = 0.

The vacuum is stable perturbatively and non-perturbatively—there simply are no lower energy states with the same charges available to which this can decay. So, this gives an example of an exactly stable, non-supersymmetric vacuum in string theory, albeit without four-dimensional gravity.21 Moreover, since in this case there are no tachyons in the brane/antibrane system, this should have a consistent limit where we decouple gravity and stringy modes, and are left with a pure, non-supersymmetric, confining gauge theory, with a large N dual description. This is currently under investigation [46].

21 This fact has been noted in [28].

Acknowledgements

We thank J. Heckman, K. Intriligator, S. Kachru, M. Mulligan, Y. Nomura, J. Seo, and especially C. Vafa for useful discussions. We also thank the Stony Brook physics department and the fifth Simons Workshop in Mathematics and Physics for their hospitality during the late stages of this work. The research of M.A., C.B., and B.F. is supported in part by the UC Berkeley Center for Theoretical Physics. The research of M.A. is also supported by a DOIOJI Award, the Alfred P. Sloan Fellowship, and the NSF grant PHY-0457317.

Appendix A. Matrix model computation

Using large N duality in the B model topological string [33], the prepotential F0 of the Calabi-Yau manifolds studied in this paper can be computed using a matrix model describing branes on the geometry before the transition. The same matrix model [31] captures the dynamics of the glueball fields S in the N = 1 supersymmetric gauge theory in space-time, dual to the Calabi-Yau with fluxes in the physical superstring theory. In this appendix, we use these matrix model/gauge theory techniques to compute the prepotential for Calabi-Yau manifolds which are A2 fibrations with quadratic superpotentials, as studied in Sections 3 and 6. To our knowledge, this computation has not previously been carried out.

The matrix model is a U(N1) x U(N2) quiver with Hermitian matrices and 02 which transform in the adjoint of the respective gauge groups, and bifundamentals Q and Q which correspond to the bifundamental hypermultiplets coming from 12 and 21 strings. The relevant matrix integral is then given by

Z = UTT,M 1 no* v. f d01 d02dQdQexp(-Tr W(<Pu<P2,Q,Q) vol(U(N1) x U(N2)) J \gs

where W is the superpotential of the corresponding N = 1 quiver gauge theory, given by

W = Tr W1($1) + Tr W2($2) + Tr(Q01Q) - Tr(Q@2Q) (A.1)

Tr = -^Tr(^1 - ai id^i)2,

Tr W2(02) = Tr(02 - a2 idW2)2.

The saddle points of the integral correspond to breaking the gauge group as

U(N1) x U(N2) ^ U(M1) x U(M2) x U(M3) (A.2)

N1 = M1 + M3, N2 = M2 + M3, by taking as expectation values of the adjoints and bifundamentals to be 'a!idM1 0 \ /a2idM2 0

* = , ®i* =

0 a3 idM3 0 a3 idM3

where a3 = (m1a1 + m2a2)/(m1 + m2), and

( 0 0 \ ~ / 0 0

(QQh = L ... , (QQ)* = [ ...

V0 -W1 (a3) idM^ \0 W2 (a3) idM3

where - W1 (a3) = m1(a1 - a3) = W2(a3).

Now let us consider the Feynman graph expansion about this vacuum. The end result is a very simple path integral. However, to get there, we need to properly implement the gauge fixing (A.2), and this is somewhat laborious. It is best done in two steps. First, consider fixing the gauge that simply reduces U(Nh2) to U(M12) x U(M3). This follows [35] directly. Let

*1 = (

To set the M1 x M3 block in to zero

F1 = <P1U = 0 we insert the identity into the path integral in the form

id = j dAS(F1)Det(jA^,

where the integral is over those gauge transformations not in U(M1) x U(M3). The determinant can be expressed in terms of two pairs of ghosts, B13, B31 and C31, C13, which are anticommuting bosons, as

= j dB13 dC31 dB31 dC 13 exp^g-Tr(B13^^ - €31^13)^)

x exp(^^Tr(B31^11C13 - C13^313B31)). By an identical argument, we can gauge fix the second gauge group factor U(N2) ^ U(M1) x U(M3)

to set the M2 x M3 block of $2 to zero. We do this by again inserting the identity into the path integral, but now with the determinant replaced by

°et( A =/ dB23 dC32 dB32 dC23exp^ g-Tr(B23 $¡3^2 - C32^^2B23)^j

x ex^ 1 Tr(B32$22C23 - C23$323 B32)).

Finally, since the vacuum will break the two copies of U(M3) to a single copy, we need to gauge fix that as well. To do this, will fix a gauge

F3 = Q33 - q id = 0

where Q33 refers to the 33 block of Q, and integrate over q. This is invariant under the diagonal U(M3) only. To implement this, insert the identity in the path integral, written as

id = J dA33^ dqqS(Q33 - q id)qM.

The above is the identity since

Det( \ = qM32, VSA33/ q

and we have taken the q-integral to be around q = 0. Inserting this, we can integrate out Q33, and Q33. The Q33 integral sets it to equal q. The Q33 integral is a delta function setting

$33 = $323, (A.3)

but there is a left over factor of q-M3 from the Jacobian of S(q($^3 - $233)). Integrating over q gives simply 1.

The remaining fields include a pair of regular bosons Q13, Q31 in the bifundamental representation of U(M1) x U(M3) and a pair of ghosts C13, B31, with exactly the same interactions. Consequently, we can integrate them out exactly and their contribution is simply 1. This also happens for Q32, Q23 and B23, C32, which also cancel out. We are left with the spectrum presented in Section 3 which very naturally describes branes with open strings stretching between them.

A.1. A special case

In the special case when m2 = -m1 = m, the matrix integral is one-loop exact. To begin with, the effective superpotential is given by (A.1) with

mm Tr ^($1) = -y Tr№)2, Tr ^($2) = 2 Tr($2 - a id^xN)2-

The theory now has only one vacuum, where $1 and Q, Q vanish, and

$2 = a id^2 xN2 ■

Expanding about this vacuum, the superpotential can be re-written as mm

Weff = —y Tr $2 + Y Tr $f - a Tr QQ + Tr(Q$Q - Q$2Q)■ If we now redefine

11 $>1 = $1 + -QQ, $>2 = $2 + -QQ,

mm the superpotential becomes quadratic in all variables, and the planar free energy is given by the exact expression:

^ S2 / S1 3 \ S2 / S2 3 \ a

F=y(tog mSis- 3)+i(log (-ma- 3)- SSlog m ■

There are higher genus corrections to this result, but they all come from the volume of the U(N) gauge groups, and receive no perturbative corrections.

Appendix B. Geometrical calculation of the prepotential

One can derive the same prepotential by direct integration. We only sketch the computation here. The equation for the geometry (3.2) can be rewritten

x2 + y2 + z(z - m!(i - a1))(z + m2(t - a2))

= -S1m^z + m2(t - a2^ - S2m2(z - m1(t - aO) - S3m3z. (B.1)

Here si are deformation parameters. This is a convenient rewriting of (3.2) because we will find that the periods of the compact cycles are given by Si = si + O(S2). As mentioned in the main text, the holomorphic three-form Q of the Calabi-Yau descends to a one-form defined on the Riemann surface obtained by setting x = y = 0 in (B.1). The equation for the Riemann surface is thus

1 = m\s\ + m2 S2

z(z - mi(t - ai)) z(z + m2(t - 0,2))

+-—-. (B.2)

(z - mi(t - ai))(z + m2(t - 02))

The one-form can be taken to be m = zdt - tdz. The one-form is only defined up to a total derivative; a total derivative changes only the periods of the non-compact cycles, and our choice avoids quadratic divergences in the non-compact periods. These divergences would not contribute to physical quantities in any case. The equation for the Riemann surface is a cubic equation for z(t), so the Riemann surface has three sheets, which are glued together along branch cuts. The compact periods are given by integrals around the cuts, while the non-compact periods are given by integrals from the cuts out to a cutoff, which we take to be t = A0. It is convenient to make the change of variables

-t + ai + z/mi m3

u =-, v = -z--(B.3)

a2i a2imim2

where a2i = a2 - ai . In the new variables, the equation for the Riemann surface takes the simple form

si i s2 i s3 i

i = -t---t-+ ^--(B.4)

uv A3 v(u + v + i) u(u + v + i)

with A3 = (a2 - ai)2mim2/m3 as in the main text. The change of variables is symplectic up to

an overall factor, so in the new variables the one-form becomes

m = A (udv - vdu). (B.5)

The change of variables makes it clear that we can think of the problem as having one dimen-sionful scale A, and three dimensionless quantities, si/A3, which we will take to be small. There are many other dimensionless quantities in the problem, such as mi/mj, but they do not appear in the rescaled equations so they will not appear in the periods, with one small caveat. While the equation for the Riemann surface and the one-form only depend on A and si/A3, the cutoff is defined in terms of the original variables, t = A0, so the cutoff dependent contributions to the periods can depend on the other parameters.

We sketch how to compute one compact period and one non-compact period. Though it is not manifest in our equations, the problem has a complete permutation symmetry among (si ,s2, —s3), so this is actually sufficient. One compact cycle (call it Si) is related to the region in the geometry where u and v are small, so that to a first approximation

i « S. —. (B.6)

We expand (B.4) for small u, v to get

Si S2 S3

uv = —3--3u(i - u - v) +--3v(i - u - v) +----. (B.7)

A3 A3 A3

This will be sufficient for the order to which we are working, and the equation is quadratic. We could solve for u(v) or v(u) in this regime; we would find a branch cut and integrate the one-form around it. Equivalently, we can do a two-dimensional integral

Si = A3 j du A dv (B.8)

over the region bounded by the Riemann surface (this is Stokes' Theorem). One can derive a general formula for the integral over a region bounded by a quadratic equation by changing coordinates so that it is the integral over the interior of a circle. In this case, the result is

Si = si + A (sis2 - si S3 - ^2^3) + A35) • (B.9)

The permutation symmetry of the problem then determines the other compact periods.

Now we compute the integral over the cycle dual to Si. The contour should satisfy uv « si/A3 and go to infinity. Also, the contour must intersect the compact i-cycle in a point. A contour which satisfies these criteria is to take u,v to be real and positive (this choice works as long as the si are real and positive, but the result will be general). We will need two different perturbative expansions to do this integral: one for small u and the other for small v. Since we have uv « si/A3, we will need a "small u" expansion which is valid up to u ~ si/A3, and similarly for the small v expansion.

To expand for small v, we first multiply (B.4) through by v to get

Si i S2 i S3 v

v = Tr---""T ----+ ■

A3 u A3 i + u + v A3 u(i + u + v)'

We now solve perturbatively for v(u), using the fact that throughout the regime of interest v ^ i + u. The largest that v/(i + u) gets in this regime is

i + u Va3' To zeroth order in v/(i + u),

v(Q) = A1 - A^• (B.iO)

A3 u A3 i + u

To first order,

(i) si i S2 i S3 v v( ; = -T---T-T7T7 +

A3 u A3 i + u + v(0) A3 u(i + u) which upon expanding becomes

(i)= _

(0) \ s3 v(0)

+ ^ ,, , , • (B.ii)

A3 u A^i + u (i + u)2J A3 u(i + u)

We need keep one more order in the perturbative expansion in order to get the prepotential to the desired order:

(2) si i s2 i S3 v(i)

v = —7---7-TTT + ■

A3 u A3 i + u + v(i) A3 u(i + u + v(0))

which upon expanding becomes

,(2)__Si i S2

A3 u A3

i v(i) (v(0))2

i + u (i + u)2 (i + u)3

v(i) (v(0))2

u(i + u) (i + u)2

(B.i2)

Note that using (B.i0), (B.ii), this is an explicit equation for v(u). We could similarly expand to find u(v) in the regime of small u, but actually we can save ourselves the computation by noting that the equation for the Riemann surface is invariant under u ^ v,s2 ^ -s3. We are now in a position to perform the integral of the one-form m over the contour. We use the approximation (B.i2) for the part of the integral where v is small, and the corresponding formula for u(v) for the part of the integral where u is small. We can choose to go over from one approximation to another at a point umin = vmin. Such a point will be approximately umin = ^si/A3, but we need a more precise formula. By setting u = v in the equation for the Riemann surface, and perturbing around u min = y si/A3, we find

2 _ sL _ s2 - s3 f~sj (s2 - s3)2 2si(s2 - s3) umin = A3 A3 V A3 + 2A6 + A6 +.

We spare the reader the details of the integration. The result is cutoff dependent, and we assume that the cutoff is sufficiently large so that we can drop contributions which depend inversely on the cutoff. After doing the integral, we rewrite the result in terms of the compact periods Si using (B.9). The result is:

9siF0 = (Si - S2) log umax + (Si + S3) log vmax - I Si log A3 - Si

- A3 (2 S22 + 2 S32 + Si S2 - SiS3 - 3S2S3^ + A) .

Here umax and vmax are cutoffs at large u, v. Since our cutoff is t = A0, we can solve for umax,vmax. When u is large, v is small, since uv « Si/A3. Looking back at the change of variables (B.3), we find

_ A0 _ Apm3 _ A0

umax — , vmax — — .

a21 a21m2 a31

Again, the other non-compact periods are determined by symmetry. It is now a simple matter to find F0:

1 A2 1 A2 1 A2 2niFp = -S2log-^ + -S2 log-^ + -S32 log ■ 0

2 1 a21a31 2 2 a21a23 2 3 a31a23

A0 A0 A0

- SiS2 log — + Si S3 log a! + S2S3 log a!

a2i a3i a 23

- 2A - 2) - 2A - 2) - 2A3 - 3

i / S4 \

- — (SiS| + S2S2 + SiS32 - S2S3 + S2S32 - S22S3 - 6SiS2S^ + Oi j.

This result agrees with the matrix model computation of Appendix A. Recall that we dropped terms which depend inversely on the cutoff. More precisely, we dropped contributions to the non-compact period of the form Si |ai2|/A0. This is necessary in order to match the result of the matrix model computation. In particular, in order to justify keeping the corrections we do keep, we require

Si |ai21

S3 » • (B.i3)

Appendix C. The Hessian at two loops

The equations required to analyze stability simplify if we introduce the notation u = iG a,b-

Since we are taking the ai to be pure imaginary, ua will be real and positive. Furthermore, since we are taking tab to be pure imaginary, we can replace it with the metric, tab = iGab. Then the equation of motion (3.6) takes the simple form

2iFkab{uaub - MaMb) = 0- (C.i)

At one-loop, the third derivative of the prepotential is nonzero only if all of the derivatives are with respect to the same variable, so at one-loop the solutions are ua = ±Ma. As discussed earlier, the physically relevant solutions are

ua = \Ma\. (C.2)

This is just a rewriting of the one-loop solutions (3.8) in terms of the new notation.

At two loops, we can find the solution by perturbing around the one-loop result. Let ua = |Ma | + Sa .We find that

Sk =-r-Fkab(-\Ma\\Mb\ + MaMb). (C.3)

2|Mk Fkkk kaby

Having solved the equations of motion at two loops, we proceed to the Hessian, providing less detail. Assuming the same reality conditions, the matrices of second derivatives are given by

dkdiV = d-kdfV = i(iFabki + iFcakiFdbiGcd)(uaub - MaMb), (C.4)

dkdfV = d-k diV = 2 iFcakiFdbiGcd (uaub + MaMb). (C.5)

The relations between the different mixed partial derivatives arise because we are perturbing about a real solution.

At two loops, taking four derivatives of the prepotential gives zero unless all of the derivatives are with respect to the same variable, so the first term in (C.4) can be simplified as

iFabki (uaub - MaMb) = SkiiFkkkk(ukuk - MkMk) = 2&uiFkkkk\Mk\Sk. (C.6)

Though it is not obvious at this stage, the other terms on the right-hand side can be approximated by their one-loop value in the regime of interest. This is very useful because, as mentioned previously, at one-loop the third derivatives of the prepotential vanish unless all indices are the

same. With these simplifications, the nonzero second derivatives become

(da + da )(db + db )V

= Y, 2iFaaaiFcccGaC\Ma \\Mc

(da - da)(-db + dh)V

iFbbbbS'

= J2 2iFaaaiFcccGac\Ma \\Mc

iFbbbi Fccc\Mc |

In these equations, no indices are implicitly summed over.

In order to analyze the loss of perturbative stability, we compute the determinant of the Hessian. Since the eigenvalues remain real, in order to go from a stable solution to an unstable one, an eigenvalue should pass through zero. We therefore analyze where the determinant is equal to zero. Up to possible constant factors, the determinant is given by

and so in order to vanish, one of the last two determinants must go to zero.

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