Extended Supersymmetric Moduli Space and a SUSY/Non-SUSY Duality

We study N=1 supersymmetric U(N) gauge theories coupled to an adjoint chiral field with superpotential. We consider the full supersymmetric moduli space of these theories obtained by adding all allowed chiral operators. These include higher-dimensional operators that introduce a field-dependence for the gauge coupling. We show how Feynman diagram/matrix model/string theoretic techniques can all be used to compute the IR glueball superpotential. Moreover, in the limit of turning off the superpotential, this leads to a deformation of N=2 Seiberg-Witten theory. In the case where the superpotential drives the squared gauge coupling to a negative value, we find that supersymmetry is spontaneously broken, which can be viewed as a novel mechanism for breaking supersymmetry. We propose a new duality between a class of N=1 supersymmetric U(N) gauge theories with field-dependent gauge couplings and a class of U(N) gauge theories where supersymmetry is softly broken by nonzero expectation values for auxiliary components of spurion superfields.


Introduction
The last decade has seen great progress in our understanding of the dynamics of N = 1 supersymmetric gauge theories, with string theory playing a large role in these developments thanks to its rich web of dualities. In particular, motivated by string theoretic considerations [1], a perturbative approach was proposed for the computation of glueball superpotentials in certain N = 1 supersymmetric gauge theories using matrix models [2], which leads to non-perturbatively exact information for these theories at strong coupling.
Further evidence for this proposal was provided through direct computations [3], as well as from consideration of N = 1 chiral rings [4].
The simplest class of gauge theories considered in [1] involve an N = 1 supersymmetric U (N ) gauge theory with an adjoint superfield Φ together with a superpotential TrW (Φ) = k a k TrΦ k .
In this paper, we consider further deforming this theory by the most general set of singletrace chiral operators. This is accomplished by the introduction of superpotential terms where W α is the field strength superfield. In string theory, these theories are constructed by wrapping D5 branes on vanishing cycles in local Calabi-Yau three-folds, where the addition of a background B-field which depends holomorphically on one complex coordinate of the three-fold leads to the above deformation, with We show how the strongly coupled IR dynamics of these theories can be understood using both string theoretic techniques (large N duality via a geometric transition) and a direct field theory computation as in [3]. Moreover, following [5], we can consider the limit where W (Φ) is set to zero, in which case we recover an N = 2 supersymmetric theory with Lagrangian given by The prepotential F (Φ) is related to α(Φ) by where Φ is an adjoint-valued N = 2 chiral multiplet. In this limit, our solution reduces to that of the extended Seiberg-Witten theory with general prepotential (1.1). Our results are in complete agreement with the beautiful earlier work of [6], which uses Konishi anomaly [4] and instanton techniques [7] to study these same supersymmetric gauge theories. 1 The stringy perspective which we develop, however, sheds light on nonsupersymmetric phases of these theories, which will be our main focus. In particular, it turns out that if α(Φ) is chosen appropriately, there are vacua where supersymmetry is broken. The idea is that a suitable choice of higher-dimensional operators can lead to negative values of g 2 YM for certain factors of the gauge group. Motivated by string theory considerations, we will show that strong coupling effects can make sense of the negative value for g 2 YM , and at the same time lead to supersymmetry breaking. In the string theory construction, this arises from the presence of antibranes in a holomorphic B-field background. When g 2 Y M is negative in all the gauge group factors, we propose a complete UV field theory description of these vacua. This is another U (N ) gauge theory, already studied in [9,10,11], with an adjoint fieldΦ and superpotential where W (Φ) = k (a k + 2it k θθ)Φ k .
Note that since the spurion auxiliary fields have nonzero vevs t k , this theory breaks supersymmetry.
A duality between supersymmetric and nonsupersymmetric theories may appear contradictory. The way this arises is as follows (see figure 1). We have an IR effective N = 1 theory which is valid below a cutoff scale Λ 0 . The IR theory is formulated in terms of chiral fields which we collectively denote by χ (for us, these are glueball fields). The theory depends on some couplings t, and for each value of t we find two sets of vacua -one which is supersymmetric, and one which is not. However, for any given values of t, only one of these vacua is physical, in that the expectation value of the chiral fields is below the cutoff scale | χ | < Λ 0 . The other solution falls outside of this region of validity. In particular, in one regime of parameter space, only the supersymmetric solution is acceptable. As we change t, the supersymmetric solution leaves the allowed region of field space, and at the same time the nonsupersymmetric solution enters the allowed region. We obtain in this way a duality between a supersymmetric and a nonsupersymmetric theory. Moreover, we are able to identify two dual UV theories. However, unlike the effective IR theory, which is valid for the entire parameter space, each UV theory is valid only for part of the full parameter space. The supersymmetric IR solution matches onto a supersymmetric UV theory, and the nonsupersymmetric IR solution matches onto another UV theory where supersymmetry is broken softly by spurions. The organization of this paper is as follows: In section 2 we establish the basic field theories which will be studied. In section 3 we show how these field theories can be realized in type IIB string theory on local Calabi-Yau three-folds.
In section 4 we show how this string theory construction leads to a solution for the IR dynamics of the theory. In section 5 we derive the same result directly from field theory considerations. In section 6 we specialize to the N = 2 case.
In section 7 we consider these field theories when some of the gauge couplings g 2 YM become negative. We explain why this leads to supersymmetry-breaking and propose a dual description. Some aspects of the effective superpotential computation are presented in an appendix.

Field Theory
Consider an N = 2 supersymmetric U (N ) gauge theory with no hypermultiplets. Classically, this theory is described by a holomorphic prepotential F (Φ) which appears in the N = 2 Lagrangian, where Φ is an adjoint-valued N = 2 chiral multiplet, and Above, t 0 determines the classical gauge coupling and θ angle and the integral in (2.1) is over a chiral half of the N = 2 superspace. The low energy dynamics of this theory were studied in [12], where it was shown that the theory admits a solution in terms of an auxiliary Riemann surface and one-form.
This theory admits a natural extension via the introduction of higherdimensional single-trace chiral operators, which deform the theory in the ultraviolet. One effect of these new terms is that the effective gauge coupling at a given point in moduli space now depends explicitly on the expectation value of the scalar component φ of the superfield Φ, We therefore define In this paper, we will solve for the low energy dynamics of this extended Seiberg-Witten theory.
We will also study deformations of the theory (2.1) to an N = 1 supersymmetric theory by the addition of a superpotential, 5) for the N = 1 chiral multiplet Φ that sits inside Φ. In N = 1 language, the full superpotential of the theory then becomes where W α is the gaugino superfield.
Classically, the superpotential W (Φ) freezes the eigenvalues of φ at points in the moduli space where For generic superpotential, we can write with e i all distinct, so the critical points are isolated and the choice of a vacuum breaks the gauge symmetry as for the vacuum with N i of the eigenvalues of φ placed at each critical point As long as the effective gauge couplings of the low-energy theory are posi- the general aspects of the low energy dynamics of this theory are readily apparent. In the vacuum (2.9), at sufficiently low energies, the theory is pure N = 1 super-Yang-Mills, which is expected to exhibit confinement and gaugino condensation.
When the original N = 2 theory has canonical prepotential (2.2), the condition (2.10) is satisfied trivially, and in this case the problem of computing the vacuum expectation values of gaugino condensates in the N = 1 theory, has been studied extensively from both string theory [1] and gauge theory [3,4] perspectives. The question can be posed in terms of the computation of an effective glueball superpotential [1], whose critical points give the supersymmetric vacua of the theory. In this paper, we will show how to compute W eff for the N = 1 theory with the more general prepotential (2.4). Note that physically inequivalent choices of α(Φ) correspond to polynomials in Φ of degree at most n − 1. This is because, for the supersymmetric theory, any operator of the form is trivial in the chiral ring [4].
In section 7, we will ask what happens when (2.10) is not satisfied and it appears that some of the gauge couplings of (2.9) become negative in the vacuum.
We will show that in this case, the theory (2.6) generically breaks supersymmetry. Moreover, the supersymmetry-breaking vacua still exhibit gaugino condensation and confinement, and we will be able to compute the corresponding expectation values (2.11) as critical points of a certain effective scalar potential

The String Theory Construction
In this section we give the string theory realization of the above gauge theory. To begin with, we consider type IIB string theory compactified on an which is fibered over the complex x-plane. This has a singularity for all x at u, v, y = 0, which can be resolved by blowing up a finite P 1 . Wrapping N D5 branes on the P 1 gives a d = 4 U (N ) N = 2 gauge theory at sufficiently low energies. The adjoint scalar φ of the gauge theory corresponds to motion of the branes in the x-plane.
In the microscopic N = 2 gauge theory we also have a choice of prepotential F (Φ). What does this correspond to geometrically? To answer this, note that the microscopic prepotential determines the bare 4d gauge coupling, which arises in the geometry from the presence of nonzero B-fields, In the undeformed theory with the prepotential (2.2), the gauge coupling was a constant t 0 . This translates to the statement that, classically, as the ALE space is fibered over the x-plane, the Kähler modulus of the P 1 (in particular the Bfields in (3.2)) does not vary with x. In the extended Seiberg-Witten theory, the complexified gauge coupling becomes φ-dependent. Since the adjoint scalar φ parameterizes the positions of the D5 branes in the x plane, making the gauge coupling φ-dependent should correspond to letting the background B-fields in where the integral on the right hand side is over the S 2 at a point in the x plane. In order to reproduce the gauge theory, we require To summarize, the gauge theory in section 2 is realized as the low-energy limit of N D5 branes wrapped on an A 1 × C singularity with H-flux turned on, It may seem surprising that turning on H-flux does not break supersymmetry down to N = 1. 2 In the case at hand, the flux we are turning on is due to a B-field that varies holomorphically over the complex x-plane. It is known that if the B-field varies holomorphically, the full N = 2 supersymmetry is preserved [13].
As was explained in [1], turning on a superpotential TrW (Φ) for the adjoint chiral superfield, as in (2.5), corresponds in the geometry to fibering the ALE space over complex x-plane nontrivially, The resulting manifold is a Calabi-Yau three-fold and supersymmetry is broken to N = 1. After turning on W (x), the minimal S 2 's (the holomorphic P 1 's) are isolated at n points in the x-plane, x = e i , which are critical points of the superpotential, At each of these points, the geometry develops a conifold singularity, which is resolved by a minimal P 1 . The gauge theory vacuum where the gauge symmetry is broken as in (2.9) corresponds to choosing N i of the D5 branes to wrap the i'th P 1 . In particular, the tree-level gauge coupling for the branes wrapping the P 1 at x = e i is given by which agrees with the classical values in the gauge theory.
In summary, we can engineer the N = 1 theory obtained from the extended N = 2 theory by the addition of a superpotential W (Φ) with N D5 branes wrapping the S 2 in the Calabi-Yau (3.6), with background flux H 0 . In the next section, we will study the closed-string dual of this theory.

The Closed String Dual
The open-string theory on the D5 branes has a dual description in terms of pure geometry with fluxes. The gauge theory on the D5 branes which wrap the In flowing to the IR, the D5 branes deform the geometry around them so that the P 1 's they wrap get filled in, and the S 3 's surrounding the branes get finite sizes. This is a conifold transition for each minimal S 2 , after which the geometry is deformed from (3.6) to where f n−1 (x) is a polynomial in x of degree n − 1. This has n coefficients which govern the sizes of the n resulting S 3 's.
In addition, there is H-flux generated in the dual geometry, Before the transition, the S 3 's were contractible and had RR fluxes through them due to the enclosed brane charge. After the transition, they are no longer contractible, but the fluxes must remain. In other words we expect the disappearance of the branes to induce (log-)normalizable RR flux, localized near the branes' previous locations, which we denote by H RR . If we denote the S 3 that replaces the k'th S 2 by A k -cycles, then It is also natural to expect that there will be no H RR flux through the B k -cycles, as there were no branes to generate it. In other words In addition to the induced flux H RR , we have a background flux H 0 due to the variation of the background B 0 field, which was present even when there were no branes, and which we denote by H 0 = dB 0 . Thus we expect the total flux after the transition to be given by Note that before the transition, there are no compact 3-cycles, and so there is no compact flux associated with dB 0 . It is then natural to postulate that after the transition, dB 0 will have no net flux through any of the compact 3-cycles.
Moreover, far from the branes, we expect B 0 to be given by its value before the transition. For the noncompact 3-cycles in the dual geometry, denoted by B k , we can then explicitly evaluate the periods of H 0 , Because these cycles are noncompact, the integral is regulated by the introduction of a long distance cutoff Λ 0 in the geometry. As usual, we identify this scale with the UV cutoff in the gauge theory.
To summarize, the total flux H = H RR + dB 0 after the transition should be determined by the following facts: H RR is (log-)normalizable, with only nonzero A k periods (given by N k ), and far from the branes, B 0 is given by its background value (3.4), i.e., The fact that the deformed background flux is given by an exact form dB 0 emphasizes the fact that it is cohomologically trivial and has no nonzero periods around compact 3-cycles.
The striking aspect of the duality is that in the dual geometry, the gaugino superpotential W eff becomes purely classical. We will turn to its computation in the next subsection.

The effective superpotential
The effective superpotential is classical in the dual geometry and is generated by fluxes, where Ω is a holomorphic three-form on the Calabi-Yau, This has a simpler description as an integral over the Riemann surface Σ which is obtained from (4.1) by setting the u, v = 0: The Riemann surface Σ is a double cover of the complex x-plane, branched over n cuts. The 3-cycles A k and B k of Calabi-Yau three-fold descend to one-cycles on the Riemann surface Σ, with A k cycles running around the cuts and B k cycles running from the branch points to the cutoff (see figure 2). In addition, H RR descends to a one-form on Σ with periods (4.2), (4.3). Moreover, Ω descends to a one form on Σ, given by ydx, where y solves (4.5). The effective superpotential then reduces to an integral over the Riemann surface, The one-form H RR is defined by its periods and the asymptotic behavior of B 0 is determined by where ± correspond to the values of the one-form on the top and bottom sheets of Σ. The evaluation of the superpotential is now straightforward. Using the Riemann bilinear identities, we can evaluate the first term, and F 0 is the genus 0 prepotential of the Calabi-Yau. The background contribution to the superpotential is also straightforward to evaluate, since there are no internal periods for the flux, where the last equality follows from the fact that B 0 (x) = α(x) for large x by Cauchy's theorem (since the cycle around P is homologous to the sum of all the A k -cycles).
Thus, the full effective superpotential is This expression is in line with our intuition from the open-string description.
Namely, to the leading order we have where the omitted terms are higher order in S i . To this approximation, the superpotential is given by Note that the first term above comes from the classical superpotential of the gauge theory, since the A i -cycle periods S i in the geometry are identified with glueball superfields in the gauge theory. The coefficient of S i in the effective superpotential is the microscopic gauge coupling of the U (N i ) gauge group factor in the low energy effective field theory. This is precisely equal to the B-field on the S 2 wrapped by the branes (3.7).
However, this cannot be the whole story. After the deformation, the location of the P 1 is no longer well defined, as the P 1 at the point x = e k has disappeared and been replaced by an S 3 which is a branch cut on the x-plane.
The geometry has been deformed around the branes and the two sheets of the Riemann surface connect through a smooth throat. We need to specify where the gauge coupling is to be evaluated, and since the point in the x-plane has been replaced by a throat, the most natural guess is that we smear the B-field over the cuts. This is precisely what (4.7) does! In the appendix, we provide more details for the derivation of (4.7) based on the use of the Riemann bilinear identities.
In the next section, we will show that the same effective superpotential follows from a direct gauge theory computation. Moreover, we will relate the gauge theory computation to an effective matrix model. We will also give a more explicit expression for W eff , which arises from the following nontrivial identity that we prove in section 5 using the formulation of the topological string in terms of matrix models [14]: Equations (4.7) and (4.8) agree with the results of [6,8].
The form of the superpotential (4.8) suggests a dual role played by (a k , S k ) and (t k , N k ) -indeed it suggests a formulation in terms of fluxes [15] (see also [10]). We can think of the fluxes N k as turning on auxiliary fields for the S k superfields in the N = 2 effective theory, where S k is the lowest component of the superfield, The N = 1 superpotential arises by the integration over half of the chiral N = 2 superspace Similarly, we can view the background parameters a k as scalar components of non-normalizable superfields, and the t k as the corresponding fluxes leading to vevs for their associated auxiliary fields, Thus the full superpotential can be obtained from the N = 2 formulation simply by giving vevs (t k , N k ) to the auxiliary fields of (a k , S k ).

Extrema of the superpotential
With the closed-string dual of our gauge theory identified, we turn to the extremization of the flux superpotential. We wish to solve From (4.8) this can be written as where η is an n × n matrix, and τ ik is the usual period matrix, Note that for a fixed choice of Higgs branch, specified by N i , the number of parameters specifying the choice of B 0 (x) and the number of parameters determining the normalizable deformations of the geometry, given by f n−1 (x), are both equal to n. Therefore we would expect to generically have a one-to-one map. This allows us to invert the problem. Instead of asking how B 0 determines we can instead ask for which choice of B 0 (x) we obtain a given deformed ge- In this formulation, the extremization problem has a simple solution. We choose a set of complex structure moduli for the Riemann surface, by picking values for the S i (or equivalently for the coefficients of f n−1 ). This completely determines the matrices τ ij and η ij through (4.11) and (4.12). The equations (4.9), (4.10) can then be thought of as n linear equations for the n coupling constants {t i } n−1 i=0 , thus determining B 0 (x). The equations (4.9) determine the explicit form of the flux H RR +H 0 on the solution. Recall that, off-shell, H RR + H 0 was defined by its compact periods, (4.13) and asymptotic behavior for large x, The equations of motions (4.9) then imply that the one-form H RR + H 0 is holomorphic on the punctured Riemann surface Σ − {P, Q}, and given by (4.14) Above, P and Q correspond to points at infinity of the top and the bottom sheet of the Riemann surface, and ∂ ∂S k ydx are linear combinations of the n − 1 holomorphic differentials on Σ, together with x n−1 dx/y, which has a pole at infinity.
To derive this, we note that (4.9) implies that H RR + H 0 is orthogonal to the complete set of holomorphic differentials in the interior. This implies that H RR + H 0 is holomorphic away from the punctures. We can also show that (4.14) has the correct periods and asymptotic behavior. Consider the periods and the periods of The reason for the Λ i 0 term in (4.16) is that ∂F 0 ∂S i is the B i -period with boundary term subtracted. The A k periods also match -this is because the ∂ ∂a k derivative is taken at fixed S k , per definition. Using these periods and (4.10), we immediately see that (4.14) has the correct periods (4.13). It is also clear that the large x behavior is dominated by ρ i and this yields dα(x) for the large x behavior of H RR + H 0 as required.

Gauge Theory Derivation
In this section we will sketch the derivation of the effective glueball superpotential directly in the gauge theory language, and show that this exactly reproduces the results of the string theoretic derivation. In [3] the effective superpotential for the glueball superfields was computed by explicitly integrating out the chiral superfield Φ. This is possible as long as we are only interested in the chiral d 2 θ terms in the effective action. In the absence of the deformation (2.6), computation of the relevant gauge theory Feynman graphs with Φ running around loops directly translates into the computation of planar diagrams in a certain auxiliary matrix integral. We will see that this is the case even after the deformation, albeit with a novel deformation of the relevant matrix integral.
Let us review the results of [3]. For simplicity, consider the vacuum where the U (N ) gauge symmetry is unbroken. The propagators for Φ can be written in the Schwinger parameterization as where s i are the Schwinger times, p i are the bosonic momenta, and π α the fermionic momenta. The mass parameter m is given by m = W ′′ (φ 0 ). These propagators have the property that each Φ loop brings down two insertions of the glueball superfield W α . Using the chiral ring relation only those operators of the form are nontrivial as F-terms. In particular, there must be at most two insertions of W α per index loop. This implies that only planar Φ-diagrams contribute to the superpotential -nonplanar graphs have fewer index loops than momentum loops.
The integration over bosonic and fermionic loop momenta in a planar diagram with h holes gives a constant factor, The rest of the computation, namely combinatorial factors, contributions of vertices, and an additional factor of 1/m h−1 from the propagators, is captured by a zero-dimensional, auxiliary holomorphic matrix theory with path integral where Φ is an M × M matrix, and W (Φ) is the same superpotential as in (2.5).
The coefficient In the semiclassical approximation, the effective superpotential of the undeformed theory is simply In the full answer, F 0 contains a 1 2 S 2 log S piece which, in the matrix model, comes from the volume of the gauge group in (5.3).

The deformed matrix model
Now consider the gauge theory with the more general tree-level superpotential (2.6) (for a special form of the superpotential, this theory was studied in [8]). In this case, the propagators of the theory are unchanged, but there are now additional vertices coming from the first term in (2.6 where the matrix Λ stands for W α W α insertions that do not come from the propagators. Summing over planar graphs, the matrix integral now has the form where the omitted terms contain higher powers of traces of Λ that will not play any role. The effective superpotential, including the contribution of the new Note that it is manifest in the matrix model that the effective superpotential is invariant under the addition to α(Φ) of terms the form Φ k W ′ (Φ), as mentioned in section 2. These terms can be removed by a shift in Φ and as such they do not affect the matrix integral.
It is easy to generalize this to vacua of the gauge theory where the gauge group is broken as in (2.9). The superpotential in these vacua is computed by the same matrix model, but where one now considers the perturbative expansion about the more general vacuum, where the gauge symmetry of the matrix model is broken to n k=1 U (M k ) [16]. The contributions of insertions of are now captured by deforming the matrix model to where the omitted terms . . . are gauge fixing terms [16] corresponding to the choice for Φ to be block diagonal, and breaking the gauge symmetry to k U (M k ). Summing over the planar graphs returns where F 0 and G 0,k are functions of the matrix model 't Hooft couplings g top M k .
These are identified with the glueballs S i in the physical theory. The effective superpotential is now given by and all that remains is to compute the new terms in G 0,k .

Matrix model computation
Now let us compute the relevant correction from the matrix model. Since we are only interested in the planar graphs linear in TrΛ k , the contribution of interest can be extracted from the special case where we choose The matrix model partition function then becomes where the expectation value is evaluated in the planar limit of the k U (M k ) vacuum of the undeformed matrix model. These can be computed using well known large M matrix model saddle point techniques [2]. The answer can be formulated in terms of a Riemann surface, with a one-form ydx, where the coefficients of f n−1 are chosen so that Namely, the result is that Since the glueballs S k are identified with M k g top in the matrix model, we can write the corresponding contribution to the effective superpotential A look back at (4.7) shows that this agrees with the result of our string theoretic analysis. Moreover, this is consistent with the results of [4] for the expectation values of the corresponding chiral ring elements.
In the next subsection, we will use matrix model technology to derive the identity (4.8) for expressing δW eff , as a function of S k .

Evaluation of δW eff
To begin with, note that δW eff can be rewritten as where the trace is over the M × M matrix Φ. 3 The expectation value is now straightforward to compute. The problem amounts to the computation of TrΦ k , k = 0, . . . n − 1 in the matrix model. Recall that which implies that, for k = 0, . . . , n − 1 with Z M as defined in (5.3). In particular, since it follows that Thus we have derived (4.8), 6. The N = 2 Gauge Theory

Extended Seiberg-Witten theory
With the results of the previous section in hand, we are now in position to recover the solution to the extended N = 2 theory with classical prepotential The analysis of this section closely mirrors the approach taken in [5], and the results also follow from [6].
To begin with, consider a special case of the N = 1 theories studied in the previous section. We deform the extended U (N ) N = 2 theory (6.1) to N = 1 by the addition of a degree N + 1 superpotential, In particular, we now study a generic vacuum on the Coulomb branch of the theory, where the gauge symmetry is broken as This is important, because if we now take the limit of vanishing superpotential (6.2) while keeping the expectation value of the adjoint fixed, we expect to recover the N = 2 vacuum at the same point in moduli space. As discussed in section 3, this corresponds in string theory language to reverting to studying N D5 branes on the P 1 in the A 1 ALE space, but with a holomorphically varying B-field turned on. The nontrivial B-field background corresponds in the low energy theory on the branes to turning on the higher dimensional terms in the classical prepotential (6.1).
We found in section 4 that the critical point of this theory corresponds to a Riemann surface where the N parameters t k in (6.1) are determined in terms of the complex structure moduli S i of (6.4) by extremizing the superpotential (4.9). Moreover, at the critical point, the net flux H RR + H 0 is given by a holomorphic one-form on the Riemann surface (6.4), with periods It turns out that all of the holomorphic information about the N = 2 theory in the infrared can be recovered from calculations in the N = 1 theory, just as in [5]. To observe this, we note that if we extract an overall factor of g from y in (6.4) and use new g-independent functions W ≡ 1 g W and f N−1 ≡ 1 g 2 f N−1 , then and the periods of y have a trivial g-dependence. In particular, are independent of g. Consequently, the period matrix This fact can be made more manifest by considering the geometry in question, It is clear that the variation of g can just be absorbed into a rescaling of the coordinate y.
It is also crucial that in the process of sending g → 0, the values of t k for which the Riemann surface in question satisfies the equations of motion remain fixed. The superpotential is simply proportional to g, and hence its critical points are g-independent.
Lastly, we note that the Seiberg-Witten one-form on the Riemann surface can be recovered from the N = 1 analysis as well. First note that the H-flux H RR + H 0 at the critical point of the superpotential is given by a g-independent holomorphic one-form (6.5). Just as in [5], it follows that the Seiberg-Witten one-form on the Riemann surface is given by which we can read off from the N = 1 theory. This can be seen as follows.
Periods of λ SW compute the masses of dyons in the N = 2 theory. However, these dyons can be identified with D3 branes wrapping Lagrangian 3-cycles in the Calabi-Yau, or one-cycles on the Riemann surface, and their mass can be derived from string theory to be given by periods of the one-form (6.6).
In summary, we can obtain the full N = 2 curve and the Seiberg-Witten one-form λ SW that capture the low energy dynamics of the extended N = 2 theory (6.1). These results are consistent with those obtained recently in [17] using very different techniques. There, the authors formulate the solution of the N = 2 theory in terms of a hyperelliptic curve of genus N − 1 and a holomorphic one-form dΦ with the properties that but are instead equal to α(Λ 0 ). It is possible that in the definition of the B k integrals (6.1) of [17], there is a hidden subtraction of the value of the integral at infinity, which would account for the vanishing B k periods and resolve this discrepancy.

Duality and Supersymmetry Breaking
In this section we study the phase structure of the N = 1 models under consideration. We find that there is a region in the parameter space where supersymmetry is broken. This leads to a novel and calculable mechanism for breaking supersymmetry. Even though this method for supersymmetry breaking is motivated by string theoretic considerations, we will see that it can also be phrased entirely in terms of the underlying N = 1 supersymmetric gauge theory.
The organization of this section is as follows. We first discuss some general in the UV is given by As long as the gauge coupling for each factor of the gauge group is positive, i.e., for all i with N i = 0, we expect a supersymmetric theory in the IR to which the analysis of the previous section applies. This suggests the question: What is the meaning of the phase where (7.1) is not satisfied for some i? It is to this question which we now turn our attention.
One may be inclined to consider such cases as pathological, as one is not able to give a meaning to such a theory in the UV. However, we also know from various examples that the appearance of a negative g 2 YM is often the smoking gun for the existence of a dual description. Thus all we can conclude is that when Im[α(e i )] do not have the correct sign, the original UV picture is not appropriate, and we should look for an alternative description. Generically 4 , for an arbitrary choice of W (Φ) and α(Φ), Im α(e i ) will not have the same sign for all the critical points, and thus some vacua will have gauge group factors with g 2 YM < 0. We have a practical way to analyze the IR theory in these vacua directly from the field theory approach. We can start with parameters such that the UV theory makes sense, and then compute the effective IR action in terms of the glueball superfields, as discussed in the previous sections. We then change the parameters so that the UV theory would formally develop a negative value of g 2 YM for some of the gauge group factors. However, the effective IR theory still makes sense when we do this, so we can simply study the IR action, without worrying about the dual UV description. As we will show, in the IR theory this change of parameters leads to supersymmetrybreaking.
We are thus naturally led to ask: What is the corresponding UV theory in such cases? When only some of the gauge couplings are negative, we will argue that supersymmetry is broken, but we will not have a full field theory description in the UV. However, if they are all negative, we can formulate a complete UV field theory description for which supersymmetry is manifestly broken. In all cases, the UV description provided by string theory exists, and we will argue that it involves both branes and antibranes.
In the general, these theories have two sources of supersymmetry breaking.
One, which comes from any of the gauge factors with negative Im α(e i ), corresponds to giving a nonzero vev to spurion auxiliary fields. The other effect comes from the fact that when both signs of Im α(e i ) are present, the interaction between the gauge group factors are not supersymmetric, as each factor tries to preserve a different supersymmetry.
We first study the situation of the first kind -all Im α(e i ) negative -where the internal dynamics of the gauge theory softly break supersymmetry. For this case, we quantify the supersymmetry-breaking effect in terms of a dimensionless parameter which measures fractional mass splittings in the supermultiplets.
Moreover, we motivate and provide strong evidence for the existence of a dual nonsupersymmetric UV theory. We motivate this from field theory as well as describing its natural explanation in the context of string theory.
We then move to the multi-sign case and show that when some Im α(e i ) have different signs, there is an additional effect which breaks supersymmetry.
Essentially, this arises from each factor of the gauge group trying to preserve a different half of a background N = 2 supersymmetry, and charged bifundamental matter communicates supersymmetry breaking. For this case, we only have a stringy dual description in the UV.

Negative gauge couplings and duality
We now discuss, from both string theory and field theory perspectives, how a gauge coupling squared becoming negative can be sensibly understood in terms of the dual description. The simple example which we review, where both the original and the dual theories are supersymmetric, has already been studied in [18].
Consider N D5 branes on the resolved conifold geometry with a single P 1 .
As in section 3, we can view this geometry as obtained by fibering an A 1 ALE singularity over the x-plane as We turn on a constant B-field through the S 2 at the tip of the ALE space, In the language of section 2, this means that the gauge coupling is independent of φ. More generally, the effective gauge coupling of the 4d U (N )theory is given by 4π/g 2 YM = r 2 + B 2 NS /g s , where r is the physical volume of the P 1 . This is usually written in terms of single complex variable t, the complexified Kähler class, given by t = B NS + ir, as 4π/g 2 YM = |t|/g s . In the present paper we have permanently set r = 0, so t = B NS . This stringy duality is directly manifested in field theory. It turns out, as we now review, that this situation has a simple and elegant realization in terms of the glueball superfields which emerge as the IR degrees of freedom. Consider first the situation before the flop. In the IR, we have a deformed conifold geometry where S, the modulus of the deformation, is identified with the glueball superfield, S = TrW α W α . The Veneziano-Yankielowicz superpotential, which can be derived in either the field theory or the dual string theory, is given by As was already reviewed in previous sections, in the gravitational dual picture, the two terms above correspond to flux contributions to the superpotential.
One should note that this effective description is only valid for field values where |S/m| ≪ |Λ 2 0 |. Extremizing W with respect to S gives As long as the bare UV gauge coupling satisfies this is an acceptable solution in the sense that S is within the allowed region of field space. Note that in addition to the chiral superfield, the theory in the IR still has a U (1) vector multiplet, because only the SU (N ) ⊂ U (N ) is confined.
In the string theory construct, the extra U (1) is identified with the reduction of the 4-form IIB gauge potential on the deformed S 3 . In other words, this theory describes a massive chiral multiplet consisting of S and its fermionic partner ψ, as well as a massless photon A and its partner λ, Together these would form an N = 2 chiral multiplet before the supersymmetry is broken to N = 1 by fluxes. Now consider the same theory, but in the limit where Im(α) ≪ 0, which would have corresponded to 1/g 2 YM ≪ 0. Then the above solution (7.4) is not valid anymore, since |S/m| ≫ |Λ 2 0 | lies outside the regime of validity of the effective theory. Thus the original supersymmetry is broken, since we cannot set ∂ S W to zero. Even so, as was shown in [18], there are still physical vacua which minimize an effective scalar potential V eff . Moreover, the theory in these minima is exactly the same as one would expect for the IR limit of an N Let us review in more detail how the flop is manifested in the IR field theory of [18]. When Im(α) ≪ 0, we must look for critical points of the physical At leading order, the theory spontaneously breaks an underlying N = 2 supersymmetry, so the tree-level Kähler metric should be determined by special geometry. While we do not expect this to be an exact statement, we nevertheless make the assumption for the remainder of this section that the Kähler metric is that of the N = 2 theory 5 . Thus the action for the IR dual is given by where Λ 4 gets identified with M 4 string in the string context. This leads to the Kähler metric The effective potential can then be made explicit, and the critical points, ∂ S V eff = 0, are located at the solutions to 2i This can be satisfied through either The first solution preserves the manifest N = 1 supersymmetry, and corresponds to the solution of ∂ S W = 0. The second solution does not preserve the original supersymmetry as ∂ S W = 0. Only one of these two solutions is valid at a given point in parameter space if S is to be within the field theory cutoff of |S| ≪ |mΛ 2 0 |. For Im(α) > 0 the first solution is physical, and this is the supersymmetric solution we discussed above. However, for Im(α) = 1/g 2 YM < 0, it is the second solution which is physical, and we obtain This solution looks very much like the solution (7.4) for the original U (N ) confining theory, except that α →ᾱ. This is what one would expect if we were discussing the theory of N antibranes on the flopped geometry. In fact, as discussed in detail in [18] one can show that this theory is indeed supersymmetric, with supermultiplets aligned as in (7.6).

Supersymmetry breaking by background fluxes
Now consider the same geometry as in the previous subsection, but with a holomorphically varying B-field introduced. If we wrap branes on the conifold, this gives rise to the supersymmetric theories considered in sections 3-4.
However, in the case of antibranes, we will see that supersymmetry is in fact broken. This is due to the fact that, while branes preserve the same half of the background N = 2 supersymmetry as the B-field, antibranes preserve an opposite half.
As in the previous section, we will consider branes and antibranes on the conifold geometry (7.2) with superpotential given by (7.3), but now with the holomorphically varying B-field given by 6 We will study this from the perspective of the IR effective field theory of the glueball superfield S. Because of the underlying N = 2 structure of this theory, we will have a good description regardless of whether it is branes or antibranes which are present. In the next subsection, we will provide UV field theories describing both situations.
The superpotential in the dual geometry is given by (4.7), which we repeat here for convenience An explicit computation in the geometry yields an exact expression for the first term, The scalar potential is again given by (7.7) with the same metric and prepotential F 0 , but now with superpotential (7.12). There are two vacua which extremize the potential, ∂ S V eff = 0, The first solution satisfies ∂W = 0. This has solutions in the case where branes are present, with Here α is defined as α = t 0 + 2t 2 S m , and large positive values of Im[α] give |S/m| ≪ |Λ 2 0 | within the allowed region. This vacuum is manifestly supersymmetric, and we have studied it in sections 3-4.
We can instead study antibranes by allowing the geometry to undergo a flop, so Then the supersymmetric solution is unphysical, and we instead study solutions to the second equation in (7.13). We already know that the manifest supersymmetry is entirely broken in this vacuum, because ∂W = 0. Moreover the fact that the second equation in (7.13) is not holomorphic in S suggests that no accidental supersymmetry emerges here, unlike the cases in previous subsection and [18]. We can directly observe the fact that supersymmetry is broken in this vacuum by computing the tree-level masses of the bosons and fermions in the theory, and showing that there is a nonzero mass splitting.
From the N = 1 Lagrangian, we can read off the fermion masses, while the bosonic masses are given by Evaluating the masses in the brane vacuum, we see that λ is massless and acts as a partner of the gauge field A, while ψ is a superpartner to S. In other words, supersymmetry pairs up the bosons and fermions as in (7.5).
Evaluating the masses in the antibrane vacuum, ψ becomes the massless goldstino. However, there is no longer a bose/fermi degeneracy like where the background B-field was constant. Instead, 14) This mass splitting shows quite explicitly that all supersymmetries are broken in this vacuum. We can capture the strength of this breaking with a dimensionless quantity, We can get a heuristic understanding of this measure of supersymmetry breaking as follows. The reason supersymmetry is broken in this phase is that B-field varies in a way incompatible with the normalizable fluxes/branes. Thus its variation over the cut in the IR geometry is a natural way to quantify supersymmetry breaking. More precisely, we expect that measuring ǫ = ∆B across the cut should give a quantification of the supersymmetry breaking by a dimensionless number. Evaluating this explicitly yields ǫ = t 2 S/m, which is in rough agreement (up to a factor of order log |S|/N ) with the dimensionless quantity coming from the mass splittings.

A susy/non-susy duality
Motivated by the considerations of the previous example, we now formulate a duality between two field theories -one which is manifestly supersymmetric, and the other in which supersymmetry is broken softly by spurions. Consider an N = 1 supersymmetric U (N ) gauge theory with an adjoint field Φ and superpotential terms where, as before, Consider a choice of parameters (a k , t k ) such that ImB(e k ) < 0 (7. 16) for all e k with W ′ (e k ) = 0. Then this theory is not sensible in this regime as it has no unitary vacuum. However, we propose that this theory is dual to another U (N ) gauge theory already studied in [11], with an adjoint fieldΦ and superpotential term Note that since the auxiliary field in the spurion supermultiplets have vevs t k , this theory breaks supersymmetry. Also, the fermionic parts of the superspaces for these two actions are not related in any way. Indeed, they are orthogonal subspaces of an underlying N = 2 superspace. This is indicated by the first theory being formulated in terms of coordinates θ α 1 , and the second in terms of θ α 2 -two different N = 1 superspaces. Note that this is natural from the string theory perspective. In the regime of parameters where (7.16) holds, one should describe the physics in terms of the flopped geometry, and ask how the antibrane theory perceives the geometry.
Since the background B-field is holomorphic, it breaks supersymmetry. Indeed the tension of the antibranes will vary as they change position in the x-plane (and we do not expect a canceling term as would be the case for branes). We thus expect the potential to depend on x through a term proportional to the B-field, V eff ∼ ImB(x). (7.18) Indeed, the soft supersymmetry-breaking term in (7.17) gives precisely this contribution when we identify the eigenvalues of Φ with positions in the xplane. Moreover, note that in going from (7.15) to (7.17) we have flipped the sign of Im(t 0 ) ∼ 1/g 2 YM , which is consistent with the fact that (7.17) describes the same physics from the antibrane perspective. As an aside, note that in this section (unlike in much of the rest of the paper), t 0 and t k>0 enter on different footings.
We now provide evidence for this duality. We will show that both theories (7.15) and (7.17) have the same IR description in terms of glueball fields.
The effective superpotential for the supersymmetric theory we have already discussed, and is given by The effective glueball theory for the nonsupersymmetric theory, in which auxiliary spurion fields have nonzero vevs, has been studied in [10,11]. As shown in [11], turning on soft supersymmetry-breaking terms that give spurionic Fterms to the a i in the UV theory has the expected effect in the IR of simply giving spurionic F-terms to a k>0 in that theory, We will see that the two effective glueball theories are in fact identical! As we reviewed in section 4, one way to arrive at the effective IR theory is via a dual gravity theory. Both theories (7.19) and (7.20) originate from the same Calabi-Yau after the transition, and so have the same underlying N = 2 theory with prepotential F 0 (S, a) at low energies, 7 Im with appropriate fluxes or auxiliary spurion fields turned on. In fact, it was shown in [20,21] that turning on fluxes is also equivalent to giving vevs to auxiliary fields, so both (7.19) and (7.20) can be thought of as originating from the N = 2 theory with prepotential F 0 (S, a), where auxiliary fields for the glueball fields S i and the background fields a k are subsequently given vevs.
This breaks supersymmetry explicitly to N = 1 in the case of (7.19), and to N = 0 in case of (7.20).
To be more precise, (7.19) can be obtained by shifting the auxiliary fields of the N = 2 multiplets containing S and a according to and integrating over θ 2 . Meanwhile, (7.20) arises from instead shifting and integrating over θ 1 .
These two situations differ in how they shift the auxiliary fields F i 11 and F i 22 = F i 11 which lie in the N = 2 chiral multiplet containing S i , Shifts of fields alone cannot affect any aspect of the physics if the shift can be undone by an allowed field redefinition. Indeed, the difference between the shifts of (7.19) and (7.20) is an allowed auxiliary field redefinition, so these theories are equivalent! Put another way, in integrating out the auxiliary fields, we end up summing over all of their values, so any difference between the two theories will disappear. Note that, if F i 11 and F i 22 were independently fluctuating degrees of freedom, we could use this argument to say that both theories were equivalent to the original N = 2 theory. They are not, however, since the auxiliary field shifts we made cannot be undone by a field redefinition obeying F i 22 = F i 11 , which the fluctuating part of the auxiliary fields must satisfy.
To make this duality more explicit, we will show that both theories give rise to the same IR effective potential, V eff (S i ). For the supersymmetric theory (7.19), the superpotential (7.15) is which leads to an effective potential where in the summation t k η ki , we have removed the m = 0 term and written it explicitly. This will be convenient for the manipulations below, where we will continue to use this summation convention. We can rewrite V eff grouped by order in t k , Now we will show that the effective potential of the nonsupersymmetric theory (7.17) agrees with (7.19). The Lagrangian can be written in N = 1 superspace, c. , (7.22) and the superpotential of this nonsupersymmetric theory is simply Let F i be the auxiliary field in the S i superfield. Performing the d 2 θ integral for the superpotential term (the last terms of (7.22)) and picking out the spurion The remaining terms come from the Kähler potential term (the first term of (7.22)). This gives G ij F iFj before spurion deformation, while the spurions produce additional terms, giving a total contribution With the full F-term Lagrangian, it is now easy to check that integrating out the auxiliary fields F i , produces precisely the effective potential (7.21), which arose from the supersymmetric theory (7.19).
We have seen that the tree-level effective potentials for the supersymmetric theory (7.15) and the nonsupersymmetric theory (7.17) agree exactly, corroborating the proposed the duality between the two theories.

Multi-cut geometries and supersymmetry breaking
In the previous subsections we have focused on the case where all gauge couplings have the same sign, positive or negative. We now shift to consider the more general case in which both signs are present. For simplicity, we will focus on the case where the superpotential has two critical points, with a brief discussion of the generalization to an arbitrary number of critical points reserved for the end of this subsection.
In particular, we now consider the UV theory where the superpotential appearing in the geometry (7.2) is given by and the holomorphic variation of the B-field gives rise to an effective fielddependent gauge coupling The two critical points of the superpotential are given by Φ = ±m, at which points the gauge coupling takes values We wish to study the case where the imaginary parts of gauge couplings have opposite signs (see figure 4). Without loss of generality, we then consider Im(α − ) ≪ 0 ≪ Im(α + ). (7.23) We will consider the vacuum where the U (N ) gauge group is broken to U (N 1 )× U (N 2 ) with N i both nonzero. It is clear from the discussion in section 7.2 that this theory is that of N 1 branes wrapping the S 2 at e 1 and N 2 antibranes wrapping the flopped S 2 at e 2 .
There are now two sources of supersymmetry breaking present. First, for the N 2 antibranes (even if N 1 = 0), supersymmetry is broken due to the holomorphic variation of the B-field, as discussed in section 7.3. However, this effect is secondary to that which arises from the fact that branes and antibranes are both present and preserve disparate halves of the background supersymmetry. This more dominant source of supersymmetry breaking was studied in a slightly simpler context in [18,22,23]. We now show that this stringy UV picture is borne out in the dual IR theory. The superpotential for the closed-string dual geometry is given by (4.8) In the large N i limit, it is a sufficient approximation to work to 1-loop order in the associated matrix model. For the geometry in question, the superpotential then takes the form where a 1 = −m 2 g and F 0 was computed in [1], ) .

(7.24)
Note that at this order, the effect of the varying B-field is just to change the effective coupling constants in the superpotential from α 0 to α ± . As a result, the only supersymmetry-breaking effects which appear are due to the presence of antibranes.
This theory has no physical supersymmetric vacua, so in order to study its low energy dynamics, we minimize the physical scalar potential, where again the Kähler metric is determined by N = 2 supersymmetry, The critical points are given by solutions to At one-loop order in the matrix model, F ijk only has nonzero diagonal elements, in which case the vacuum equations simplify. In particular, for the case at hand they simplify to and using the expression for the Kähler metric arising from (7.24), we obtain following explicit solutions In addition, we can compute the vacuum energy, and find it to be The first term we identify as the brane tension due to antibranes on the flopped P 1 , which agrees with (7.18), while the second term suggests a Coulomb repulsion between brane stacks preserving opposite supersymmetries. A similar expression for the potential energy between branes and antibranes can be found in [18,22,23].
We can further study the masses of the bosonic and fermionic excitations about the nonsupersymmetric vacua. At the current order of approximation, most of the expressions from [18] still hold. We obtain four distinct bosonic masses, given by [18] (m ±,c ) and Λ is a mass scale in the action (7.8). The tree-level fermionic masses can also be computed from the off-shell N = 1 Lagrangian. As in [18], they are given by 8 The presence of two massless fermions can be thought of as representing two goldstinos due to the breaking of off-shell N = 2 supersymmetry. Alternatively, this fermion spectrum can be viewed as the natural result of breaking supersymmetry collectively with branes and antibranes. There is a light gaugino localized on both the branes and the antibranes. However, since these preserve different supersymmetries, we see the gauginos as arising one from the gaugino sector and one from the sfermion sector with respect to a given N = 1 superspace.
For a generic choice of parameters, supersymmetry breaking is not small, and there is no natural way to pair up bosons and fermions in order to write a 8 Note that the relation [24] boson m 2 − fermion m 2 = Tr(−) F m 2 = 0 holds for our system, as well as for (7.14).
mass splitting as a measure of how badly supersymmetry is broken. However from the mass formula we have given, it is clear that in the limit v → 0, the spectrum becomes supersymmetric, and there does emerge a natural pairing of bosonic and fermionic excitations. In this limit, v becomes a good dimensionless measure of the mass splitting, and we can write it in terms of parameters This vanishes and supersymmetry is restored for large separation mt 1 ≫ 1, corresponding to the extreme weak-coupling limit. One can also consider another extreme where N 1 ≫ N 2 . In this limit we again expect supersymmetry to be restored. Indeed, in this limit v ∝ N 2 /N 1 , and so vanishes.
It should be noted that, unlike the case where all gauge couplings are negative and the background flux is small, in this case the dimensionless parameter v does depend explicitly on the cutoff Λ 0 . This may be related to the fact that, in this case, there is no field theory description in the UV. Namely, even though we know that this system should be described by branes and antibranes, these brane configurations do not admit a good field theory limit. Nevertheless the arguments of the previous section can be used to show that below the scale of gauge symmetry breaking, there is an effective field theory description in terms of a i U (N i ) gauge theory which breaks supersymmetry and captures the same IR physics. In this theory, the gauge group factors with positive gauge couplings have an effective field dependent gauge coupling, while those with negative gauge couplings have supersymmetry softly broken by spurions.
However, this is not a satisfactory description for the full dual UV theory.
Before concluding this section, let us briefly consider the generalization of the previous discussion to the n-cut geometry. Here, the superpotential in (7.2) is given by Starting with D5 branes wrapped on n shrinking P 1 's at x = e i , we perform a geometric transition and study the dual closed-string geometry with n finite S 3 's. The distance between critical points are ∆ ij ≡ e i − e j .
From the period expansion of [1] we have following expressions in a semiclassical regime 2πiτ ii =2πi Generalizing the vacuum condition from the two cut geometry, the physical minima of effective potential are then determined by The expectation values of S i are expressed explicitly below, The vacuum energy density formula is now given by where the first term is the brane tension contribution from each flopped P 1 with negative g 2 YM (matching with (7.18)), and the second term suggests that opposite brane types interact to contribute a repulsive Coulomb potential energy (as in the cases considered in [23])

Decay mechanism for nonsupersymmetric systems
It is straightforward to see how the nonsupersymmetric systems studied in this section can decay. This is particularly clear in the UV picture. If the gauge coupling constants are all negative, the branes want to sit at the critical point which minimizes |ImB(e i )|, as this will give the smallest vacuum energy according to (7.27). Thus we expect that in this case the system will decay to one which is the U (N ) theory of antibranes in a holomorphic Bfield background. This still breaks supersymmetry, but it is completely stable.
Considering that RR charge has to be conserved, no further decay is possible.
If there are some critical points where ImB(e i ) is positive, there is no unique stable vacuum. Instead, there are as many as there are ways of distributing N branes amongst the critical points x = e i where ImB(e i ) > 0. Thus, we find numerous supersymmetric vacua which could be the end point of the decay process, each one minimizing the potential energy to zero. As in [18], these decays can be reformulated in the closed-string dual in terms of Euclidean D5 brane instantons, which effectively transfer branes/flux from one cut to another.

Appendix A. Computation of W eff
Here we provide more detail on the derivation of (4.7) using the Riemann bilinear identity and its extension to a noncompact Riemann surface Σ. In particular, we wish to compute the integral Σ χ ∧ λ (A.1) for closed one-forms χ and λ which are now allowed to have arbitrarily bad divergences at infinity. We need to be extra careful due to this worse-thanusual behavior at infinity. In particular, the contribution of the interior of the Riemann surface will be exactly the same as the usual case, with the only difference coming from a careful treatment of contributions coming from the boundary at infinity. where we have used the fact that P = − Q and that H RR + H 0 ∼ dB(x) for large x (and so at the contour around P ). Combining all contributions, the superpotential can indeed be rewritten as Note that when f has at worst a logarithmic divergence at P and Q, and λ has at worst a simple pole, then we can write