Geometric Transitions and D-Term SUSY Breaking

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


Introduction
In the interest of finding controllable, realistic string vacua, it is important to find simple and tractable mechanisms of breaking supersymmetry in string theory. A powerful method which has been put forward in [1] consists of geometrically engineering metastable vacua with D-branes wrapping cycles in a Calabi-Yau manifold, and using geometric transitions and topological string techniques to analyze them. In [1], metastable vacua were engineered by wrapping D5 branes and anti-D5 branes on rigid 2-cycles in a Calabi-Yau.
The non-supersymmetric vacuum obtained in this fashion was argued to have a simple closed-string dual description, in which branes and antibranes are replaced by fluxes. In [2], geometric transitions were used to study the physics of D-brane theories that break supersymmetry dynamically. Namely, the authors showed that the instanton generated superpotential that triggers supersymmetry breaking can be computed by classical means in a dual geometry, where some of the branes are replaced by fluxes (For an alternative approach see [3]).
In this paper, we propose another way to use geometric transitions to study supersymmetry breaking. As in [1], we consider D5 branes wrapping rigid cycles in a noncompact side has a simple interpretation in the dual low-energy effective theory as turning on a more generic set of FI parameters than hitherto considered in this context, but which are allowed by the N = 2 supersymmetry of the background. On-shell, this breaks some or all of the N = 2 supersymmetry. Geometrically, this corresponds to not only turning on H NS and H RR fluxes on the Calabi-Yau, but also allowing for dJ = 0 [12,13]. 3 Moreover, we show that the Calabi-Yau geometries with these fluxes turned on have non-supersymmetric, metastable vacua, as expected by construction in the open-string theory.
The paper is organized as follows: In section two, we review the physics of D5 branes on a single conifold and the dual geometry after the transition, paying close attention to the effect of Fayet-Iliopoulos terms. In section three, we consider the case of an A 2 geometry where misaligned central charges lead to supersymmetry breaking. We provide evidence that the dual geometry correctly captures the physics of the non-supersymmetric brane system. We show that the results are consistent with expectations from the gauge theory, to the extent that these are available. We also comment on the relation of this work to [2], and point out some possible future directions. In an appendix, we lay out the general case for larger quiver theories. We show that in the limit of large separations between nodes, the theory has metastable, non-supersymmetric vacua in all cases where they are expected.

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

The D-brane theory
To begin with, let us recall the well-known physics of N D5 branes wrapping the S 2 tip of the resolved conifold. This geometry can be represented as a hypersurface in The geometry has a singularity at the origin of C 4 which can be repaired by blowing up a rigid IP 1 . This gives the IP 1 a complexified Kähler class (2.1) The theory on the D5 branes at vanishing j reduces in the field theory limit to a d = 4, N = 1, U (N ) gauge theory with an adjoint valued chiral superfield of mass m. The bare gauge coupling is given by 4π for positive b NS . In string theory, the tension of the branes generates an energy density related to the four dimensional gauge coupling by Turning on a small, nonzero j can be viewed as a deformation of this theory by a Fayet-Iliopoulos parameter for the U (1) center of the gauge group [8,9]. This deforms the Lagrangian by where D is the auxiliary field in the N = 1 vector multiplet and where the factor of g s comes from the disk amplitude. This deformation (2.4) breaks the N = 1 supersymmetry which was linearly realized at j = 0. In particular, turning on j, increases the energy of the vacuum: integrating out D from the theory by completing the square in the auxiliary field Lagrangian, raises the vacuum energy to Supersymmetry is not broken however. At nonzero j, a different N = 1 supersymmetry is preserved 4 -one that was realized nonlinearly at vanishing j [15,16]. Which subgroup of the background N = 2 supersymmetry is preserved by the branes is determined by Z in (2.1), the BPS central charge in the extended supersymmetry algebra. 5 For any Z, the open-string theory on the branes has an alternative description which is manifestly N = 1 supersymmetric, with vanishing FI term and with a bare gauge coupling related to the magnitude of the central charge [7], Geometrically, this is just the quantum volume of the resolving IP 1 . As such, the central charge also determines the exact tension of a single D5 brane at nonzero j, so For small Kähler parameter, this agrees with the vacuum energy in the field theory limit (2.6).
For any j, the theory is massive; it is expected to exhibit confinement and gaugino condensation at low energies, leaving an effective U (1) gauge theory in terms of the center of the original U (N ) gauge group. We'll show next that the strongly coupled theory has a simple description for any value of Z in terms of a large N dual geometry with fluxes.

The geometric transition at general Z
We'll now discuss the large N dual geometry for general values of the central charge Z.
Special cases (either vanishing j or vanishing b NS ) have been considered in the literature, but the present, expanded discussion is, to our knowledge, new. 6 We'll see that the dual geometry exactly reproduces the expected D5 brane physics. From the perspective of the low-energy effective action, the consideration of general central charge corresponds to turning on a more general set of N = 2 FI terms than previously considered in this context. Geometrically, this will lead us to consider generalized Calabi-Yau manifolds, for which dJ is nonvanishing in addition to having H NS and H RR fluxes turned on. This 5 Strictly speaking, the central charge of N branes is N Z. In this paper, we will always take the number of branes N to be positive, so that we interpolate between the branes and antibranes by varying Z. 6 See related discussion in [7].
will provide a local description of the physics for each set of branes in the more general supersymmetry-breaking cases of sections three and four.
To begin, let us recall the large N dual description of the D5 brane theory at vanishing j. This is given in terms of closed-string theory on the deformed conifold geometry, uv = z(z − mt) + s.
The B-cycle is cut off at the scale, Λ 0 , at which α is measured. 7 The dependence of α on the IR cutoff in the geometry corresponds to its renormalization group running in the open-string theory.
If it were not for the fluxes, the theory would have N = 2 supersymmetry, with S being the lowest component of an N = 2 U (1) vector multiplet. That theory is completely described by specifying the prepotential, F 0 (S), which can be determined by a classical geometry computation, 7 For simplicity, the IIB axion is set to zero in this paper.
The presence of nonzero fluxes introduces electric and magnetic Fayet-Iliopoulos terms in the low-energy theory for the U (1) vector multiplet and its magnetic dual [17,18,19,20].
The effect of the fluxes (2.10),(2.11) can also be described in the language of N = 1 superspace as turning on a superpotential for the N = 1 chiral superfield with S as its scalar component, For the background in question, this takes the form In terms of the parameters of the D-brane theory, S is identified with the vev of the gaugino condensate. One way to see this is by comparing the superpotentials on the two sides of the duality. The αS superpotential on the closed string side corresponds to the classical superpotential term α 4 T rW α W α on the gauge theory side. What does the FI term deformation of the D-brane theory correspond to in the closedstring theory? To begin with, let us address this question from the perspective of the lowenergy effective action. We know that the U (1) gauge field after the transition coincides [6,21] with the U (1) gauge field that is left over after the SU (N ) factor of the gauge group confines. This suggests that we should simply identify Fayet-Iliopoulos D-terms on the two sides. More precisely, the Lagrangian of the theory after the transition can be written in terms of N = 1 superfields, (2.14) as an N = 2 action deformed to N = 1 by the superpotential (2.12), The Fayet-Iliopoulos deformation (2.4) should produce an additional term in this Lagrangian, corresponding to an FI term as in (2.5). Note that on the D-brane side, the center of mass U (1) corresponds to 1/N times the identity matrix in U (N ), so that the normalization of (2.16) precisely matches (2.4).
We now show that the deformation (2.16) leads to precisely the physics that we expect on the basis of large N duality. After turning on the FI term, the effective potential of the theory becomes We have also shifted the potential by an (arbitrary) constant, which we choose to be the tension of the branes at vanishing j, for convenience. We can then rewrite (2.17) as and the constant has shifted. As expected from the D-brane picture, the effective potential of the theory with the FI term turned on and with gauge coupling (2.2) is the same as that of the theory without the FI term and with gauge coupling (2.7).
In this simple example, the prepotential is known to be given exactly by The vacuum of the theory is determined by the minimum of (2.18), which occurs at 19) or, in terms of the expectation value of the gaugino bilinear, at Finally, we note that the energy in the vacuum (2.20) is larger than that in the j = 0 vacuum by the constant that enters (2.18), so which is precisely the tension of the brane after turning on j. This is a strong indication that we have identified parameters correctly on the two sides of the duality.
It is easy to see that in the vacuum, neither the F-term where X is a matrix of F-and D-terms, shifted by an imaginary part due to the presence of a "magnetic" FI term (see for example [1] and references therein) where the N = 2 auxiliary fields are identified with the auxiliary F-term of S in (2.13) and the D-term of the gauge field in (2.14) according to The supersymmetry manifest in (2.15) corresponds to ǫ 1 , and it is clearly broken in the vacuum for nonvanishing j, since neither the F-nor the D-term vanish. However, the determinant of X vanishes, and so there is a zero eigenvector corresponding to a preserved supersymmetry.
So far, we have identified turning on j with turning on an FI term in the low-energy effective action. It is natural to ask what this corresponds to geometrically in the Calabi-Yau manifold? In [17,18] (following [22,23]) it was shown that turning on a subset of the FI terms of the low-energy N = 2 theory arising from IIB compactified on a Calabi-Yau manifold corresponds to turning on H NS and H RR fluxes in the geometry. This is what we used in (2.12). The question of what corresponds to introducing the full set of FI terms allowed by N = 2 supersymmetry was studied, for example, in [12,13]. To make the SU (2) R symmetry of the theory manifest, we can write the triplet of the N = 2 FI terms These are given in terms of ten dimensional quantities by 8 Note that this agrees precisely with what we have just derived using large N duality.
Just as the bare gauge coupling S 2 B NS /g s = b NS /g s gets mapped to B H NS /g s after the transition due to running of the coupling, SU (2) R covariance of the theory demands that turning on S 2 J/g s = j/g s before the transition get mapped to turning on B dJ/g s after the transition. Moreover, we saw in this section that the latter coupling gets identified as a Fayet-Iliopoulos D-term for the U (1) gauge field on the gravity side. This exactly matches the result of [12,13], since E 3 is the Fayet-Iliopoulos D-term parameter. It is encouraging to note that [12,13] reach this conclusion via arguments completely orthogonal to ours. Finally, we observe that an SO(2) ⊂ SU (2) R rotation can be used to set the Fayet-Iliopoulos D-term E 3 = j/g s to zero, at the expense of replacing E 1 = b NS /g s by E 1 = b 2 NS + j 2 /g s , and this directly reproduces (2.18).

An A 2 Fibration and the Geometric Engineering of a Metastable Vacuum
By wrapping D5 branes on rigid IP 1 's in more general geometries with b 2 (X) > 1, we can engineer vacua which are guaranteed to be massive and break supersymmetry by choosing the central charges of the branes to be misaligned. Since the D-brane theories experience confinement and gaugino condensation at low energies, we expect to be able to study the dynamics of these vacua in the dual geometries where the branes are replaced by fluxes.
In this section, we'll consider the simple example of an A 2 singularity fibered over the complex plane C[t]. This is described as a hypersurface in C 4 , This geometry has two singular points at u, v, z = 0 and t = 0, a. The singularities are isolated, and blowing them up replaces each with a rigid IP 1 . The two IP 1 's are independent in homology, and the local geometry near each of them is the same as that studied in the previous section.
Consider now wrapping N 1 D5 branes on the IP 1 at t = 0 and N 2 branes on the IP 1 at t = a. If the central charges of the branes, are aligned (e.g., if the Kähler parameters j i both vanish), the theory on the branes has N = 1 supersymmetry. At sufficiently low energies, it reduces to a U (N 1 ) × U (N 2 ) gauge theory with a bifundamental hypermultiplet Q,Q, a pair of adjoint-valued chiral fields Φ 1,2 and a superpotential given by For a small relative phase of the central charges, e.g., by deforming the theory at vanishing Kähler parameters j i by we expect this to have a pure gauge theory description at low energies in terms of the supersymmetric theory with Fayet-Iliopoulos terms for the two U (1)'s.
Misaligning the central charges such that for any positive, real constant c 12 , should break all the supersymmetries of the background.
Nevertheless, for large enough m and a, the vacuum should be stable. Since the theory is massive, we expect it to exhibit confinement at very low energies, with broken supersymmetry. Nevertheless, as we'll now argue, the dynamics of the theory can be studied effectively for any j i in the dual geometry, where the branes have been replaced by fluxes.

Large N dual geometry
The Calabi-Yau (3.1) has a geometric transition which replaces the two IP 1 's by two The complex structure of the geometry after the transition is encoded in its description as a hypersurface, where c, d are related to the periods, S 1,2 , of the 3-cycles, A 1,2 , corresponding to the two S 3 's, As before, B i are the noncompact 3-cycles dual to A i , and F 0 is the prepotential of the N = 2 theory. The prepotential in this geometry is again given by an exact formula, The theory with N i D5 branes on the IP 1 i before the transition is dual to a theory with N i units of RR flux through S 3 i after the transition: There are additional fluxes turned on through the noncompact, dual B-cycles, corresponding to running gauge couplings, and B i dJ/g s = j i /g s corresponding to Fayet-Iliopoulos terms. The fluxes generate a superpotential, and Fayet-Iliopoulos D-terms, where D i are auxiliary fields in the two N = 1 vector multiplets.
Large N duality predicts that for misaligned central charges (3.5), the fluxes should break all supersymmetries, and moreover, that the non-supersymmetric vacuum should be metastable. We'll now show that this indeed the case. The tree-level effective potential of the theory is and the Kähler metric is determined by the off-shell N = 2 supersymmetry of the background. We have shifted the zero of the potential energy by the tension of the branes at vanishing j i , In the case at hand, whereas τ 12 is a constant 9 independent of the S i , It is straightforward to see that the critical points of the potential correspond to solutions of The first equation fixes the phase of S i 's, and the second their magnitude. Consider the case where the two nodes are widely separated, namely, where the sizes S i of the two S 3 's are much smaller than the separation a between them. In this limit, the equations of motion can be easily solved to obtain where θ ij is the relative phase between the central charges Z i and Z j . We can see that in the limit where the Z i are aligned, this reduces to the simple case without FI terms where the effective gauge coupling has been replaced with the parameterα i . The case of anti-aligned central charges was studied in [1,5]. The weak coupling limit of two widely separated nodes, in which our approximation is justified, corresponds to S i, * ≪ a < Λ 0 . In [1] it was conjectured that this also holds in the brane/antibrane case, where the central charges are anti-aligned and supersymmetry is maximally broken. It is natural, then, that the above limit should correspond to the the theory being weakly coupled at the scale of the superpotential (3.3).
In the same limit, the vacuum energy is given by Note that in the limit of aligned central charges, the potential energy is simply the brane tension. This is in fact true exactly, and is related to the fact (which will demonstrate later on) that in this case supersymmetry is preserved. For any other value of the angle, there is an additional attraction. In the extreme case, when we increase θ 12 from zero to π, we end up with a brane/antibrane system on the flopped geometry. We can view this as varying one of the Z i 's until the B NS field through that cycle goes to minus itself. At θ 12 = 0 the system is supersymmetric. For any other value of θ 12 , supersymmetry is broken.
Varying θ 12 continuously from zero to π produces a geometry which is related to the original A 2 geometry by a flop.
This is a flop, and by comparing to [5], it follows that the solution we found above for θ 12 = π precisely corresponds to a brane/antibrane system in the flopped geometry.
To see that supersymmetry is broken by the vacuum at nonvanishing θ 12 , we write the action (2.15) in an N = 2 invariant way in terms of N = 2 chiral multiplets A i consisting of N = 1 chiral multiplets S i , and W α i , The appropriate N = 2 Lagrangian is given by where X ab i is defined as in (2.22). Then, the relevant supersymmetry variations of the fermions are given by and at the extrema of the effective potential, The equations of motion imply that the determinants of both X 1,2 vanish. For each node, then, X i has one zero eigenvalue. It can be shown (see the general discussion of appendix A), that (3.10) and (3.11) imply that a global supersymmetry is preserved if and only if the central charges (3.2) are aligned, i.e., if a positive real constant c 12 exists such that This is exactly as expected from the open-string picture, and provides a nice test of the large N duality conjecture for general central charges.

Now we'll show that the vacuum is indeed metastable. Consider the masses of bosonic
fluctuations about the non-supersymmetric vacuum. As was found to be the case in [5], the Hessian of the scalar potential can be block diagonalized. After changing variables to bring the kinetic terms into their canonical form, the eigenvalues become where we've adopted the notation of [1] in defining with all quantities evaluated in the vacuum. In addition, we've introduced a new angle, which is defined by the equation where Y i are related to X i as in (2.22) for each i. In the limit (3.8), so we can treat this as being the same as the phase which appears in (3.7),(3.9). Note that in the limit cos θ → 1, (3.13) we recover the results for a supersymmetric system, with the masses of bosons becoming pairwise degenerate. The other extreme of anti-aligned charges can be shown to correspond to cos θ → −1, where the results of [5] should be recovered. Indeed by plugging in and rearranging terms we recover the mass formulas from page 25 of [5]. This, in turn, is ensured as long as the metric on moduli space is positive definite in the vacuum. So indeed, the system is metastable, as expected.
To get a measure of supersymmetry breaking, let's now compare the masses of the bosons and the fermions in this vacuum. The fermion masses arise from the superspace interaction which appears as For the geometry in question, the prepotential is exact at oneloop order, and the third derivatives vanish except when all derivatives are with respect to the same field. We can then write the fermion mass matrices for a given node (and non-canonical kinetic terms) as Performing a change of basis to give the fermion kinetic terms a canonical form, we can diagonalize the resulting mass matrix and obtain the mass eigenvalues. There are two zero modes, M λ 1,2 = 0, corresponding to two broken supersymmetries. In addition there are two massive fermions, which we label by ψ i , (3.14) Note that in the supersymmetric limit (3.13), the masses of ψ 1 and ψ 2 match those of φ 1,3 and φ 2,4 , which have become pairwise degenerate. For small misalignment, and large separation of the two nodes, the mass splittings of bosons and fermions are easily seen to where v goes to zero in the limit of large separation, and θ 12 measures the misalignment of the central charges.

Gauge theory limit
In the gauge theory limit (3.4), the vacuum energy (3.9) reduces to The first terms are classical contributions, as we saw in section two. The last term comes from a one-loop diagram in string theory, with strings stretched between the two stacks of branes running around the loop.
To begin with, consider the Abelian case, 10 when the gauge theory has no strong dynamics at low energies. We should be able to reproduce (3.15) directly in the field theory by computing the one-loop vacuum amplitude in a theory with FI terms turned on. We can write the classical F-and D-term potential of the gauge theory as 10 Although the rank of the gauge group is not large in this case, the geometric transition is still expected to provide a smooth interpolation between the open-and closed-string geometries. For a recent review, see [2], and references therein. It is natural to expect that for small deformations by FI terms that break supersymmetry, the two sides still provide dual descriptions of the same physics.
and φ i , q,q are the lowest components of the corresponding chiral superfields. The gauge theory quantities are related to those of the string theory construction by The identification between the field theory FI term and the string theory parameter is expected to hold only for small j i /b NS,i . For nonzero ξ 1,2 , supersymmetry appears to be broken since the two D-term contributions cannot be simultaneously set to zero with the F-terms. In fact, we know that if the central charges are aligned, this is just a relic of writing the theory in the wrong superspace.
For large m, a, this potential has a critical point at the origin of field space. At this point, all the F-terms vanish, and there is pure D-term supersymmetry breaking. The spectrum of scalar adjoint and gauge boson masses is still supersymmetric at tree-level, since the only contribution to the masses in the Lagrangian is the FI-dependent piece for the bifundamentals. This means that the only relevant contribution to the one-loop corrected potential is from the bifundamental fields. The scalar components develop a tree-level mass which is simply given by 11 while the fermion masses retain their supersymmetric value, We have defined the constant The one-loop correction to the vacuum energy density is given by 11 One can easily check that for small r, the masses agree with what we expect from string theory. The bifundamental matter is the same as for the 0 − 4 system, with small B-fields turned on along the D4 branes. See, e.g., [24,25].
where m b,f are the boson and the fermion masses, and Λ 0 is the UV cutoff of the theory.
The limit in which we expect a good large N dual is when the charged fields are very massive, r ≪ a 2 , and at low energies the theory is a pure gauge theory. Expanding to the leading order in r/a 2 , the one-loop potential is then given by We have omitted the Λ 0 independent terms which correspond to the finite renormalization of the couplings in the Lagrangian and are ambiguous. We see that this exactly agrees with effective potential (3.9), (3.15) as computed in the dual geometry, after the transition.
In the general, U (N 1 ) × U (N 2 ) case, we have a strongly coupled gauge theory at low energies. Nevertheless, since in the (N i ,N i ) sector supersymmetry is preserved, the oneloop contribution of that sector to the vacuum energy density should vanish beyond the classical contribution. Thus, we expect that only the bifundamental fields contribute to the vacuum energy at this level. The one-loop computation then goes through as in the Abelian case, up to the N 1 N 2 factor from multiplicity, once again reproducing the answer (3.15) from large N dual geometry.

Relation to the work of [2]
We close with a comment on the relation to the work of [2], to put the present work in context. The A 2 model at hand is the same as the geometry used to engineer the Fayet model in [2]. More precisely, the authors there engineered a "retrofitted" Fayet model.
The parameter a that sets the mass of the bifundamentals was generated by stringy or fractional gauge theory instantons, and thus was much smaller than the scale set by the FI terms, which were taken to be generic. That resulted in F-term supersymmetry breaking which was dynamical.
In the present context, we still have a Fayet-type model, but we find ourselves in a different regime of parameters of the field theory, where r/a 2 < 1, with r defined in terms of the FI parameters as in (3.17). Outside of this regime, the vacuum at the origin of field space, with Q andQ vanishing, becomes tachyonic even in the field theory, as can be seen from (3.16). Once this becomes the case, the large N dual presented here is unlikely to be a good description of the physics. For example, for N 1 = N 2 = N and r/a 2 > 1, it was found in [2] that the theory has a non-supersymmetric vacuum where all the charged bifundamental fields are massive and the gauge symmetry is broken to U (N ). This may still have a description in terms of some dual geometry with fluxes, but not the one at hand. This may be worth investigating.
Thus, unlike the models of [2], those considered here break supersymmetry spontaneously but not dynamically. It would be nice to find a way to retrofit the current models and to generate low scale supersymmetry breaking in this context. This would require finding a natural way of obtaining small FI terms. The mechanism of [2] does not apply here, since the terms in question are D-terms and not F-terms. This may be possible in the context of warped compactifications, 12 and compact Calabi-Yau manifolds, perhaps along the lines of [29].
(see [7] and references therein). In fibering this over the t plane, the z i become polynomials z i (t). At a generic point in the t plane, the ALE space is smooth, with singularities resolved by blowing up r independent 2-cycle classes where r is the rank of the corresponding Lie algebra. This corresponds to turning on Kähler moduli where the latter sum runs over nodes that are linked.
For vanishing j i , the structure of the vacua of the theory was computed in [7]. For Turning on generic Fayet-Iliopoulos terms for the U (1) centers of the gauge group factors, breaks supersymmetry while retaining (meta)stability of the vacuum as long as j i is much smaller than the mass of all the bifundamentals in the vacuum.
The ALE fibrations have geometric transitions in which each IP 1 is replaced by a minimal S 3 . The leading order prepotential F 0 for all these singularities was computed in [30], and is given by The theory on this geometry without fluxes is an N = 2, U (1) k gauge theory, where k is the number of S 3 's. The effect of the fluxes on the closed-string theory in this background was determined in [12,13]. The result is a set of electric and magnetic N = 2 Fayet-Iliopoulos terms, which enter the N = 2 superspace Lagrangian, with and where the auxiliary fields Y a are shifted by the magnetic FI term, The auxiliary field Lagrangian then has the form and integrating out the auxiliary fields sets them equal to their expectation values, or, to be more precise, We can make contact with the more familiar form of this action and its scalar potential by reducing to N = 1 superspace. There, the auxiliary fields Y a get identified with auxiliary fields of the vector and chiral multiplets corresponding to the a'th S 3 , and the fluxes give rise to the usual flux superpotential W = a α a S a − M a ∂ S a F 0 (S).
In addition, there are Fayet-Iliopoulos terms for the U (1)'s, and the total scalar potential is given by where α a = b RR,a + ib NS,a /g s .
There are vacua at the field values which satisfy ∂ a V ∼ F abc G ce G bd (α e −τ em M m )(ᾱ d −τ dn M n ) + j e j d /g 2 s = 0.
At one-loop order, the prepotential has nonvanishing third derivatives only when all derivatives are with respect to the same field. The vacuum condition can be simplified to this order, and upon considering the equation as two real equations for the real and imaginary part, the conditions become The first of these can be solved easily for the phases of the S a . Moreover, we see that it is equivalent to the condition that the real part of the auxiliary fields Y a 2 vanish for all a in the N = 2 superspace Lagrangian, In light of that result, the second condition can be written as Since the supersymmetry transformations are 3) is precisely the condition that there exists some supersymmetry transformation on each node which is locally preserved by the vacuum. Of course, for supersymmetry to be conserved globally, these supersymmetry transformations must match for all nodes. The condition for this to be the case is which, along with the requirement that the metric on moduli space be positive definite in the vacuum, requires that Z a = c ab Z b for a positive, real constant c ab . This conforms to our intuition from the open-string picture that preserving supersymmetry should require that the complex combination of the FI terms and gauge couplings should have the same phase on each node.
We can also see that the vacuum we just found is metastable, as we expect based on large N duality. Consider the Hessian of the potential, and similarly for complex conjugates. The eigenvalues of the Hessian are manifestly positive in the limit where G ab vanishes for a = b, which corresponds to widely separated nodes, and where the matrix ∂∂V is diagonal. Moreover, the determinant of the Hessian is strictly positive for any G ab , so the one-loop Hessian remains positive definite for any G ab .
Finally, we can compute the value of the vacuum energy in the limit where the branes are far separated. The relevant limit in this more general case is S a, * ≪ a bc < Λ 0 .
The vacuum energy is then given by which reduces to the one-loop value in the gauge-theory limit, as in the A 2 case.