# Ippp/09/16 Dcpt/09/32 Electric/Magnetic Duality with Gauge Singlets

Steven Abel, James Barnard
Institute for Particle Physics Phenomenology, Durham University, Durham DH1 3LE, UK
s.a.abel or
###### Abstract

We demonstrate how gauge singlets can be used to find new examples of Kutasov duality (i.e. where the matching of the dual theories relies on a non-zero superpotential) in SQCD with flavours of quark and multiple generations of adjoints, or antisymmetrics, or symmetrics. The role of the singlets is to simplify greatly the truncation of the chiral ring whilst maintaining an -symmetry, and at the same time allowing an unambiguous identification of the elementary mesons of the magnetic theory. The dual theories satisfy all the usual tests, including the highly non-trivial ’t Hooft anomaly matching conditions.

## 1 Introduction and background

An interesting possibility in the context of electric-magnetic duality [1] is that the supersymmetric Standard Model is the magnetic dual of some unknown electric theory [2, 3, 4]. This would have a bearing on many important questions. For example it would accommodate Landau poles in models of direct gauge mediation of supersymmetry (SUSY) breaking. The dynamical scales could also explain hierarchies in the SUSY breaking sector and even in the Yukawa couplings. More recently [4] it was argued that this possibility may, rather counterintuitively, be compatible with unification. That argument was based on known examples of Kutasov duality, i.e. duality that relies on a non-zero superpotential [5]. The vacuum structure of these models was analysed in the simplest SQCD plus a single adjoint case by Kutasov, Schwimmer and Seiberg (KSS); it can be that of a broken GUT theory with electric and magnetic dual descriptions [6, 7]. Further related studies of these theories were made in Refs.[8, 9, 10, 11].

The observation made in Ref.[4] was that (as shown in Figure 1.1) unification in such theories is preserved in the mapping from electric to magnetic descriptions, with the gauge couplings of the magnetic theory appearing to unify at the same energy scale but at unphysical (imaginary) values of the gauge couplings. This unphysical gauge unification also happens in a supersymmetric Standard Model with a large number of messengers in complete representations, and so in Ref.[4] it was suggested that the latter would be a strong hint of a magnetic dual GUT theory. Furthermore it was argued that this would explain why the supersymmetric Standard Model appears to unify, but the proton does not decay. The proton decay operator (being generated at the GUT scale) is a baryon operator that is generated in the electric theory. However the decay takes place at low energies and is therefore computed in the magnetic description with the relevant operator being mapped to the corresponding magnetic baryon: consequently the decay rate is suppressed by the many powers of associated with the mapping of electric to magnetic baryons.

Unfortunately the arguments of Ref.[4] are limited by the fact that no convincing electric dual description of the supersymmetric Standard Model or indeed the Georgi-Glashow model is known. In particular, in for example an framework, one would like to be able to build chiral models, models with three generations, with adjoint GUT Higgs fields and antisymmetrics and so on. Although some of these ingredients (for example chiral models) are present in the models in the literature (related examples with gauge groups can be found in Refs.[7, 12, 13, 14]), the particle content is very constrained and is far from that of the Georgi-Glashow model. Motivated by this fact, we will in this paper demonstrate how one can extend the Kutasov class of electric-magnetic dual theories using gauge and flavour singlets.

In very general terms our approach can be described as follows. First recall the three basic tests of electric-magnetic duality [15]: the two dual theories should

• have the same moduli space of vacua, and in particular a one-to-one map between the degrees of freedom represented by meson and baryon operators,

• they should share the same global symmetries and have the same global anomalies (i.e. the ’t Hooft anomaly matching conditions),

• their moduli spaces should still match under all deformations of the theories by chiral operators.

The KSS-like theories can be shown to pass all these tests, thanks to the IR dynamics being simplified by the addition to the superpotential of various operators. In its original formulation, the KSS theory consisted of SQCD plus a single adjoint, , and the superpotential operator

 W⊃Xk+1 (1.1)

for some integer . This term alters the infra-red dynamics (it is a dangerously irrelevant operator, as one says) for appropriate choices of and flavours, and truncates the chiral ring, allowing one to match the degrees of freedom in the moduli space relatively easily. Further examples of this kind of duality were discovered in Refs.[13, 14] involving more adjoint fields, and/or symmetric and antisymmetric representations. An obstacle restricts the whole class of models however: as more fields are added, the chiral ring becomes more difficult to truncate and additional operators have to be added to the superpotential to do this; on the other hand each new operator projects out some global symmetry and weakens the ’t Hooft anomaly matching test. In particular once the -symmetry is explicitly broken the ’t Hooft anomaly matching test becomes virtually meaningless, so this symmetry at least one needs to preserve. Our approach in order to overcome this problem is to use an additional singlet to enable us to add the extra operators needed to truncate the chiral ring while preserving a meaningful -symmetry. The role of this singlet is to break the -symmetry spontaneously by acquiring a VEV, thereby generating the -symmetry breaking couplings necessary to truncate the chiral ring.

At first sight one might suppose that there is little to be gained from doing this. Indeed the -charged couplings (and hence the singlet) will generally appear in the elementary mesons that are a feature of the spectrum of the magnetic dual theory, so it might seem that the -charges in the magnetic theory are going to be arbitrary, and the correct anomaly matching condition impossible to identify. This is not so however: the matching of the composite mesons of the electric description to the elementary mesons of the magnetic one entails an assignment of singlet contribution that is unambiguous. Moreover a nontrivial test is that the equivalent assignment in the baryons makes their global charges match automatically. With the -charges of the elementary mesons of the magnetic dual theory duly fixed, anomaly matching can then proceed as normal, and it provides the usual powerful test of the duality.

In the following section we begin by showing how this can be done in the original KSS (SQCD plus single-adjoint) model of Refs.[6, 7]. An important feature of this model is as we have stated the possibility of adding deformations that would break the GUT symmetry. Although these operators are not crucial for truncating the chiral ring (being relevant deformations rather than drastically changing the IR behaviour as the original operator did), it is still interesting to see how they can be made compatible with a preserved -symmetry by the addition of a gauge singlet. The procedure required to identify the elementary mesons of the magnetic description in this simple case will serve as a useful guide for later models. We find that anomaly matching in the deformed theory works in two cases; the first corresponds to the original deformed KSS model with the singlet VEV merely parameterizing the size of the deformation. The second model is different from the original model, and as we shall see is (unlike the first) not associated with the flow to a fixed point.

In sections 3, 5 and 6 we consider models with two or more adjoints and models with extra antisymmetric and symmetric representations respectively. In each case we can add to the superpotential enough operators to truncate the chiral ring while still maintaining an -symmetry and therefore highly non-trivial ’t Hooft anomaly matching conditions.

Section 4 deals in more detail with the RG flow. Concentrating on the adjoint models, we construct a complete picture of the flow in models with one adjoint plus a singlet using -maximization [16, 17]. The additional singlet alters the RG flow, and gives various fixed points; the flow is to the KSS model with the singlet appearing in a deformation term, however the theory can flow between fixed points corresponding to different deformations. We use this picture to deduce the RG flow behaviour for the more general models with additional adjoints. For the models with an odd number of adjoint fields, we argue that the endpoint of the flow is also the deformed KSS model. Models with an even number of adjoints flow to normal SQCD. In both cases the duality at the endpoints of the flow is well understood.

We close the Introduction by making a parenthetical remark about the phenomenology. Recall that our ultimate goal is to be able to describe the supersymmetric Standard Model (or a Georgi-Glashow like GUT theory) as a magnetic dual. These theories generally do not allow anomaly-free global -symmetries because of the complexity of their Yukawa couplings and Higgs mass terms. It is therefore likely that any -symmetry would have to be spontaneously broken anyway in order to have reasonable phenomenology, and using gauge singlets to do this would be the simplest way. We consider this to be an additional motivation for the study in this paper.

## 2 R-symmetry in the deformed KSS model

Let us first revisit the model of KSS in order to see how singlets can be introduced. As we have said, one of the appealing features of those models was the ability to match the electric and magnetic theories under deformations of the electric superpotentials

 Wel=k−1∑i=0tik+1−iTr[Xk+1−i]+λTr[X] (2.1)

where is an adjoint field of the gauge group and we have chosen our basis for such that there is no term as explained in Ref.[7]. The undeformed theory has only the term. The deformations spontaneously break the gauge symmetry and lead to a rich vacuum structure; it is then possible to show that there exists a similar deformation in the magnetic theory that produces the same vacuum. There is an aspect of this procedure that we will address in this section as a warm-up exercise which is this. By adding the extra terms in Eq.(2.1), one breaks the -symmetries of the model. In principle therefore the ’t Hooft anomaly matching conditions apply only to the undeformed theory. However it is often useful to think of couplings such as as background -charged fields that acquire VEVs. Thus a way to match the anomalies directly in the deformed theory would be to consider these fields as singlets in the spectrum and to do the anomaly matching on the complete theory. The singlet VEVs can then be fixed at the end to generate the required -breaking terms in Eq.(2.1) spontaneously. This is what we will investigate here.

We commence by summarizing the models of KSS [6, 7] in more detail. They are based on an gauge group with flavours of quarks and anti-quarks, and an adjoint field of the denoted by . The symmetry content is

 SU(FQ)L×SU(FQ)R×U(1)B×U(1)R1×U(1)R2. (2.2)

When there is only the term in the superpotential, the global symmetry is partially broken but retains a symmetry. The matter content is then summarised by Table 2.1.

When there are also non-zero couplings, the -symmetry is completely broken. The -term equation for the adjoint in this case can easily be solved by diagonalizing the using rotations; the equation for a single entry on the diagonal is

 W′=0≡k−1∑i=0tiχk−i+λ. (2.3)

This is a ’th order polynomial so there are roots: hence

 ⟨X⟩=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝χ1Ip1χ2Ip2⋱χkIpk⎞⎟ ⎟ ⎟ ⎟ ⎟⎠wherek∑i=1pi=N. (2.4)

The gauge symmetry is broken as

 SU(N)→SU(p1)×SU(p2)…SU(pk)×U(1)k−1. (2.5)

Note that the parameter is used as a Lagrange multiplier to fix .

To get the corresponding magnetic theory we need to identify a set of elementary meson fields associated with composite operators of the electric model. A crucial aspect of the superpotential is that it truncates the chiral ring; that is the equation of motion for (in for example the undeformed theory) sets (ignoring the Lagrange multiplier term) along the -flat directions. This means that when matching the moduli spaces, one need only consider operators up to (in the deformed theory is mapped to a polynomial in of order less than ). Thus there are types of meson operator that we denote ;

 mj=~QXjQ,j=0,…,k−1. (2.6)

The object is the meson of usual Seiberg duality. Indeed, the model is just the original Seiberg SQCD model if one integrates out the adjoint field. The field content of the magnetic theory is , , and , where is an adjoint in the magnetic gauge group, and where the elementary magnetic mesons are directly and unambiguously identified with the composite operator . Assuming for the moment that there is only the term in the electric theory then, as in Refs.[6, 7], the baryon matching implies that the gauge group of the full (unbroken) magnetic theory is

 SU(n)=SU(kFQ−N). (2.7)

The matter content is summarised in Table 2.2.

The superpotential in the deformed magnetic theory is of the form

 (2.8)

This form of superpotential was deduced in Ref.[7] by equating the vacuum structures of the electric and magnetic theories. The broken model in the magnetic theory is

 SU(n)→SU(FQ−p1)×SU(FQ−p2)…SU(FQ−pk)×U(1)k−1. (2.9)

Note that the matching of the sub-theories is generically the standard SQCD duality.

Let us now turn to the anomaly matching of the deformed theories and, as we have proposed, introduce a singlet field whose role is to generate the couplings by acquiring a VEV: we therefore write for some exponents . The coefficients in the magnetic superpotential are directly related to those in the electric one which, in turn, directly relates the singlet structure in the magnetic theory to that in the electric one. As a simple example consider the electric superpotential

 Wel=ϕρ0Tr[Xk+1k+1]+ϕρlTr[Xk+1−lk+1−l]. (2.10)

We will henceforth omit the Lagrange multiplier term, and also assume and . We may read off the magnetic superpotential from Eq.(2.8):

 Wmag = −ϕρ0Tr[xk+1k+1]−ϕρlTr[xk+1−lk+1−l]+ (2.11) (ϕρ0μ2k−1∑j=0mj~qxk−1−jq+ϕρlμ2k−1−l∑j=0mj~qxk−1−j−lq).

Note that all the couplings of the magnetic theory, in particular the powers of , are already determined by the vacuum structure. However the parameter is an unknown free parameter that itself carries charge. For example, it determines the matching conditions for the dynamical scales of the electric and magnetic theories ( and respectively) which are [6]

 Λ2N−FQ¯Λ2n−FQ=(μt0)2FQ. (2.12)

The -symmetry is anomaly free (with respect to the mixed gauge- anomalies) so the dynamical scales do not carry -charge or depend on . To be consistent the -charge of has to be and moreover must scale as . Both electric and magnetic theories now have an -symmetry that is spontaneously broken by the VEV of .

Indeed let us now lay out the charges of the fields explicitly. By virtue of the electric superpotential and the absence of mixed anomalies, the matter content of the electric theory is given by Table 2.3.

Since we know the charges of the mesons from Eq.(2.6) we can independently derive the magnetic superpotential by writing down terms consistent with the -symmetry. This turns out to yield precisely as required, so that the superpotential deduced from the vacuum structure is consistent with that deduced from the -symmetries. At the classical level allows arbitrary values of and . However we will now see that the ’t Hooft anomaly matching conditions constrain them.

To do the anomaly matching we first need to identify correctly the elementary fields of the magnetic theory, and in particular deal with some arbitrariness about the definition of the elementary mesons. Specifically, the mesons’ role is to mirror (and project out via their equations of motion) the corresponding directions in the moduli space of the magnetic theory. For example, consider the leading meson term of the magnetic superpotential. Remembering that this can be written as

 ϕ−ρ0k−1∑j=0mj~qxk−1−jq≡ϕ−ρ0k−1∑j=0(~QXjQ)(~qxk−1−jq) (2.13)

where we have exhibited the composite objects of the electric theory that are the erstwhile elementary mesons. The equations of motion of therefore project out the corresponding directions in the magnetic moduli space. The arbitrariness lies in the singlet content of what we call the elementary meson of the magnetic theory. Because the singlets carry -charge, unless we determine this correctly, anomaly matching is impossible.

We therefore propose the following principle: the singlets are to be distributed among the elementary and composite magnetic mesons in the same way. That is, rewriting the leading interaction as

 ϕ−ρ0k−1∑j=0mj~qxk−1−jq≡k−1∑j=0(~Q(ϕαX)jQ)(~q(ϕαx)k−1−jq), (2.14)

where

 α=−ρ0k−1, (2.15)

allows us to identify the elementary mesons of the magnetic theory as

 Mj≡~Q(ϕαX)jQ. (2.16)

As we have said, identifying this singlet contribution to the meson for non-zero is an important step since it fixes the charges of the elementary mesons and hence the anomaly matching.

An independent check of this identification of the magnetic mesons is that the same identification is appropriate for the mapping between electric and magnetic baryons. In detail, the baryons of the electric theory in the deformed KSS model are given by

 B{rj}∼k−1∏j=0(XjQ)rj=QNX∑k−1j=0jrjwherek−1∑j=0rj=N (2.17)

(for concision we have omitted colour indices, which are contracted with a rank epsilon tensor). The magnetic baryons to which these are mapped were found (by mass flow and by equating the global charges including -charge) in Refs.[6, 7] to be

 b{rj}∼t0kFQ/2(μ2)−n/2k−1∏j=0(xjq)FQ−rj∼(ϕρ0)(kFQ−2n)/2qnx∑k−1j=0j(FQ−rj). (2.18)

Using , we find that the -charges of the baryons are equal if

 NRQ−nRq+(k−1)(N−kFQ2)RX+ρ0(n−kFQ2)Rϕ=0 (2.19)

which is satisfied for any (the specific -symmetry of the undeformed models [6, 7] is, of course, equivalent to taking ). Using Eq.(2.15) one finds that this is equivalent to

 NRQ−nRq+(αRϕ+RX)(k−1)(N−kFQ2)=0. (2.20)

In other words, the prefactor

 t0kFQ/2(μ2)−n/2∼(ϕρ0)(kFQ−2n)/2 (2.21)

which is required to match the -charges of the electric and magnetic baryons, is equivalent to associating with each precisely the same factor as we did did for the mesons

 B{rj} ∼ k−1∏j=0[(ϕαX)jQ]rj=QN(ϕαX)∑k−1j=0jrj b{rj} ∼ k−1∏j=0[(ϕαx)jq]FQ−rj=qn(ϕαx)∑k−1j=0j(FQ−rj). (2.22)

All that remains is to determine the charges of the magnetic quarks. They can be fixed by demanding that the charges of the electric and magnetic baryons match up:

 Bq=−B~q=NnBQ. (2.23)

We may now proceed to anomaly matching. The mixed anomalies are found to be

 U(1)B×SU(FQ)2L : NBQ U(1)B×SU(FQ)2R : −NBQ U(1)R×SU(FQ)2L : −2N2(ρl−ρ0)FQ[ρl(1+k)−ρ0(1+k−l)] U(1)R×SU(FQ)2R : −2N2(ρl−ρ0)FQ[ρl(1+k)−ρ0(1+k−l)] U(1)B×U(1)2R : 0 U(1)R×U(1)2B : −4N2B2Q(ρl−ρ0)ρl(1+k)−ρ0(1+k−l) U(1)B : 0 U(1)R : (2N2+k+3)(ρ0−ρl)−l(ρ0−2)[ρl(1+k)−ρ0(1+k−l)], (2.24)

in both theories. The last two correspond to mixed -gravity anomalies – there is a contribution from the diagrams with gravitinos in the loop that is obviously universal and has been ommitted. Three of the cubic anomalies also match

 SU(FQ)3L : N SU(FQ)3R : −N U(1)3B : 0. (2.25)

Up to this point the matching occurs for arbitrary and . However, the final anomaly only matches for the three combinations of and that are presented in Table 2.4.

There is also the special case which satisfies the anomaly matching for any and . Note that, because of its complexity, the existence of simple solutions to the matching of the anomalies is certainly not a foregone conclusion.

In the first case the -symmetry is clearly that of the original KSS model with the singlet acting to preserve it under the addition of a deformation, but it is interesting that there are two other consistent choices. The second case

 ρl=(k−l−1k−1)ρ0 (2.26)

is equivalent to

 RX=1andRϕ=1−kρ0. (2.27)

The last case has , ergo . It is therefore less interesting, since the singlet has simply introduced an -symmetry of its own under which none of the matter superfields are charged, and will be discarded.

The simplest choice of exponents in the model (where ) is the superpotential

 Wρ0≠0=ϕk−1Tr[Xk+1k+1]+ϕk−1−lTr[Xk+1−lk+1−l] (2.28)

with and . Note that as far as the anomalies are concerned, we are at liberty to choose which generates a constant in the superpotential

 Wρ0≠0=ϕk−1Tr[Xk+1k+1]+ϕ−2 (2.29)

when gets a VEV. Such a term could conceivably be generated dynamically. In the model the -symmetry is precisely that of the KSS model. In this case is unconstrained and the superpotential is

 Wρ0=0=Tr[Xk+1k+1]+ϕρlTr[Xk+1−lk+1−l] (2.30)

with . Clearly both cases allow for generally deformed superpotentials with more than two terms; i.e. the general -symmetric superpotentials (with these particular -charges) can be written

 Wρ0≠0 = ϕk−1Tr[Xk+1k+1f(ϕX)] Wρ0=0 = Tr[Xk+1k+1f(ϕρl/lX−1)] (2.31)

where is an arbitrary polynomial of order .

There are various limits that one can consider. In the theory for example, one can continuously take the limit. This simply corresponds to turning off the deformation of the KSS model. The magnetic superpotential is

 Wmag=Tr[xk+1k+1]+ϕρlTr[xk+1−lk+1−l]+1μ2k−1∑j=0Mj(~qxk−1−jq)+ϕρlμ2k−1−l∑j=0Mj(~qxk−1−j−lq) (2.32)

and goes continuously to the undeformed model as required. In the oppositie limit, we may define a rescaled and similarly rescaled magnetic adoints, so that the electric superpotential becomes

 Wρ0=0=ϕ−ρl(k+1)/(k+1−l)Tr[^Xk+1k+1]+Tr[^Xk+1−lk+1−l] (2.33)

and the magnetic superpotential is

 Wmag = ϕ−ρl(k+1)/(k+1−l)Tr[^xk+1k+1]+Tr[^xk+1−lk+1−l]+ (2.34) ϕ−ρl(k+1)/(k+1−l)^μ2k−1∑j=0^Mj(~q^xk−1−jq)+1^μ2k−1−l∑j=0^Mj(~q^xk−1−j−lq)

where

 (2.35)

Thus in the large limit, keeping constant, we have KSS duality with . In terms of the original scales this corresponds to dialing down the Landau poles with respect to . Taking, for example, , the matching relation tells us that

 Λ=μt0∼ϕ−ρl/(k+1−l)^μ (2.36)

in this case. We shall later consider the RG flow in the theory.

## 3 Models with more adjoints

Now that we have demonstrated the use of singlets in electric/magnetic duality, we can go on to apply this technique to models for which the dualities are not currently known. All of the models presented below have electric theories based on SQCD, with gauge group and chiral quark flavour symmetry . We will try always to denote fields and parameters in upper case for the electric theory and in lower case for the magnetic theory. In each case the magnetic gauge group is for some value of which must be determined.

The first model we consider extends SQCD with two adjoints and , and an singlet . Electric/magnetic duality with two adjoints was first discussed by Brodie and Strassler in Refs.[12, 13], however the form of their model is much more restricted than that which we are about to present. As our electric superpotential we take

 Wel=ϕρXTr[XkX+1kX+1]+ϕρYTr[YkY+1kY+1]+Tr[XY]. (3.1)

for some and , which allows a non-anomalous, global symmetry group

 SU(FQ)L×SU(FQ)R×U(1)B×U(1)R (3.2)

where is a unique -symmetry. To be exhaustive, we will consider both positive and negative values of and (with negative values possibly being generated dynamically).

The -terms for the adjoints give

 ϕρXXkX+Y = 0 ϕρYYkY+X = 0 (3.3)

These equations truncate the chiral ring for all non-zero values of . To see this, one uses the first equation to set then substitutes into the second equation to find

 (−1)kYϕρXkY+ρYXkXkY=−X. (3.4)

Hence truncation occurs at or, equivalently, .

Eq.(3.4) resembles the usual truncation equation in KSS duality (see §2). It is natural to suppose therefore that the duality proceeds in the same way, but with the effective value of

 k∗=kXkY. (3.5)

This implies the meson structure

 mj∼~QXjQ,j=0,…,k∗−1 (3.6)

and consequently a dual gauge group with

 n=k∗FQ−N. (3.7)

Note that here it is convenient to use one of the adjoints to be ‘active’; thus , where and .

Let us now investigate the theory a little more closely. The non-anomalous global symmetries and the matter content of the electric theory are displayed in Table 3.1.

If one assumes the dual gauge group Eq.(3.7), the equivalent information for the magnetic theory is as given in Table 3.2, with the full properties of the elementary magnetic mesons yet to be determined. The magnetic charges are again fixed by the baryon map which we will discuss shortly. Following KSS duality, Table 3.2 also assumes that the magnetic superpotential takes the form

 Wmag=ϕρXTr[xkX+1kX+1]+ϕρYTr[ykY+1kY+1]+Tr[xy]+meson terms. (3.8)

The gauge singlet does not interact with the gauge group so is oblivious to the duality and must retain the same charges throughout.

Now let us turn to the elementary mesons of the magnetic theory. In §2 we found that powers of appeared in their definition, and the same occurs here. To calculate the appropriate powers of we once again write for and some to be determined, and then consider the meson terms in the magnetic superpotential. These are

 Wmeson=k∗−1∑j=0Mj~q(ϕαx)k∗−1−jq (3.9)

where the power of is again shorthand for the corresponding power of and . Recall that no explicit parameter is required now since all the -charge is carried by the singlets. Note that the magnetic mesons still mirror the electric mesons, hence both use the same value of . Demanding that all of the terms have -charge leads to a unique solution for

 α=ρX+ρYkX1−k∗. (3.10)

The final form for the electric mesons is therefore

 Mj=~Q(ϕ(ρX+ρYkX)/(1−k∗)X)jQ. (3.11)

Equivalently, we could have used instead of as the active adjoint to describe the mesons, in which case we would have found

 Mj=~Q(ϕ(ρY+ρXkY)/(1−k∗)Y)jQ. (3.12)

As usual, these mesons are to be added into the magnetic theory as elementary fields. Their properties are summarised in Table 3.3.

We can now test the duality with ’t Hooft anomaly matching. As before the , and anomalies are all zero; henceforth we will take them as read. The non-zero mixed anomalies are

 U(1)B×SU(FQ)2L : 1 U(1)B×SU(FQ)2R : −1 U(1)R×SU(FQ)2L : −N2/FQ U(1)R×SU(FQ)2R : −N2/FQ U(1)R×U(1)2B : −2 U(1)R : (3.13)

They are the same in both the electric and magnetic theories. The cubic anomalies all match: the non-zero ones are

 SU(FQ)3L : N SU(FQ)3R : −N U(1)3R : −2+N2−2N4/F2Q+3Rϕ−3R2ϕ+R3ϕ (3.14)

with as given in Table 3.1. As the anomalies match for all values of and , any values giving well defined -charges are allowed, and there is no constraint coming from the anomaly. This is because the terms mean that and consequently the and fermions have equal and opposite -charge and don’t contribute here. The quark -charges are independent of and for the same reason (i.e. their -charges are determined by the required absence of anomalies and this gets no contribution from and either). The only field whose -charge depends on and is therefore the singlet which contributes the same in both electric and magnetic theories. Note that demanding a well defined -charge for the singlet eliminates the solution .

The point is singular. Assuming and are both positive and that therefore is negative, then the -term equations Eq.(3.1) have solution at , and the chiral ring truncates trivially. The behaviour of the theory and no adjoint mass is not well understood, but it is obvious that here ultimately the electric theory flows to that corresponding to normal SQCD with no adjoints, since we can integrate out the adjoint fields and below their (Dirac) mass. The assignment of mesons other than also breaks down at this point. Correspondingly the theory is ill-defined in the magnetic superpotential, which contains

 Wmeson=k∗−1∑j=0Mj~q(xϕα)k∗−1−jq, (3.15)

and which therefore diverges. This is hardly surprising because we are using to encode both the couplings in the superpotential and the dimensionful parameter that defines the elementary mesons in the magnetic description (i.e. the equivalent of the parameter of the KSS model).

#### 3.1.1 Baryon matching

The form of the mesons Eq.(3.11) strongly suggests that the adjoints should also come with these powers of the singlets in the baryon map between electric and magnetic theories. Applying this observation we construct electric baryons

 B{rj}=k∗−1∏j=0[(ϕαX)jQ]rj (3.16)

and magnetic baryons

 b{rj}=k∗−1∏j=0[(ϕαx)jq]FQ−rj (3.17)

with and as in Eq.(3.10). Again we are for convenience using to be the ‘active’ adjoint; thus the dressed quarks correspond as , where and . Our experience with KSS duality then leads us to propose the baryon map . This map is totally consistent with all global symmetries; in particular the powers of the singlets are precisely those needed to match the -charges of the baryons.

#### 3.1.2 Generalising the mass term

It may not always be desirable for the electric superpotential Eq.(3.1) to contain the mass term . We will therefore outline how this term can be generalised. In order to do this we shall consider the matching of the two theories in the absence of this term, and then deduce that many alternative terms would be consistent with the duality. These terms would not necessarily truncate the chiral ring in which case a simple classical matching of the moduli spaces would not be possible. Nevertheless the fact that all the other tests of the duality are still satisfied is compelling.

In the absence of the term, Eq.(3.1) are modified and the chiral ring does not truncate in the same way. Furthermore, the -symmetry is relaxed to

 (3.18)

for arbitrary in the electric theory and

 (3.19)

in the magnetic theory. These facts lead us to reconsider the meson structure of the theory.

The new modification of Eq.(3.1) means that we are generally unable to define an ‘active’ adjoint as we did previously, so are forced to consider the more general meson

 Mab=~Q(ϕαXX)a(ϕαYY)bQ (3.20)

with and . The parameters and are undetermined functions of . The meson terms in the magnetic superpotential now look like

 Wmeson=kX−1∑a=0kY−1∑b=0Mab~q(ϕαXx)kX−1−a(ϕαYy)kY−1−bq. (3.21)

Demanding that these terms all have -charge 2 allows one to write as a function of and , but we are not able to remove the dependence of the ’s. We thus conclude that the meson terms in the magnetic superpotential are inconsistent with more than one -symmetry. Hence if we remove the term from the electric superpotential we are forced to replace it with a different third term that yields a single, non-anomalous -symmetry in order to maintain the duality. The precise value of is completely arbitrary as long as it is non-zero. We thus have a very large amount of freedom in choosing the third term in the electric superpotential.

Having fixed the -symmetry and the meson structure there is still one free parameter left in the theory; . The only anomalies that might be sensitive to are those involving the -symmetry. It turns out that the linear and the mixed anomaly matching does not depend on . The cubic -symmetry anomaly matching, however, depends quadratically on . If we choose one of the two values of that set the cubic -symmetry to zero all anomalies therefore match.

Using generalised dressed quarks we can construct electric baryons

 B{rab}=kX−1∏a=0kY−1∏b=0[(ϕαXX)a(ϕαYY)bQ]rab (3.22)

and magnetic baryons

 b{rab}=kX−1∏a=0kY−1∏b=0[(ϕαXx)a(ϕαyY)bq]FQ−rab (3.23)

with . The baryon map is then consistent with all global symmetries for any choice of . The only aspect of the theory that now needs testing is the meson sector. We have not established how (or even if) the chiral ring truncates when the term is replaced by some other interaction, which makes it difficult to see whether the mesonic degrees of freedom match up on either side of the proposed duality. However, the fact that all other tests of duality, including the highly non-trivial ’t Hooft anomaly matching and the baryon matching (itself, even less trivial than it was in the presence of the term) are still passed suggests that the mesons will match up. It may be that we have to resort to investigating the quantum chiral ring, rather than the classical chiral ring we have considered thus far, to see this.

The electric theory of this model is similar to the electric theory of the two adjoint model in §3.1, but with an extra adjoint and an extended superpotential

 Wel=ϕρXTr[XkX+1kX+1]+ϕρYTr[YkY+1kY+1]+ϕρZTr[ZkZ+1kZ+1]+Tr[XY]+Tr[YZ]. (3.24)

For arbitrary values of , the electric theory has no non-anomalous -symmetry. One must choose

 ρY=ρX(1−kYkZ)−ρZ(1−kXkY)kZ−kX (3.25)

in order for the theory to posses one. This is markedly different from the two adjoint case in which all ’s were independent. With fixed as above, the non-anomalous, global symmetry group is once again

 SU(FQ)L×SU(FQ)R×U(1)B×U(1)R. (3.26)

The -terms for the adjoints give

 ϕρXXkX+Y = 0 ϕρYYkY+X+Z = 0 ϕρZZkZ+Y = 0 (3.27)

One can use the first equation to set , then the second equation to set . Upon substitution, the third equation reduces down to

 ϕρZ[−X−(−1)kYϕρXkY+ρYXkXkY]kZ=ϕρXXkX (3.28)

i.e.

 XkXkYkZ∼O(XkXkYkZ−kXkY+1). (3.29)

As long as this implies that the chiral ring is truncated at or, equivalently, . It is not generally possible to use as the ‘active’ adjoint as one is unable to express both and purely in terms of .

Using as the active adjoint (the following argument would be equivalent if we used instead) we now follow the procedure of §3.1. Defining

 k∗=kXkYkZ (3.30)

we ought to be able to use the dual gauge group with

 n=k∗FQ−N. (3.31)

The mesons are

 Mj=~Q(ϕαX)jQ,j=0,…,k∗−1 (3.32)

but with a new value of

 α=ρZ(k∗−kX)−ρX(k∗−kZ)(1−k∗)(kZ−kX) (3.33)

The magnetic superpotential is

 Wmag = ϕρXTr[xkX+1kX+1]+ϕρYTr[ykY+1kY+1]+ϕρZTr[zkZ+1kZ+1] (3.34) +Tr[xy]+Tr[yz]+k∗−1∑j=0Mj~q(ϕαx)k∗−1−jq

as expected. We summarise the matter content for the electric and magnetic theories in Tables 3.4 and 3.5 respectively. All charge assignments can be seen to be consistent with both the meson structure and the magnetic superpotential as required.

The ’t Hooft anomaly matching reveals another significant difference between the two adjoint and three adjoint models. The non-zero mixed anomalies are

 U(1)B×SU(FQ)2L : 1 U(1)B×SU(FQ)2R : −1 U(1)R×SU(FQ)2L
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