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*D. Kutasov*

Enrico Fermi Institute and

Department of Physics

University of Chicago

Chicago, IL 60637, USA

.2in

*A. Schwimmer*

Department of Physics of Complex Systems

Weizmann Institute of Science

Rehovot, 76100, Israel

.2in

*N. Seiberg*

Department of Physics and Astronomy

Rutgers University

Piscataway, NJ08855, USA

.3cm

We study in detail the space of perturbations of a pair of dual $N=1$ supersymmetric theories based on an $SU(N_c)$ gauge theory with an adjoint $X$ and fundamentals with a superpotential which is polynomial in $X$. The equivalence between them depends on non-trivial facts about polynomial equations, i.e. singularity theory. The classical chiral rings of the two theories are different. Quantum mechanically there are new relations in the chiral rings which ensure their equivalence. Duality interchanges “trivial” classical relations in one theory with quantum relations in the other and vice versa. We also speculate about the behavior of the theory without the superpotential.

The recent progress in the understanding of the dynamics of supersymmetric theories (for recent reviews and an extensive list of references see ) uncovered the crucial role played by electric-magnetic duality \[\] in understanding the strong coupling dynamics. In scale invariant theories like the $N=4$ \[\] and finite $N=2$ theories \[\] duality provides two different descriptions of the same physical system that are equivalent at all distance scales. In asymptotically free theories the underlying degrees of freedom are visible at short distance and therefore the existence of different descriptions that are equivalent at all scales (exact duality) is impossible. Nevertheless, duality may be generalized to such theories \[\], relating different quantum field theories with the same long distance behavior. When this long distance behavior is described by a non-trivial superconformal quantum field theory the dual theories are in the same universality class – they flow to the same fixed point of the renormalization group. When one of these theories is infra-red free it gives a simple description of the long distance physics of its strongly coupled dual.

No proof of this duality is known but there is a lot of evidence supporting it. There are three kinds of independent tests:

[1.]{} The two dual theories have the same global symmetries and the ’t Hooft anomaly matching conditions for these symmetries are satisfied.

[2.]{} The two theories have the same moduli space of vacua. These are obtained by giving expectation values to the first components of chiral superfields.

[3.]{} The equivalence is preserved under deformations of the theories by the $F$-components of chiral operators. In particular the moduli spaces and chiral rings agree as a function of these deformations.

It is important to stress that in every one of these tests the classical theories are different and only the quantum theories become equivalent. There is also a crucial difference in the physical interpretation of the deformations of the two theories along the moduli space and by the chiral operators. Often, when one theory is Higgsed and becomes weaker, its dual is confining and becomes stronger. This is one of the reasons for interpreting the relation between these theories as electric-magnetic duality.

The last two tests above are closely related. The rings of chiral operators can be thought of as functions on the moduli space ${\CM}_0$. Hence, one might think that test two above implies test three. However, such a relation is not always simple. When there are points on the moduli space with extra massless particles the situation is more involved. Then the moduli space ${\CM}_0$ is constrained also by the equations of motion of these particles. As one adds sources to the theory proportional to $F$-components of chiral operators, the expectation values of the chiral fields can move away from ${\CM}_0$. The simplest way to describe the situation in such a case is to use an enlarged field space $\CM$ which includes all the fields (including those massive fields which become massless at special points) and to write a superpotential on $\CM$. Then, the equations of motion derived from the superpotential lead to relations in the ring. These relations depend on the parameters in the theory – the sources.

In previously studied examples the structure of the chiral ring was relatively simple. In particular it was determined to a large extent by the symmetries. In general, the structure of the chiral ring can be quite involved. One may describe the ring in terms of generators satisfying certain relations. The relations in the classical chiral ring are consequences of the composite nature of the gauge invariant chiral operators . Quantum mechanically these relations can be modified \[\]. We will see that it may also happen that new relations in the chiral ring appear quantum mechanically. Then, the classical chiral ring is truncated, in some cases rather dramatically.

The purpose of this paper is to study some qualitative and quantitative features of the duality of  in a class of examples that exhibit the phenomena mentioned above and a rich duality structure which helps in analyzing them. In the rest of this section we will describe the models we will study and state the main results. Derivations and many additional details appear in subsequent sections.

*The electric theory*

Consider a $G=SU(N_c)$ gauge theory with an adjoint field $X$ and $N_f$ quarks $Q^i$ and $\tilde Q_{\tilde i}$ ($i,\tilde i=1,...,N_f$) in the fundamental and anti-fundamental representations of the gauge group, respectively. These theories are still not fully understood (see however a discussion below in section [*7*]{}). When a superpotential is turned on, the dynamics simplifies. At first sight the fact that the high order polynomials appearing in  can have any effect on the physics is surprising. Indeed, the presence of these non-renormalizable interactions seems irrelevant for the long distance behavior of the theory, which will be our main interest below. Nevertheless, these operators have in general strong effects on the infrared dynamics. They are examples of operators that in the general theory of the renormalization group are known as [*dangerously irrelevant*]{}. Some comments about the properties of such operators appear in Appendix A.

To simplify the analysis of  with a traceless matrix $X$, we may view $X$ as an arbitrary matrix and represent the constraint by a Lagrange multiplier term $\lambda \Tr X$ in the superpotential. Physically, this amounts to adding two massive chiral fields to our problem, $ \lambda $ and $\Tr X$. Clearly, this does not affect the long distance behavior. Then we can shift $X$ by a term proportional to the identity matrix $X_s=X + {s_1 \over s_0k}1 $ to set the coefficient of $\Tr X_s^k$ in  to zero; $\lambda$ is also shifted by a suitable constant. Such a shift removing the first subleading term in the superpotential is a standard manipulation in singularity theory. Rewriting the superpotential  in terms of the shifted $X$ corresponds to performing an analytic reparametrization on the space of coupling constants. The new electric coupling constants will be denoted by $\{t_i\}$. The explicit coordinate transformation from $\{s_i\}$ to $\{t_i\}$ will appear below.

To find the classical moduli space of the theory one should first impose the D-flatness equations and mod out by gauge transformations. This is equivalent to moding out the space of chiral fields by $SU(N_c)_{\IC}$. Using this symmetry we can diagonalize $X$ and then impose the equation of motion from . The eigenvalues of $X$ must satisfy $W^\prime(x)=0$. For generic couplings $\{s_i\}$ there are $k$ distinct solutions $c_1,\cdots, c_k$. Vacua of the gauge theory are labeled by sequences of integers $(r_1, r_2, \cdots, r_k)$; $r_l$ is the number of eigenvalues of the matrix $\langle X\rangle$ residing in the $l$’th minimum of the potential. The gauge group is broken by the $X$ expectation value: At low energies the theory describes $k$ decoupled supersymmetric QCD (SQCD) systems with gauge groups $SU(r_l)$ and gauged baryon number. For a given choice of $\{r_l\}$ there is a moduli space of vacua associated with giving expectation values to the quarks. Therefore, the classical moduli space consists of many disconnected components parametrized by the $\{r_i\}$. One can fine tune the couplings such that some of the eigenvalues $\{c_i\}$ coincide. The resulting multicritical behavior will be analyzed in section [*2*]{}.

Quantum mechanically, not all vacua are stable. If, for example, we pick a classical vacuum  with one or more of the $r_l>N_f$, the resulting SQCD theory is destabilized by quantum effects . Hence, such classical vacua are not present in the quantum moduli space. Similarly, some of the vacua in the multicritical case are destabilized and are removed from the quantum moduli space.

The classical chiral ring can be thought of as the ring generated by the operators $\Tr X^j$ subject to two classes of constraints. The first comes from the equation of motion which follows from the superpotential , $W^\prime=0$. The second comes from the characteristic polynomial of $X$, and is an example of a relation following from the composite nature of gauge invariant operators mentioned above.

Quantum mechanically one expects on general grounds to find new relations in the chiral ring corresponding to the quantum reduction of the moduli space described above. It is in principle possible to construct these relations by requiring that imposing them has the effect of removing exactly the vacua that we know from our previous discussion should be removed, but this is very difficult in practice as well as unmotivated. Duality provides an elegant general solution to the problem, explaining why such new relations appear in the quantum chiral ring and providing a constructive way of determining them.

*The magnetic theory.*

It was shown in that in the presence of a superpotential  there exists a simple dual magnetic description. It is similar to the original electric theory but based on the gauge group $SU(\bar N_c=kN_f-N_c)$ with an adjoint field $Y$ and $N_f$ quarks $q_i$ and $\tilde q^{\tilde i}$ in the fundamental and anti-fundamental representations as well as some gauge invariant fields $(M_j)^i_{\tilde i}$ which correspond to the composite operators where the suppressed color indices are summed over (when $s_1 \not= 0$ we find it convenient to define $M_j$ in terms of $X_s$). The magnetic theory has a superpotential where the shifted field $Y_s$ is defined similarly to $X_s$ so that the coefficient of $\Tr Y_s^k$ in the magnetic superpotential vanishes. The auxiliary scale $\mu$ is needed for dimensional reasons; note that even though the fields $M_j$ are elementary in the magnetic description, the identification  implies that they are assigned scaling dimension $j+1$. Therefore, $\mu$ in  has indeed dimensions of mass. One could redefine $M_j$ by powers of $\mu, s_0$ to make their kinetic terms canonical. The numerical coefficients of the various terms in  can be calculated using flows, and will be derived below. The function $\alpha_s(t)$ will be computed too.

The magnetic superpotential  can be used to find the operator map relating the operators $\Tr Y_s^j$ to the electric ones. Differentiating the generating functional with respect to the couplings $t_{k+1-j}$ one can derive the map of the operators The function $\alpha_s$ in  is important because it contributes to the map of the operators .

Note that  describes the duality map in the coordinates on theory space described above, $\{t_i\}$. $Y_s$ is the adjoint field in those coordinates (see section [*3*]{}). The relatively simple form of the map is in fact the main motivation behind the coordinate transformation $s\to t$. In the original $s$ variables the duality map is significantly more complicated. Its precise form will be exhibited below.

The discussion of the classical and quantum moduli space and chiral ring for the theory  is essentially identical to the electric case, replacing $N_c, r_l\to \bar N_c, \bar r_l$. The classical electric and magnetic moduli spaces are different; they have different numbers of disconnected components (labeled by $\{r_i\}$, $\{\bar r_i\}$ respectively). Similarly, the classical chiral rings are different. Since the matrices $X$, $Y$ are of different size, the effects of the characteristic polynomial are different in the two cases.

Quantum mechanically, the moduli spaces agree. After removing the subspaces of moduli space corresponding to unstable vacua we find, rather remarkably, the same number of components on both sides, with the same physical properties. Some of these properties will be investigated below.

Duality suggests a natural candidate for the quantum deformation of the chiral rings in the two theories. It is very natural to add to the classical relations coming from the equation of motion and the characteristic polynomial in the electric theory an additional set of relations following from the [*magnetic*]{} characteristic polynomial using the duality map . These relations would be described as quantum strong coupling effects in the electric theory (we will see that they are trivial when the electric theory is weakly coupled), while in the magnetic language they correspond to classical ‘trivial’ relations following from compositeness of $\Tr Y^j$. Similarly, the magnetic chiral ring is modified by quantum relations obtained via the duality map from the electric characteristic polynomial. One of our goals in this paper will be to establish that the quantum modification of the chiral ring just described does indeed take place.

The detailed map between the electric theory and the magnetic theory ,  satisfies a number of non-trivial constraints:

[1.]{} The electric theory has a complicated vacuum structure which depends on the details of the superpotential . The magnetic theory should have the same vacuum structure.

[2.]{} The expectation values of the operators $\Tr X^j$ can be calculated in all the vacua of the electric theory. They should agree with the corresponding calculation done using the magnetic variables and the map of operators .

[3.]{} As various perturbations are turned on, some fields become massive and can be integrated out both in the electric and in the magnetic theories. The two resulting low energy theories are calculable. They should be dual to one another. In particular, powerful constraints arise from requiring consistency of the operator map implied by duality with various deformations.

Additional constraints on duality follow from the requirement that certain scale matching relations are consistent with all deformations. These relations are discussed in the next subsection.

It is highly non-trivial and surprising that there exists a duality transformation which satisfies such a large number of consistency checks.

An important chiral operator in the electric theory is the kinetic term of the gauge fields $W_\alpha^2$. Its coefficient is $P \log \Lambda$ where $\Lambda$ is a dynamical scale related to the gauge coupling by dimensional transmutation, and $P$ is the coefficient of the one loop beta function (in an $SU(N_c)$ gauge theory with $N_f$ flavors of quarks in the fundamental representation $P=3N_c-N_f$). Even though $W_{\alpha}$ is chiral (annihilated by $\bar D_{\dot
\alpha}$), it is usually not a chiral primary field at the IR fixed point. The reason is that the anomaly equation often relates it to another primary field $\CO$ as $W_\alpha^2=\bar D^2 \CO$. Therefore it is not in the chiral ring. Nevertheless, its coefficient in the electric Lagrangian should be related to its counterpart in the magnetic theory, $\bar P \log \bar \Lambda$ ($\bar P$ is the coefficient of the one loop beta function in the magnetic theory). In our case we will show that the relation has the form

The relation  shows in a quantitative way how one theory becomes stronger as the other becomes weaker. Its consistency with various flows will provide rather stringent quantitative tests of duality.

$\Lambda$ ($\bar\Lambda$) is usually thought of as the scale at which the behavior of the electric (magnetic) theory crosses over from being dominated by the short distance fixed point to the long distance one (the typical mass scale of the theory). This raises a number of questions; it is not clear why $\Lambda$, $\bar\Lambda$ defined in such a way should satisfy a relation like  when the theories are only equivalent in the extreme infrared. Also, this definition leaves ambiguous a numerical factor, especially in theories with more than one scale.

One can think about eq.  and the scales appearing in it purely in the extreme IR theory. Since the scaling dimensions of various operators in the infrared are not the same as in the UV (see  and discussion below), one needs a dimensionful parameter to relate the UV operators to the IR ones. That dimensionful parameter, which can be defined for example through two point functions of such operators, is $\Lambda$. Similarly, in the magnetic theory one has $\bar\Lambda$, and additional parameters such as $\mu$ . The meaning of the scale matching relation is that $\bar\Lambda$ and $\mu$ must be chosen to obey  in order for the correlation functions of the two theories to agree including normalization. At any given point in the space of theories we can absorb the scales $\Lambda$, $\bar\Lambda$, $\mu$ into the definitions of the operators, thus making the scale matching relation  seem trivial. The non-trivial content in  is its consistency with duality under all possible deformations of both theories. Indeed, we will see that consistency leads to highly non-trivial checks of duality. The situation is reminiscent of the Zamolodchikov metric in two dimensional field theory, which is trivial at any given point in the space of theories, but whose curvature carries invariant geometrical information. The scale matching also describes an invariant relation between the geometries of the spaces of electric and magnetic theories.

We plan to discuss two main issues. The first is the structure of the quantum chiral ring and moduli space, and their transformation under duality. The second is non-trivial quantitative tests of duality, which are possible because of the large number of vacua that the system possesses in general. In all these tests one uses symmetries to write down certain duality relations. This leaves in general some undetermined functions of the coupling constants. These functions can be calculated by assuming duality in some vacua of the theory. Since there are in general many more vacua (and independent tests) than unknown functions, the agreement of the resulting structures with duality in all vacua is a non-trivial check.

In section [*2*]{} we review the results of . We describe the class of theories we will study, describe the duality map, and discuss their classical and quantum chiral rings and moduli spaces of vacua. In particular we establish the existence of new quantum relations in the chiral rings of these theories corresponding to classical relations in the duals.

In section [*3*]{} we construct the detailed map of the superpotential . We describe the transformation of coordinates $\{s\}\to \{t\}$ and show that the duality map is rather simple in the $\{t\}$ coordinates. After constructing this map we change coordinates back to the physically natural ones.

In section [*4*]{} we turn to the gauge coupling constant matching relation . We show that it is preserved under the various deformations. In particular, with arbitrary coupling constants $t_i$ one can compute the effective Lagrangian of the low energy theory both in the electric and in the magnetic variables. These lead (generically) to dual pairs of supersymmetric QCD like theories whose scales are related by the appropriate scale matching relation. The couplings $s_i$ can also be fine tuned to yield multicritical infrared behavior, whose consistency with duality puts further constraints on the structure described in sections [*2, 3*]{}.

In section [*5*]{} we study the baryon operators in the theory and show that they are mapped correctly between the electric and magnetic theories. This provides additional non-trivial checks of duality and the explicit coefficients in the magnetic superpotential.

In section [*6*]{} we illustrate the general results with a few examples. In particular, we show that in some cases the quantum relations in the chiral ring lead to qualitative changes in the structure of the chiral ring. We conclude with some comments about the theory with no superpotential in section [*7*]{}. Two appendices contain a discussion of dangerously irrelevant operators and some useful identities about polynomial equations which are used in the text.

We start with a review of the results of on supersymmetric Yang-Mills theory with gauge group $G=SU(N_c)$ coupled to a single chiral matter superfield $X_\alpha^\beta$ in the adjoint representation of the gauge group, and $N_f$ flavors of fundamental representation superfields, $Q^i_\alpha$, $\tilde Q_j^\beta$; $\alpha,
\beta=1,\cdots, N_c$; $i,j=1,\cdots, N_f$. This theory is in a non-Abelian Coulomb phase for all $N_f\ge1$. Its anomaly free global symmetry is The two $SU(N_f)$ factors act by unitary transformations on $Q$, $\tilde
Q$ respectively; baryon number assigns charge $+1(-1)$ to $Q$ ($\tilde
Q$), while under the $R$ symmetries the superspace coordinates $\theta_\alpha$ are assigned charge $1$, $Q$, $\tilde Q$ charge $B_f$, and $X$ charge $B_a$; anomaly freedom implies that: Without a superpotential this model is not currently understood beyond the vicinity of $N_f\simeq 2N_c$ where perturbative techniques are reliable \[\].

One of the main points of was that the theory simplifies if we add a superpotential This superpotential corresponds, for generic $k$, to a dangerously irrelevant perturbation of the theory with $W=0$ (see Appendix A), and thus cannot be ignored despite being irrelevant near the (free) UV fixed point of the theory. This superpotential has the effect of truncating the chiral ring of the theory, imposing the constraint which follows from the equation of motion for $X$; it also removes many of the flat directions of the original theory. In addition, the superpotential  breaks one of the two $R$ symmetries in . It is useful to think of $s_0$ (and other couplings to be introduced below) as background superfields whose lowest components get expectation values \[\]. The superfield $s_0$ is then seen to transform under the $U(1)_R$ symmetries  with charge $B_0=2-(k+1)B_a$. Therefore, only the $U(1)_R$ symmetry under which the charge of the adjoint field $X$ is $B_a=2/(k+1)$ leaves the vacuum with $\langle s_0\rangle= s_0$ invariant, and corresponds to a good symmetry. The unbroken global symmetry of the model is: The matter fields $Q$, $\tilde Q$ and $X$ transform under this global symmetry as follows:

We will also be interested in more general superpotentials: $\lambda$ is a Lagrange multiplier enforcing the condition ${\rm Tr}
X=0$. The coupling constant $s_i$ has dimension $2-k+i$ and $U(1)_R$ charge $B_i=2-(k+1-i)B_a$. For non zero $\{s_i\}$  breaks both $R$ symmetries . The space of theories labeled by the $\{s_i\}$ describes rather rich dynamics. Using transformations in $SU(N_c)_{\IC}$ one can diagonalize the matrix $X$. For generic $\{s_i\}$, such that with all eigenvalues $c_i$ different from each other, the theory splits in the infrared into a set of decoupled SQCD theories. Ground states are labeled by sequences of integers $r_1\le
r_2\le\cdots\le r_k$, where $r_l$ is the number of eigenvalues of the matrix $X$ residing in the $l$’th minimum of the potential $V=|W^\prime(x)|^2$. Clearly, $\sum_{l=1}^kr_l=N_c$. $\lambda$ in  is determined by requiring that the sum of the eigenvalues (which depends on $\lambda$) vanishes,

In each vacuum $X$ has a quadratic superpotential, i.e. it is massive and can be integrated out. The gauge group is broken by the $X$ expectation value: $SU(r_l)$ is the gauge group of the $l$’th decoupled SQCD theory one finds in the infrared. The existence of a large number of vacua  (labeled by partitions of $N_c$ into $k$ or less integers) after resolving the singularity  is the key fact leading to simple but non-trivial quantitative checks of duality.

For generic $\{s_i\}$ (and fixed $\{r_i\}$) the classical infrared behavior of the theory, a decoupled set of SQCD theories with gauge groups , is insensitive to the precise values of $\{s_i\}$. Quantum mechanically , this is not the case; each of the low energy SQCD theories has a scale $\Lambda_i$, $i=1,\cdots, k$. These $k$ scales are functions of the scale $\Lambda$ of the underlying theory with the adjoint superfield, and the $k-1$ couplings $s_1, \cdots, s_{k-1}$. The functions $\Lambda_i(\Lambda, s_i)$ will be computed below (in section [*4*]{}).

A similar discussion holds if the $\{s_l\}$ are fine tuned such that some of the $c_i$ in  coincide: where $\sum_i n_i=k;\;\;m\le k;\;\; n_i\ge1$. Vacua are labeled by sequences $(r_1,\cdots, r_m)$, $\sum_i r_i=N_c$, corresponding to different ways of partitioning the $N_c$ eigenvalues among the $m$ critical points $c_i$ . In this case one finds in the infrared a decoupled set of theories with gauge groups $SU(r_l)$ and superpotentials of the form  with $k=n_l$ (see ). Therefore, the deformations  connect theories with different $k$ in  and unify them into a single framework.

It is often possible to deduce non-trivial properties of the theory  by studying the deformed theories . As a simple example , turning on small $s_i$ in  and using the fact that the resulting SQCD theories have stable vacua iff all $r_i\le N_f$ one deduces that the theory  has a vacuum iff We will see other examples below.

The main result of was that the strongly coupled infrared physics of theory  can be studied using a dual description in terms of a “magnetic” supersymmetric gauge theory with gauge group $\bar G=SU(\bar N_c)$, $\bar N_c=kN_f-N_c$, and the following matter content: $N_f$ flavors of (dual) quarks, $q_{\bar i}^{\bar\alpha}$, $\tilde q^{\bar j}_{\bar \beta}$, an adjoint superfield $Y_{\bar\alpha}^{\bar\beta}$, and gauge singlets $(M_j)_{\tilde i}^i$ representing the generalized mesons, of the original, “electric” theory. The mesons $M_j$ have in the magnetic theory standard kinetic terms $\int d^4\theta
M_j^\dagger M_j$, rescaled by powers of $s_0$, $\mu$. The magnetic superpotential is: The auxiliary scale $\mu$ (mentioned in the introduction) is needed for dimensional reasons . In the next sections we will see that it is very natural to normalize $Y$ such that $\bar s_0=-s_0$; this choice leads to all the coefficients in the sum over $j$ in  being $1$, as indicated.

The transformation properties of the magnetic matter fields, $q$, $\tilde q$, $Y$ and $M_j$ under the global symmetries are:

The auxiliary scale $\mu$ in  is actually not an independent parameter. The scale of the electric theory $\Lambda$, that of the magnetic theory $\bar \Lambda$, and $\mu$ satisfy the scale matching relation described in the introduction: It is easy to check that  is invariant under all global symmetries including those under which $s_0$, $\mu$, $\Lambda$ transform. It is in fact uniquely fixed by these symmetries. As in , the scale matching condition  implies that when the electric theory is weakly coupled the magnetic one is strongly coupled, and vice versa. Differentiating the actions with respect to $\Lambda$ holding $s_0$ and $\mu$ fixed, the electric gauge field strength $W_\alpha^2$ is related to the magnetic one $\bar W_\alpha^2$ by the relation: It is highly non-trivial, and will be shown in section [*4*]{}, that the scale matching relation  which is completely determined by global symmetries, is consistent with all possible deformations of the theory, such as turning on masses for the quarks $Q^i$, giving expectation values to the mesons $M_j$ , and deforming the theory to non-zero $s_i$ . The numerical constant $C$ in  is also fixed by the flows, $C=1$, and its value and in particular its independence of $N_f$, $N_c$, $k$ lead to additional tests of duality.

We view the consistency of  with deformations as strong evidence for the validity of the electric-magnetic duality hypothesis of .

When the electric superpotential is deformed to  the electric theory develops a large number of vacua labeled by ($r_1$, $\cdots$, $r_m$), with $\sum r_i=N_c$ (see the discussion after ). The $SU(r_l)$ theory contains an adjoint field with superpotential $W\simeq
X^{n_l+1}$. The magnetic theory has a similar structure obtained by analyzing vacua of the superpotential . Its vacua are labeled by integers ($\bar r_1, \cdots, \bar r_m$), $\sum \bar r_i=\bar N_c$. Classically, the moduli spaces do not agree. However when we include the quantum stability constraints  we find a one to one correspondence of the quantum vacua in the two theories. The vacuum map is non trivial:

A class of operators that was discussed in , and will be revisited in section [*5*]{}, are the baryon-like operators: where $\sum_i n_i=N_c$. The duality map relating the baryons $B$ , and the analogously defined dual baryons $\bar B$ is: where $\bar n_l=N_f-n_{k+1-l}$, $l=1,\cdots, k$, and $P$ is a phase that will be discussed in section [*5*]{}. The form of  is determined by global symmetries, while the overall numerical constant is uniquely fixed by the flows, and it is non-trivial that it is consistent with the deformations, such as . This compatibility will be established in section [*5*]{}.

In addition to the mesons  and baryons , the chiral ring of the electric (magnetic) theory contains the generators ${\rm Tr} X^j$ (${\rm Tr} Y^j$), $j=2,\cdots, k$. Global symmetries require (up to terms depending on the quarks which we will determine below) that with $f_j$ calculable numerical constants. In the next section we will calculate $f_j$ and describe the generalization of  to the deformed theories .

The duality map described in this section may be used to study the physics of the theory  at strong coupling using a weak coupling description in terms of magnetic variables. For example, one finds that this theory may exhibit at strong coupling a “free magnetic phase,” with the magnetic variables governed by a non-asymptotically free gauge theory. Some additional features are described in .

The classical chiral ring of the electric theory is generated by the generalized mesons $M_j$ , baryons  and traces of the adjoint matrix, $\Tr X^j$. The latter, which we will focus on here satisfy classically two sets of constraints, following from the equation of motion $W^\prime=0$,  and from the characteristic polynomial. Defining $f(p)\equiv\det(p-X)$, the constraint is $f(X)=0$. The two together give rise to a set of algebraic equations for the generators of the classical chiral ring. The solutions of these equations describe the ring of functions on the classical moduli space.

One can repeat the same construction for the magnetic theory. The discussion proceeds in complete parallel to that of the moduli space. Classically the chiral rings of the electric and magnetic theories cannot be the same: while the equation of motion $W^\prime=0$ gives similar relations for the generators, the characteristic polynomials give different sets of constraints, since the sizes of the matrices $X$ and $Y$ are different. However, the discussion of the correspondence of the quantum moduli spaces points to the correct modification of the chiral rings in the quantum theory needed to restore duality and the relation between the (quantum) chiral rings and moduli spaces. For generic values of the couplings $\{s_i\}$ the quantum restriction on the $\{r_i\}$, $r_l\le n_l N_f$ turns under duality  to the trivial condition $\bar r_l\ge 0$ (and vice versa). This means that if we add the [*classical*]{} characteristic polynomial relations on the magnetic chiral ring as [*quantum*]{} relations in the electric theory, we are guaranteed that they will have precisely the right effect on the electric moduli space, eliminating the unstable electric vacua and leaving all other parts of the electric moduli space unchanged.

To summarize, we have the following structure of the electric and magnetic chiral rings. Both are generated by the operators $\Tr X^j$ (or $\Tr Y^j$ – the two sets of operators are related by the transformation  which we will make more explicit below) satisfying three sets of constraints:

[1.]{} The equation of motion, $W^\prime=0$.

[2.]{} The vanishing of the electric characteristic polynomial, $f_{\rm el}=0$.

[3.]{} The vanishing of the magnetic characteristic polynomial, $f_{\rm mag}=0$.

The first set of relations appears classically in both the electric and magnetic theories and gives similar constraints in both. The second set of relations is classical in the electric theory (and is due to the compositeness of $\Tr X^j$), but it is a non-trivial quantum effect in the magnetic theory. The third is classical in the magnetic theory and quantum in the electric one.

To complete the discussion of the previous section we must construct the duality map taking the electric theory  to the magnetic one  when the electric theory is deformed to . One expects the magnetic superpotential  to be deformed as well, and in this section we will discuss the detailed way in which this happens. We will start with a discussion of the deformation of the first term on the r.h.s. of , the superpotential for $Y$, and then turn to the second term, proportional to $M_j$. In the process we will verify all the numerical coefficients in  and learn some qualitative things about the duality map. We start with a discussion of the $Y$ superpotential.

The electric superpotential  describes a space of theories parametrized by the couplings $s_i$. On general grounds one expects a magnetic superpotential $\bar s_i=\bar s_i(s)$ are the magnetic coupling constants and $\alpha(s)$ is a constant. The purpose of this section is to find $\bar
s(s)$ and $\alpha(s)$.

In complete analogy with the discussion of the electric theory in section [*2*]{}, the magnetic theory  exhibits for generic $\bar
s_i$ a large number of vacua, parametrized by integers $\bar r_l$ corresponding to the number of eigenvalues of the matrix $Y$ with the value $\bar c_l$, the $l$’th minimum of the magnetic superpotential $|\bar W^\prime|^2$. The $\{ \bar c_l\}$ are defined by a magnetic analogue of . Clearly $\sum_l\bar r_l= \bar N_c=kN_f-N_c$. The low energy magnetic theory is a direct product of decoupled copies of SQCD with $N_f$ flavors of quarks, with the gauge group broken according to:

The picture proposed in  was that the original duality between and  reduces for the deformed theories ,  to a direct product of the SQCD dualities of  for the separate factors in , . That means that the magnetic multiplicities $(\bar r_1,\cdots,
\bar r_k)$ are related to the electric ones via the SQCD duality relation (compare to ): The fact that  is a one to one map of the sets of vacua of the electric and magnetic theories follows from results of on vacuum stability in SQCD.

Furthermore, it was argued in  that when two or more of the critical points of $W$ coincide, as in , the same number of critical points of $\bar W$ should coincide. If, using the notation of equation , the order of a critical point $c_i$ (and therefore that of $\bar c_i$ as well) is $n_i$, the degeneracies $r_i$ and $\bar r_i$ of this critical point in the electric and magnetic theories are related by The duality of section [*2*]{} induces in this case a duality of a similar kind, between an electric theory with gauge group $SU(r_i)$ and a superpotential $\Tr X^{n_i+1}$, and a magnetic one with gauge group $SU(\bar r_i)=SU(n_iN_f-r_i)$ with a similar superpotential.

For the above scenario to be realized, the electric and magnetic superpotentials must be closely related. In particular, the fact that whenever any number of critical points $c_i$ coincide, the same number of dual critical points $\bar c_i$ must coincide as well is a very strong constraint on the dual couplings $\bar s_i(s)$. A naive guess for the solution would be a proportionality relation between the electric and magnetic couplings, with $c$ a constant. Throughout this paper we will be using the convention which defines the normalization of $Y$ relative to $X$. This convention would set $c=-1$. Eq.  implies that $\bar c_i=c_i$ and automatically satisfies the degeneration of singularities requirement described above. However, the mapping $\bar s_i(s)$ cannot (in general) be as simple as  because of the non-trivial mapping of electric to magnetic multiplicities, . Indeed, in a vacuum with given $\{r_l\}$, the tracelessness of $X$ implies that $\sum_{l=1}^k
c_lr_l=0$, whereas, assuming $\bar c_i=c_i$ and using , the tracelessness of $Y$ is the condition $\sum_{l=1}^k c_l(N_f-r_l)=0$. The two are incompatible unless $s_1$ in  vanishes. For non zero $s_1$ we conclude that the mapping $\bar s(s)$ must be non trivial; we will construct it below.

The duality map relating $\Tr X^j$ to $\Tr Y^l$ is closely related to the mapping $\bar s(s)$. Define the free energy of the model as where $s_i$ are background chiral superfields. Then, correlation functions of the operators ${\rm Tr} X^j$ are given by derivatives of the free energy $F$ with respect to the superfields $s_i$: and similarly in the magnetic theory, in terms of the dual free energy $\bar F$: where duality implies Since $\Tr X^j$, $\Tr Y^j$ are tangent vectors to the space of theories, ,  we find that the electric and magnetic operators are related by: Taking the expectation values of both sides of eq.  we find that the mapping $\bar s_i(s)$ must satisfy a constraint in addition to the previously described one on the degeneration of eigenvalues. The expectation values of the left and right hand sides of  which depend in a highly non-trivial way on the particular vacuum chosen (the set of $r_i$) must satisfy a relation that is [*independent*]{} of the particular vacuum chosen. Clearly, the special form of the mapping  and the large number of vacua in which it should hold presents a formidable constraint on their form.

It is convenient to think of $X$, $Y$ as general $U(N)$ matrices, with a dynamical Lagrange multiplier $\lambda$ ($\bar\lambda$) imposing the tracelessness of $X$ ($Y$). Consider the electric theory, described by . It is convenient to define a shifted $X$, denoted by $X_s$ as: with The shift ,  cancels the first subleading term in $W$, leading to the superpotential: where $W_s(X_s)\equiv W(X)$, Note that $t_0=s_0$ and $t_1=0$. The transformation , corresponds to an analytic coordinate transformation on the space of theories . A similar transformation can be performed on , replacing $Y$ by $Y_s$ and $\bar\lambda$ with $\bar\lambda_s$. The $k-1$ independent couplings $s_i$, $i=1,\cdots, k-1$ are replaced by the $k-2$ couplings $t_i$, $i=2,\cdots, k-1$, and $b$ . In the $X_s$ variables the coefficient of the first subleading term $\Tr X_s^k$ in the superpotential  always vanishes. The information about that coefficient in the original description  is in $b$ . In a sense, the transformation ,  allowed us to trade the operator $\Tr X^k$ for the operator $\lambda$, which is possible since by the $X$ equation of motion (ignoring $D$ terms as usual)

Since in the $t_i$, $b$ parametrization of the space of theories the first subleading term in $W$ vanishes by construction, it is natural to postulate that the duality map for the eigenvalues of $X_s$ , $a_i$ defined analogously to  is simply With the convention  this means that (compare to ): The second equation in  is an operator identity; it can be thought of as arising from the coupling relations: using . Equations ,  specify the mapping $\bar s_i(s)$ completely.

We now determine $\alpha_s$. Using ,  we find the operator equation ($i=2,\cdots, k-1$). The expectation values of the l.h.s of  in a vacuum specified by a set of $\{r_l\}$ is while in the magnetic theory: where (see also Appendix B): Comparing  and  we see that for consistency of the picture advocated above we must be able to write the $u_j$ as: Indeed, one can check that eq.  is satisfied with $\alpha_s$ given by: The proof of the fact that  satisfies  uses the following property of the $u_n$ : This and other properties of the $u_n$ are reviewed in Appendix B.

Summarizing, the main results of this subsection are the mapping of the couplings in the electric superpotential $t_i$, $b$ defined by ,  to their magnetic counterparts, given by equations , . This simple transformation law induces a transformation  for the operators $\Tr X_s^j$ with $j=2, \cdots, k-1$. The operator $\Tr X^k$ that was conjugate to the coupling $s_1$ is eliminated in favor of the Lagrange multiplier $\lambda$ which also has a simple transformation given by the second equation in .

Of course, the simple transformation laws described in this subsection become more complicated when we translate them back to the original coordinates $s_i$.

After the discussion of the previous subsection it is not difficult to describe the duality map for the perturbed superpotential  in the original coordinates $s_i$. The main point is that while as we shall see the mapping $\bar s_i(s)$ (and therefore the operator map ) is somewhat complicated in this case, the mapping of the eigenvalues remains simple. Indeed, using the simple relation between the eigenvalues of $X_s$ and of $Y_s$  and the relation between $X$ and $Y$ and their shifted counterparts $X_s$ and $Y_s$ ,  we conclude that the mapping of the eigenvalues of $X$, $Y$, ($c_i$, $\bar
c_i$) is: Similarly, we derive the map of the coupling constants and the operators (using ):

The function $\alpha$ defined in  can be expressed in terms of $\beta$ and $\alpha_s$ defined in ,  as:

Interestingly, when all $s_i$ except $s_0$ vanish (i.e.the superpotential is ) there is nevertheless a non-trivial operator matching following from equations , : From eq.  one can read off the values of the coefficients $c_j$ of section [*2*]{} .

So far our discussion focused on the way the first term in the magnetic superpotential  is deformed as we deform the electric superpotential . In this subsection we will use these results to determine the deformation of the second term in $W_{\rm mag}$. We will work in the parametrization of the space of theories described in subsection [*3.2*]{}.

When one turns on non-vanishing couplings $t_i$ in , the magnetic superpotential  can in principle receive contributions proportional to $t_j$, $t_jt_l$, etc, consistently with the global symmetries. The way to fix all these terms is to require that duality act in the way described after eq. . Namely, for generic $t_i$ we expect the magnetic theory to split into an approximately [*decoupled*]{} set of SQCD theories that are dual to the different decoupled factors in .

This requirement of decoupling is rather non-trivial since the second term on the r.h.s. of  tends to couple the different $SU(\bar r_i)$ theories. Indeed, denote the first $r_1$ components (in color) of the electric quarks $Q$ by $Q_1$, the next $r_2$ by $Q_2$ and so on. Similarly, the first $\bar r_1$ components of $q$ are denoted by $q_1$, the next $\bar r_2$ by $q_2$, etc.. Then expanding around $\vev{X_s}$ we find (recall that for generic $s_i$ we defined $M_j=\tilde Q X_s^{j-1}Q$) $\tilde Q_l Q_l$ are the mesons of the $l$’th electric SQCD theory with gauge group $SU(r_l)$. The color $SU(r_l)$ indices are as usual suppressed and summed over. Similarly we write: where as in [*3.2*]{} we denote the shifted $Y$ field appropriate for the $\bar t_i$ coordinate system on theory space by $Y_s$. In the above formula we used the fact that in the coordinates $t_i$ the duality map is trivial, $\bar a_i=a_i$ .

The second term in $W_{\rm mag}$  has to be corrected in such a way that the different SQCD theories do not couple – there should not be any cross terms coupling $\tilde q_jq_j$, $\tilde Q_i Q_i$ with $i\not
=j$. The unique solution to this requirement is All the numerical coefficients in the second term on the r.h.s. of  are fixed by the requirement that when we substitute , into it, cross terms such as $\tilde Q_1 Q_1\tilde q_2 q_2$ vanish. Indeed, the coefficient of the above operator is proportional to: which vanishes because $a_1$, $a_2$ are roots of $W^\prime$ (see ). Thus, with the choice of couplings in  the magnetic theory reduces for generic $t_i$ into [*decoupled*]{} SQCD theories as required by duality.

The form  which at this stage of the discussion is completely fixed, must satisfy additional consistency conditions. The simplest of these involves getting the right behavior when some of the roots $a_i$ coincide. For example, if $a_1=a_2$ and all other $a_i$ are different,  is replaced by: with $X_{s1}$ the fluctuating deviation of the shifted adjoint field of $SU(r_1)$ from its v.e.v. Similarly,  is replaced by: Using ,  in  we find the correct superpotential for decoupled SQCD and an $SU(\bar r_1)$ sector with $k=2$ as required by the duality. All these checks give the expected results. More generally, when some eigenvalues $a_i$ coincide as in , one finds that the terms that must vanish are always proportional to derivatives of $W$ at $a_i$ which vanish.

Additional consistency conditions on the detailed form of  will appear in the next section. We see again that consistency of the deformed theory with duality fixes uniquely coefficients in the superpotential of the unperturbed theory .

It is also useful to note at this point that the $t_l$ dependence of the magnetic superpotential  implies a correction to the dual of $\Tr
X_s^j$ given in . Indeed, differentiating the free energies of the electric and magnetic theories (see ,  – ) one finds: ($i=2,\cdots, k-1$). The second term on the r.h.s. mixes the operators $\Tr X_s^j$ with the generalized magnetic mesons.

In section [*2*]{} we mentioned the relation  between the scale of the electric theory, $\Lambda$, that of the magnetic theory, $\bar
\Lambda$, and the dimensionful parameter $\mu$ entering , : It is interesting to check whether this relation is consistent with the various deformations that the model possesses. These include adding terms proportional to $M_j$  to the superpotential, and turning on $s_i$ . In this section we will check the compatibility of  with two kinds of flows:

[1.]{} Adding a mass term to one of the flavors (e.g. $m(M_1)_{N_f}^{N_f}$).

[2.]{} The general $s_i$ perturbations.

In doing that we should stress that the relation  does not depend on the masses or $s_i$. This follows from the symmetries.

We will see that, remarkably, there is complete detailed agreement of the two kinds of flows with . We start with a summary of the conventions we will be using.

Before discussing the flows we must specify the threshold corrections relating the scales of the theory when massive particles are integrated out. When we integrate out a massive fundamental chiral superfield $Q$ with mass $m$, the $SU(N_c)$ gauge theory with an adjoint and $N_f$ flavors goes in the infrared to one with an adjoint and $N_f-1$ flavors, and: The analogous relation in SQCD is: When we integrate out a chiral adjoint field $X$ of mass $m$, we have: where the scale on the l.h.s. corresponds to the theory with an adjoint, and the scale on the r.h.s to SQCD. Finally, when we integrate out a massive vector superfield of mass $m$ in the fundamental representation of the gauge group (i.e. when part of the gauge group is Higgsed) we have in the theory with an adjoint and $N_f$ fundamentals: whereas in SQCD: Notice that in the theory with the adjoint  $N_f$ does not decrease under Higgsing. This is because one massless chiral superfield is eaten by the massive gauge field but another appears from decomposing the adjoint of $SU(N_c)$ w.r.t. $SU(N_c-1)$.

Out of  –  only three definitions are independent (e.g. , , ). With these conventions the scale matching condition in SQCD is : Here $\mu$ is an auxiliary scale similar to that in . It is defined such that the magnetic superpotential in SQCD is : With the conventions in hand we next turn to examine the flows.

Consider adding to the electric theory a mass term: The theory loses a flavor in the infrared; the scales of the high and low energy theories ($\Lambda_{N_c, N_f}$ and $\Lambda_{N_c, N_f-1}$ respectively) are related by : In the magnetic theory, the superpotential  is modified to: One next needs to set the massive fields to solutions of their equations of motion and integrate them out. It is easy to see that all the fields $(M_j)_i^{N_f}$, $(M_j)_{N_f}^i$ ($i=1,\cdots, N_f$), and components of $q$, $\tilde q$, $Y$ in $k$ of the $\bar N_c$ directions are massive. The expectation values of $\tilde q^{N_f}$, $q_{N_f}$, $Y$ satisfy: Taking into account the $D$ terms (which fix the relative normalization of $\tilde q, q, Y$) and $Y$ equation of motion leads to: We can think of the effect of $m$ in  in two stages. First, the magnetic gauge group $SU(\bar N_c)$ is broken by the Higgs mechanism to $SU(\bar N_c-k)$. At this stage, $q_{N_f}$, $\tilde q^{N_f}$, $Y^\alpha_m$ $m=2,\cdots, k$, $Y_\alpha^s$ $s=1,\cdots, k-1$ ($\alpha=k+1,\cdots\bar N_c$) gain a mass and join $k$ massive vector superfields in the fundamental representation of the unbroken, $SU(\bar
N_c-k)$ gauge group. According to our convention  this generates a factor of the $k$’th power of the mass squared of the vector superfields, $\left[\left({m\mu^2\over\bar s_0}\right)^{2\over
k+1}\right]^k$.

In a second stage, $Y_1^\alpha$ and $Y_\alpha^k$ get a mass from expanding the superpotential: In  we have used the fact that in expanding $\Tr Y^{k+1}$ to leading order in $Y_1^\alpha Y^k_\alpha$ we must take these two $Y$’s next to each other; terms like $\Tr \langle Y^n\rangle Y_1 \langle
Y^m\rangle Y^k$ ($n,m\not=0$) do not contribute such mass terms. Since $Y_1^\alpha$, $Y_\alpha^k$ can be thought of as massive chiral superfields in the fundamental representation of $SU(\bar N_c-k)$ with mass $\bar s_0\left({m\mu^2\over\bar s_0}
\right)^{k-1\over k+1}$ (see , ), we use  for the scale matching.

Finally, combining the two stages we have: Using ,  we conclude that  leads to: or: which is exactly the right scale matching relation for the theory with $N_f-1$ flavors. We therefore conclude that the scale matching relation  is consistent with the mass perturbation .

An interesting element of the preceding analysis is the fact that the overall relative coefficient between the first and second terms in the magnetic superpotential  was important for quantitative agreement of the matching conditions. This relative coefficient was not fixed by the discussion in section [*3.4*]{} and we can view the analysis of this subsection as a way to determine it. We will soon see a non-trivial check of its value from the $s_i $ flows .

In this subsection we will be deforming the superpotential for $X$ ($Y$) in the electric (magnetic) theory, as in , . Recall the duality where $a_i$ are the eigenvalues of $X_s$ defined as in  and $\bar
a_i$ are similarly related to $Y_s$.

To test  in the deformed theory  we proceed in two stages. First consider generic $t_i$ such that all eigenvalues $a_i$  are distinct. Then the IR electric theory is a direct product of decoupled SQCD theories with gauge groups $SU(r_i)$ (see discussion following ). The scale of the $i$’th SQCD theory is related to that of the high energy theory by: One can think of  as following from a two step process. First we turn on an expectation value for $X_s$: where $a_1^{r_1}$ means the eigenvalue $a_1$ appears $r_1$ times, etc. This makes some vector fields massive and using  we get the factor of $\left[\prod_{j\not=i}(a_i-a_j)^{ 2r_j}\right]$ in . Then, at a second stage, we integrate out the massive adjoint field in the $SU(r_i)$ theory using . This gives rise to ${1\over
\left[W^{\prime\prime}(a_i)\right]^{r_i}}$.

We evaluate $W^{\prime\prime}(a_i)$ using  and find for $\Lambda_i$: Repeating the same arguments for the magnetic theory using  we find:

At this stage we have no independent check on ,  separately (although one will appear in the next section when we discuss baryons), but multiplying the left and right hand sides of  and  and using ,  we find which means that Recall that $\mu_i$ is defined through the magnetic superpotential in the $SU(\bar r_i)$ theory (compare to ): The scale parameters $\mu_i$ can be independently calculated by the analysis described in section [*3.4*]{}. By decomposing $M_j$, $\tilde q Y^l q$ into their $SU(r_i)$, $SU(\bar r_i)$ components ,  and evaluating the coefficient of $\tilde Q_i Q_i \tilde q_i q_i$ in the magnetic superpotential  we find: which is exactly the right value . Since the magnetic superpotential  is completely fixed by the considerations of section [*3*]{}, the agreement between  and  gives another non-trivial quantitative check of duality.

We conclude that, at least for generic $t_i$, the deformation  is consistent with the scale matching relation . This still leaves the question of whether we get a consistent picture when some $\{a_i\}$ coincide, which we briefly address in the next subsection.

Consider for simplicity the special case discussed in section [*3.4*]{}, where two of the $a_i$, say $a_1$ and $a_2$, coincide. Then instead of $k$ copies of SQCD, one gets in the IR $k-2$ copies of SQCD, corresponding to $a_3,\cdots, a_k$, and one copy of the model  with $k=2$. The discussion of the $k-2$ SQCD vacua is exactly as in the last subsection. The only new feature here is the discussion of the scale of the $SU(r_1)$ $k=2$ model corresponding to $a_1$, and its dual, the $SU(\bar r_1)$ model with $k=2$ and $\bar r_1=2N_f-r_1$. The scale $\Lambda_1$ of that low energy model is related to $\Lambda$ by: by an argument analogous to that following . Similarly:

Defining $t_0^{(l)}$ and $\mu^{(l)}$ to be the analogues of $t_0$, $\mu$ in the low energy theory, we have the scale matching relation: Multiplying  and  and using ,  we find: One can again perform an independent check of by calculating $t_0^{(l)}$, $\mu^{(l)}$ directly. Starting with $t_0^{(l)}$ we use  near $x\simeq a_1$: so that: To get $\mu^{(l)}$ we use the decompositions ,  and calculate the coefficient of $\tilde Q_1 Q_1\tilde q_1 Y_{s1} q_1$ (or equivalently $\tilde Q_1 X_{s1}Q_1\tilde q_1 q_1$). We find that Combining ,  we see that  is indeed valid and the structure of flows  is consistent with the scale matching relation .

In section [*2*]{} we proposed an exact mapping  between the baryon operators in the electric and magnetic theories. The mapping involves powers of the dimensionful couplings that are fixed by global symmetries and numerical factors that can be fixed by consistency with the various flows. In this section we will outline the checks of consistency of  with various deformations. We will see that these consistency checks lead to additional highly non-trivial checks of duality. In particular, since $\Lambda$ appears in the map , we will have a more direct check on the expressions for the scales $\Lambda_i$ ,  derived in section [*4*]{}.

There is an inherent phase ambiguity in  related to the possibility of making field redefinitions in the two theories corresponding to global symmetries of the problem. Performing a baryon number transformation in the magnetic theory introduces an arbitrary phase in  and an opposite phase in the relation between $\tilde B$ and its dual $\bar {\tilde B}$. We can use this freedom to make the phases in the mappings of $B$ and $\tilde B$ identical. This still leaves a sign ambiguity in all our formulae for mapping of $B$, $\tilde B$ separately, which disappears in the mapping of $B\tilde B$. We will not write explicitly these $\pm$ signs, but instead leave the branches of square roots appearing in $B$ and $\tilde B$ unspecified. Phase factors appearing in the matching of $B\tilde B$ have an absolute meaning and therefore will be calculated.

In the first subsection we will discuss the consistency of  when turning on masses for quarks . In the second subsection we will verify the consistency of  with deformations of the superpotential $W(X)$ .

As in section [*4*]{} we add a mass term for $Q^{N_f}$, $\tilde Q_{N_f}$ . The electric theory now has $N_f-1$ flavors. The electric baryon  splits into components with the indices $i_{p_1},j_{p_2},\cdots, z_{p_k}$ taking values between $1$ and $N_f-1$ which stay massless, and the rest that become massive. In the magnetic theory we deform the superpotential by an $mM_1$ term . The gauge group breaks from $SU(\bar N_c)$ to $SU(\bar N_c-k)$ and the number of flavors decreases by one, $N_f\to N_f-1$. One notes that:

[1.]{} It follows from  that:

[2.]{} All components of the magnetic quarks $q$, $\tilde q$ and the adjoint field $Y$ with color indices between $1$ and $k$, as well as $q^{N_f}$, $\tilde q_{N_f}$ (all color components) become massive.

[3.]{} Due to the structure of the r.h.s. of  if any one or more of the indices $i_{p_1}, \cdots, z_{p_k}$ equals $N_f$ the corresponding magnetic baryon (the r.h.s. of ) is massive, since it is not possible to saturate the color indices $\bar\alpha=1,
\cdots, k$ corresponding to broken generators except with massive quarks. This is in agreement with the behavior of the electric theory (the l.h.s. of ) as mentioned above.

[4.]{} If all indices $i_{p_1}, \cdots, z_{p_k}$ are less than $N_f$ the r.h.s. of  does lead to massless baryons in the magnetic theory. The broken color indices $\bar\alpha=1, \cdots, k$ can now be saturated by the expectation values of the $Y^jq^{N_f}$. The only way to saturate the broken color indices by the expectation values  is to replace in $\bar B$  one of each group of indices $\bar i, \cdots, \bar z$ by the appropriate expectation value in . This leads to the following transformation of the r.h.s. of  under the mass flow: The factors of $\bar n_i$ come from the number of possibilities of placing the expectation value among the $\bar n_i$ $X^{i-1}Q$ operators. The phase is due to a certain reordering of the $k$ color indices $1,\cdots, k$ that is needed to bring the baryon into standard form after symmetry breaking. A similar factor does not appear in the analogous mapping for $\tilde B$. Clearly, the only meaningful quantity is the phase appearing in the transformation of $B\tilde B$.

Inserting  into  we see that all the factors generated in the breaking magically arrange to give the form  again, with $N_f\to N_f-1$. In particular, one notes that $\bar n_i\to \bar n_i-1$, and the scale absorbs the factor of $m$ (see ). Eq.  and the analogous relation following from the analysis of the mapping  under turning on an expectation value for $M_k$ in the electric theory allow one to determine the dependence on $N_f$ and $N_c$ of the unambiguously defined phase in the mapping relation for $B\tilde B$. One finds that the phase, called $P$ in  is: As discussed above, only $P^2$, which is a sign, is meaningful.

When a general superpotential  is present the gauge group breaks generically to . Correspondingly, the baryon operator  can be expressed in terms of the baryons of the SQCD theories . Similarly, the baryon of the magnetic $SU(\bar N_c)$ theory is decomposable into the baryons of the magnetic SQCD theories . The mapping of baryons in the high energy theory, , should reduce in the deformed theory to the baryon mapping in the individual SQCD theories: Eq.  has the same sign ambiguity discussed above.

The general discussion of the reduction of  under the deformations  is unfortunately rather complicated. To get a flavor of the issues involved we will discuss here the special case of a theory with a cubic superpotential ($k=2$ in ); furthermore, we will only discuss the special baryons with $n_1=N_f$, $n_2=N_c-N_f$ (see ). The main reason for considering these baryons is that they are dual by  to the simplest magnetic baryons, those with $\bar
n_1=2N_f-N_c=\bar N_c$ and $\bar n_2=0$. Using  and the phase $P$ found in the previous subsection we expect for these baryons the following duality mapping: We will check the validity of  when a general superpotential $W(X)={s_0\over3}\Tr X^3+{s_1\over2}\Tr X^2$ is introduced. In this particular case $W$ has two critical points $a_1$, $a_2$, and the electric and magnetic color groups are broken as in , to: $SU(r_1)\times SU(r_2)\times U(1)$ and $SU(\bar r_1)\times
SU(\bar r_2)\times U(1)$ respectively. The consistency check of involves comparing two paths:

[1.]{} Decompose $B^{j_1\cdots j_{n_2}}$ in terms of the electric baryons of SQCD and then use the SQCD mapping  to obtain an expression in terms of the magnetic SQCD baryons.

[2.]{} Use  first, and then decompose the magnetic baryon into the SQCD baryons of the broken theory.

Clearly, the two expressions obtained this way should coincide for agreement with duality. The electric baryon decomposes after the breaking as: $Q_1$, $Q_2$ here are the SQCD quarks defined in section [*3.5*]{}. The non-trivial phases are due to the rearrangement of the $\epsilon$ symbol of $SU(N_c)$ in the order corresponding to a product of $\epsilon$ symbols of $SU(r_1)\times SU(r_2)$. Next we replace the electric SQCD baryons in  by their magnetic counterparts, using : and an analogous relation for the baryon constructed out of $Q_2$. The scales $\Lambda_i$, and the parameters $\mu_i$ are given by , . Another $s$ dependent phase (in addition to that in ) appears from the rearrangement of the flavor indices in the various $\epsilon$ tensors. After summing over the $i$ flavor indices the sum over $s$ can be performed producing $(a_1-a_2)^{N_f-N_c}$. Putting all the factors together we obtain: It is straightforward to see that the same result for $B^{j_1\cdots j_{n_2}}$ is obtained by decomposing the r.h.s. of eq.  into baryons of $SU(\bar r_1)$ and $SU(\bar r_2)$.

In this section we will study some of the consequences of the general phenomena discussed in previous sections in some particular cases.

We start with the general cubic superpotential,  with $k=2$. The only coupling that exists in this case is $s_1$. For generic $s_1$, $W$ has two critical points, $c_1$, $c_2$. Vacua are labeled by integers $r_1$, $r_2$ corresponding to placing $r_1$ of the eigenvalues in the first critical point, and the remaining $r_2=N_c-r_1$ in the second one. For $r_1\not=0, N_c$ the gauge group breaks to (compare to ): Eq.  and the tracelessness condition can be replaced by the two linear equations with solution

The chiral ring is generated by $\Tr X^2$, whose expectation value in a vacuum with multiplicities $(r_1,r_2)$ is: Similarly, in the magnetic theory: and Comparing to  we see that to match the $r$ dependent terms in $\langle\Tr X^2\rangle$ and $\langle\Tr Y^2\rangle$ we should choose which is a special case of . Comparing the constant terms of ,  then leads to Eq.  is a special case of . The operator relation  takes in this case the form: which should be compared to . There are two things to note here:

[1.]{} Considering the deformed theory with $s_1\not=0$ allows one to uniquely determine the numerical coefficients in the operator mapping  including the numerical constants $f_j$  which are defined in the theory with $s_1=0$.

[2.]{} While $c_i$  and $\bar c_i$  depend on $r_i$, they satisfy a simple linear relation This linear relation is a special case of .

For even $N_c$ we encounter here an example of an interesting general phenomenon which occurs whenever $N_c$ and $k$ are not relatively prime. The solution for the eigenvalues $c_1$ and $c_2$  is singular when $r_1=r_2=N_c/2$ and $s_1 \not=0$ and therefore these vacua are absent. On the other hand, for $s_1=0$ the values of $c_1$ and $c_2$ seem to be ambiguous. Indeed, in this case the superpotential has a flat direction with $\vev{X}={\rm diag}(c,...,c,-c,...,-c)$ (up to gauge transformations). The physics along this flat direction is exactly that of the “missing vacuum” with $r_1=r_2=N_c/2$. The adjoint field is massive and the gauge symmetry is broken: $SU(N_c)\to\left[SU(N_c/2)\right]^2\times U(1)$. The only difference is that the mass of the adjoint field and the scales of the two SQCD theories depend on the Higgs parameter $c$ instead of $s_1$.

How does the chiral ring look in this case? The only chiral operator in the electric theory that can be written in terms of $X$ alone using the equation of motion  is $\Tr X^2$. The electric characteristic polynomial gives rise to a relation for $X$ that depends on the parity of $N_c$. For odd $N_c$ this relation takes the form: where $b_l$ are easily calculable coefficients which depend on $s_1/s_0$. When $s_1=0$ all $b_l$ go to zero, and the relation is $(\Tr X^2)^{N_c+1\over2}=0$ in agreement with the fact that moduli space is in this case a point (setting $M_j=0$ as usual). For generic $s_1$  has $(N_c+1)/2$ solutions in one to one correspondence with the moduli space .

For even $N_c$ the relation following from the characteristic polynomial is For non zero $s_1$ there are $N_c/2$ solutions in one to one correspondence with the solutions of . In particular, the solution with $r_1=r_2$ which is singular does not appear. For $s_1=0$ the characteristic polynomial  does not provide any constraints on $\Tr X^2$, in agreement with the presence of the flat direction described above.

Quantum mechanically, we have learned that one needs to add to  (or ) another relation which is obtained from the characteristic polynomial of the magnetic theory using the operator map . For odd $N_c$, $\bar N_c=2N_f-N_c$ is odd too, and one finds a relation There are two cases to discuss:

[1.]{} $N_c>N_f>\bar N_c$. In this case the electric theory is more strongly coupled than the magnetic one. The relation  is non trivial in the electric chiral ring, reducing the number of distinct vacua from $(N_c+1)/2$ to $(\bar N_c+1)/2$. This is in perfect agreement with the counting of vacua  which satisfy $r_1, r_2\le
N_f$. The expectation values calculated from  agree with  for the appropriate vacua.

[2.]{} $\bar N_c>N_f> N_c$. The electric theory is more weakly coupled than the magnetic one. The relation  is satisfied on all electric vacua satisfying the classical relation  and therefore the electric moduli space is not modified quantum mechanically. This is in agreement with weak coupling intuition in the electric theory.

For $k=1$ (SQCD with $N_c$ colors and $N_f=N_c+1$ flavors) the theory is known to confine . The low energy degrees of freedom are the mesons $M^i_{\tilde j}$ and baryons $B_i$, $\tilde B^{\tilde j}$. The classical constraint relating these fields is lifted quantum mechanically and is replaced by the superpotential: This picture, which is due to strong coupling effects in the electric theory, is much more simply understood in the weakly coupled dual magnetic theory . The mesons are the gauge singlets that appear in the dual. The baryons $B_i$, $\tilde B^{\tilde j}$ are proportional to the magnetic quarks $q_i$, $\tilde q^j$. The first term in the superpotential  is the tree level magnetic superpotential (see e.g. ). The last term (proportional to $\det M$) is due to non-perturbative instanton effects that arise in the process of complete breaking of the magnetic gauge group.

For $k>1$ we are in a position to use duality to make predictions. The operators generating the classical chiral ring of the $SU(kN_f-1)$ gauge theory  are $M_j$ , $\Tr X^j$, $j=2,\cdots, k$, and the baryons , with $i=1,\cdots, N_f$, as well as antibaryons $\tilde B^{(j)}$. However, unlike the case $k=1$, a long distance description which includes all these fields does not satisfy ‘t Hooft anomaly matching. Duality suggests another solution to the problem. The only independent fields in a macroscopic description of the theory are $M_j$ and $B^{(k)}$, $\tilde B^{(k)}$. The baryons $B^{(k)}$, $\tilde B^{(k)}$ which we will denote by $B$, $\tilde B$ in this subsection, are mapped by duality  to the magnetic quarks. The other generators of the chiral ring either vanish or can be expressed in terms of $M_j$, $B$ and $\tilde B$. For example, from  we deduce that: where $\rho=-(-)^{{k(k-1)\over2}N_f}s_0^{-N_c-1}\Lambda^{1-(2k-1)N_c}$. Clearly the chiral ring at long distances is drastically modified from its classical structure. The superpotential of the theory can be read off : As a check note that the auxiliary scale $\mu$ present in disappears in the electric variables $M$, $B$, $\tilde B$. The meson fields $M_1$, $M_2$, $\cdots$, $M_{k-1}$ do not appear in the superpotential. In general one expects additional dangerously irrelevant terms in the superpotential  which do depend on all $M_j$ and whose effect would be to lift some flat directions and break some accidental symmetries. We have not analyzed these terms in detail.

The theory has rather different descriptions at short distances, where it is described in terms of gauge fields $W_\alpha$, and quarks $Q$, $X$, and long distances where the quarks are confined and the appropriate degrees of freedom are the mesons $M_j$ and baryons $B$, $\tilde B$. It is natural to ask what is the long distance, macroscopic description of deforming the superpotential $W$ by the $s_i$, as in . The operator map  and deformed magnetic superpotential  with $Y$ set to $0$, tell us that in terms of the low energy degrees of freedom the structure is always essentially the same. The superpotential is The combination of mesons in brackets couples to the baryon field, and the other $k-1$ combinations of generalized mesons remain free.

This is in agreement with the microscopic picture, according to which turning on a generic superpotential  leads to the appearance of $k$ distinct critical points $a_i$ as described above. But because $N_c=\sum_i r_i=kN_f-1$, there is a unique vacuum; we must assign $N_f-1$ eigenvalues to one of the critical points (e.g. $r_1=N_f-1$) and $N_f$ eigenvalues to each of the remaining $k-1$ ($r_i=N_f$ for $i\ge2$). Thus, a microscopic physicist would expect to see at low energies a set of almost decoupled SQCD theories, $k-1$ with $N_c=N_f$ and one with $N_c=N_f-1$, coupled by irrelevant interactions. According to , in the present context (remembering the gauged $U(1)^{k-1}$ – see ), each of the theories with $N_f=N_c$ gives rise to a free meson field, whereas the factor with $N_f=N_c+1$ gives rise to a meson and baryon coupled via the superpotential .

The combination of mesons in brackets in eq.  corresponds to the meson of the SQCD with $N_f=N_c+1$. The baryons $B$, $\tilde B$ correspond to the appropriate baryons of that theory. The other $k-1$ combinations of generalized mesons give the $k-1$ free mesons needed for the theories with $N_f=N_c$. Comparing to  we learn that  is missing a term proportional to $\det\left(
\sum_lt_lM_{k-l}\right)$ which is allowed by symmetries and can therefore appear much like in SQCD in the process of completely breaking the magnetic gauge group. Its coefficient in the superpotential is constrained by the symmetries to be a polynomial of the form $t_{k-1}+\cdots + t_1^{k-1}$. We leave the detailed understanding of this term for future work.

It is not difficult to repeat the analysis above for the case when the microscopic theory is deformed by  with $s_i$ fine tuned such that some of the critical points $a_i$ coincide. In fact, by requiring the consistency of the resulting superpotential with all $\{s_i\}$ deformations puts extremely strong constraints on the corrections to the superpotential  and probably (over-) determines it. It would be interesting to find this superpotential.

The theory with $N_c=kN_f-2$ is the simplest theory with a non-Abelian dual gauge group – $SU(2)$. Consider first the case $k=2$. The electric theory has gauge group $G=SU(2N_f-2)$ and superpotential: The magnetic one has gauge group $SU(2)$ and: The magnetic theory has no superpotential for $Y$ (setting other fields to their expectation values). Thus, there is a flat direction corresponding to Along it the gauge group is broken: $SU(2)\to U(1)$. $Y$ is massive except for one massless field, $\Tr Y^2$, parametrizing small fluctuations of the expectation value $a$.

Remarkably, the electric theory also has a flat direction, since $N_c$ (which is even) is divisible by $k$ ($=2$). This flat direction, which was discussed in [*6.1*]{} is described by: It too breaks the (electric) gauge symmetry: $X$ is massive except for one massless field, $\Tr X^2$, parametrizing small fluctuations of $b$. The non-Abelian dynamics in  is SQCD with $N_f=N_c+1$. We expect two sets of mesons and baryons coupled by the superpotential . Substituting  in  we indeed find the required structure. Linear combinations of the two meson fields $M_1$, $M_2$ couple to the two components of $q,
\tilde q$ which serve as baryons. The couplings look like the first term in . We are again not analyzing the $\det M$ terms due to instantons that must be added to the magnetic superpotential  in the weakly coupled magnetic theory. The $U(1)$ dynamics also agrees and the massless field $\Tr X^2$ corresponds to $\Tr Y^2$ completing the duality map.

At the origin ($a=b=0$ in , ) we find a strongly coupled dual of the $SU(2)$ theory with $W=0$.

For even $k>2$ the structure is similar but richer. Theory space has many components depending on the number of distinct critical points. Consider, for example, the case when all $k$ critical points are distinct. Since we want the magnetic superpotential for $Y$ to vanish we add only odd powers of $X$ to the electric superpotential  (i.e.all $s_{2l+1}$ vanish). Therefore, the critical points appear in pairs, $\pm a_1$, $\pm a_2$, $\cdots$, $\pm a_{k\over2}$. There is a unique vacuum with The magnetic $SU(2)$ gauge theory is coupled to $k$ singlet fields $M_j$, $j=1,\cdots, k$ and $$W_{\rm mag}=\sum_{j=1}^k M_j\tilde q Y^{k-j} q.$$ There is still a flat direction , along which $k-2$ combinations of $M_j$ do not couple (in the IR at the origin of moduli space $\langle M_j\rangle=0$), while two combine as before with $q, \tilde q
\to B, \tilde B$ to give the right structure for two SQCD theories with $N_f =N_c+1$. The non-renormalizable part of the superpotential again remains to be analyzed. It is not hard to repeat these considerations for the case when some eigenvalues $a_i$ coincide. We leave the details to the reader.

It is appropriate to end this section by returning to the truncation of the chiral ring, and explain why the quantum chiral ring is smaller than the classical one. In the example discussed here, in the quantum chiral ring there are relations: which are absent classically.

Consider first the classical theory where as usual we deform $W$ to resolve the singularity. One way of seeing that $\Tr X^j$, $j=2,\cdots, k$ are all independent classically is to note that the theory with generic $\{s_i\}$  has many classical vacua in which $\langle \Tr X^j \rangle$ are all different. There are no relations among the $\langle\Tr X^j\rangle$ that hold uniformly in all vacua. The key point is that as discussed previously many of the classical vacua are unstable and disappear in the quantum theory, due to the $N_f\ge
N_c$ constraint in the low energy SQCD theories (see section [*2*]{}). As we saw, in the case $N_c=kN_f-2$ analyzed here most of the vacua disappear, leaving behind a unique vacuum (for generic $s_i$). Relations among the operators need to hold only in the quantum vacua, and therefore many more exist in general quantum mechanically than classically. One can easily convince oneself that the relations  in particular are valid in all the vacua that are stable quantum mechanically and break down in vacua that are unstable. Therefore they only exist in the quantum theory and not in the classical one.

The $SU(N_c)$ theory with an adjoint $X$, $N_f$ fundamentals $Q$ and $N_f$ anti-fundamentals $\tilde Q$ without a tree level superpotential is very interesting. Unfortunately, our understanding of this theory is very limited. In the previous sections we followed and added the tree level superpotential ${1 \over k+1} s_0 X^{k+1}$ to simplify the analysis. In this section we will point out a few observations about the theory without the superpotential ($s_0=0$).

No dynamically generated superpotential can lift the classical flat directions. To see that, note that at a generic point in the moduli space the gauge group is broken to an Abelian subgroup (or completely broken). Therefore, instanton methods should be reliable. Repeating the analysis of , it is easy to see that instantons have too many fermion zero modes to generate a superpotential. Therefore, no superpotential can be generated and the theory has a moduli space of stable vacua for any $N_f$. The same conclusion can be reached by examining the symmetries and the way they restrict a dynamically generated superpotential.

Furthermore, the perturbations of this theory by various tree level superpotentials (see below) relate it to other theories where the moduli space is understood. This leads to further constraints on the quantum moduli space and its singularities. The conclusion is that the quantum moduli space is identical to the classical one. The only singularities are at points where classically the unbroken gauge symmetry is enhanced. The most singular point is at the origin. This does not mean that the physical interpretation of the singularities in the quantum theory is as in the classical theory.

When $N_f \roughly< 2N_c$ a simple two loop calculation similar to that in  reveals a non-trivial fixed point of the beta function. This calculation can be justified rigorously at large $N_c$. Therefore, the origin is in a non-Abelian Coulomb phase. The physics at that point is similar to the physics seen in perturbation theory, describing interacting quarks and gluons.

The dimensions of the chiral operators are constrained by the superconformal algebra to satisfy $D={3 \over 2 }R$ where $R$ is the charge of the $U(1)_R$ symmetry in the superconformal algebra. In simpler examples this $U(1)_R$ symmetry is easily identified . In our case, it is ambiguous. Anomaly freedom constrains the $R$ charge of $Q$ and $\tilde Q$, $B_f$ and the $R$ charge of $X$, $B_a$ to satisfy  $N_f B_f+N_c B_a=N_f$. For $\epsilon={2N_c-N_f \over 2N_c} \ll 1 $ the fixed point is visible in perturbation theory with the conclusion $B_a={2\over 3}(1- \epsilon {N_c^2\over 2 N_c^2-1} + \CO(\epsilon^2))$; $B_f={2\over 3}(1- {\epsilon \over 2}{N_c^2-1 \over 2N_c^2-1} +
\CO(\epsilon^2))$.

As $N_f$ is reduced, the fixed point becomes more strongly coupled and the dimensions of the operators become smaller. Since the dimensions of the the spin zero fields cannot be smaller than one \[\], at some point this description must be modified. We would expect, by analogy with , that these fields become free and decouple. Then, the $U(1)_R$ in the superconformal algebra is an accidental symmetry which assigns $R={2 \over 3}$ to these fields. This observation might suggest that all $u_j=\Tr
X^j$ with $j=2,...,N_c$ should be included as elementary fields in a dual description.

A useful tool in analyzing the dynamics of supersymmetric theories is the confining phase superpotential. It is obtained by coupling the generators of the chiral ring to external sources and computing the effective action for these sources. Upon a Legendre transform this leads to an effective action for these composites which gives a good description of the confining phase of the theory . This procedure was used in  to study the $N_c=2$, $N_f=1$ theory. The authors of \[\] studied the $N_c=2$ problem for larger values of $N_f$ and gave partial answers for larger $N_c$. The massless modes at the generic point in the moduli space are among these fields and therefore this effective superpotential gives a good description of the theory on the moduli space. The effective superpotential derived this way exhibits a singularity at the origin for every value of $N_c$ and $N_f$. This means that at the origin more degrees of freedom are needed. This fact is in accord with our interpretation of the origin as being in a non-Abelian Coulomb phase. However, we could not find a useful description of the theory at the origin which gives rise to the singularity in these effective superpotentials.

The theory with a tree level superpotential $\lambda_i^{\tilde i}
\tilde Q_{\tilde i} X Q^i= \Tr \lambda M_2$ can be analyzed easily. For $\lambda_i^{\tilde i}= \delta_i^{\tilde i}$ the theory becomes $N=2$ supersymmetric. This theory for $N_c=2$ was analyzed in  and for larger values of $N_c$ in . The moduli space of the theory has a Coulomb branch which has only massless photons at generic points. At special singular points on the moduli space there are more massless particles: massless monopoles, massless dyons, massless gluons and quarks, and even points with interacting non-trivial $N=2$ superconformal field theories . More quantitatively, this branch of the theory is described in terms of a hyperelliptic curve. The characteristic scale on this Coulomb branch is the only dimensionful parameter in the theory $\Lambda$ which appears as a parameter in the curve.

It is easy to extend the curve away from this $N=2$ theory; i.e. for arbitrary values of $\lambda_i^{\tilde i} \not= \delta_i^{\tilde i}$. As in using the symmetries of the theory, this is achieved by replacing every factor of $\Lambda^{2N_c-N_f} $ in the curve by $(\det \lambda)
\Lambda^{2N_c-N_f}$. Therefore, as $\lambda \rightarrow 0$, all the features on the Coulomb branch approach the origin (more precisely, they approach points where classically there is an unbroken non-Abelian gauge symmetry). Conversely, by turning on $\lambda$ the singularity at the origin splits to several singularities with various massless particles. Since these particles are not all local with respect to one another, there is no local Lagrangian which includes all of them. Therefore, it is impossible to write a local field theory which describes the deformation by $\lambda$ along the entire Coulomb branch in weak coupling.

A similar situation was encountered in $SO(N_c)$ gauge theories . There the theory at the origin was given several different dual descriptions. Each gave a weak coupling description of another deformation or another region of the moduli space. An attempt to imitate this procedure here will necessarily lead to a very large number of dual theories to describe the different phenomena in the Coulomb branch.

This is the theory we studied in this paper. Removing this perturbation by letting $s_0$ go to zero is a singular operation as it changes the asymptotic behavior of the potential. This can also be seen from the matching equation $\Lambda^{2N_c-N_f} \bar \Lambda^{2\bar N_c-N_f} =
\left({\mu\over s_0}\right)^{2N_f} $ which becomes singular as $s_0$ goes to zero. Alternatively, we might attempt to remove this perturbation by letting $k$ go to infinity.

As $k$ becomes large, the operator $\Tr X^{k+1}$ becomes irrelevant at the long distance theory at the origin and therefore it does not affect the dynamics. As we noted above, this operator is dangerously irrelevant and cannot be ignored. However, as $k$ goes to infinity the potential it leads to becomes very flat and it is it reasonable that the theory without a superpotential is achieved.

For large $k$ the dual gauge group $SU(\bar N_c= kN_f-N_c)$ becomes large. This theory is strongly coupled and therefore one might think that it does not lead to a useful dual description. However, as we saw in the previous sections, this theory gives a weak coupling description of some of the deformations. Therefore, if there is a unique good dual theory at the origin, it should include this $SU(\bar N_c)$ for arbitrarily large $\bar N_c$. Such an $SU( \infty)$ theory is expected to behave like a string theory. Therefore, we might speculate that the dual theory at the origin is not a field theory but a string theory. Perhaps, if this is indeed the case, there will not be a need for a large number of dual descriptions as suggested by the structure of the Coulomb phase.

[**Acknowledgments**]{}

We would like to thank T. Banks, K. Intriligator, S. Shenker, and especially E. Witten for many helpful discussions. This work was supported in part by DOE grant \#DE-FG05-90ER40559, BSF grant number 5360/2, the Minerva foundation and a DOE OJI grant. A.S. would like to thank the hospitality of the Enrico Fermi Institute where part of this work was performed. D.K. thanks the Weizmann Institute, Aspen Physics Center and Department of Physics at Rutgers University for hospitality during the course of this work.

A quantum field theorist might wonder about the presence of high order polynomials such as  in the superpotential. These are non-renormalizable interactions which seem irrelevant for the long distance behavior of the theory. How is it then that they affect the physical results? In order to answer this question we should review the notion of a dangerously irrelevant (DI) operator in the theory of the renormalization group.

Consider deformations of a fixed point of the renormalization group by relevant operators or along flat directions of the potential. It might be that an irrelevant operator at the original fixed point becomes relevant after the deformation. Such an operator is called dangerously irrelevant; ignoring its presence will lead to incorrect results.

An example of a dangerously irrelevant operator is the gauge superfield $W_\alpha W^\alpha$ in a non asymptotically free gauge theory (for definiteness one may think of supersymmetric QCD with quarks in the fundamental representation of the gauge group). In such a situation the gauge coupling flows to zero at long distance; hence the operator $W_\alpha W^\alpha$ is irrelevant. Nevertheless, it is clearly important to keep the gauge coupling when describing the long distance behavior of gauge theories. If, for example, we turn on quark masses, the number of light quarks may fall below the asymptotic freedom bound and therefore the gauge coupling becomes relevant. Thus, $W_\alpha^2$ is relevant in part of theory space and irrelevant in other parts. When it is irrelevant it is referred to as being dangerously irrelevant.

The operator $\Tr X^{k+1}$ behaves in a very similar way. At the gaussian (UV) fixed point it has (for $k>2$) dimension larger than three. However, if we turn on the gauge coupling $g$, $\Tr X^{k+1}$ develops for sufficiently small $N_f$ an anomalous dimension which can make it relevant. Hence, while this operator is irrelevant near the gaussian UV fixed point, it too is a dangerously irrelevant operator.

Actually, $\Tr X^{k+1}$ is dangerously irrelevant even without the gauge coupling. While it is irrelevant when expanding around the trivial, $X=0$ vacuum, it is clearly relevant when expanding around any non-zero $X$. Thus, the presence of the superpotential  lifts some of the flat directions of the original theory; also, turning on a polynomial superpotential  one finds many minima at non vanishing $X$, where clearly all powers up to and including $\Tr X^{k+1}$ are relevant. Therefore, at the origin $\Tr X^{k+1}$ is a DI operator and cannot be ignored.

Consider a polynomial of order $k+1$, The $k$ roots of $W^\prime(x)$, $a_i$ ($i=1,\cdots k$) satisfy: where $\{s_l\}$ and $\{a_i\}$ are related by: It is also natural to define the objects: which satisfy the recursion relation: Eq.  can be thought of as determining $u_l$ in terms of $\{s_m\}$ with $m\le l$. For the few lowest cases we have: etc. In the text (section [*3*]{}) we used a remarkable property of the $u$’s: For all $j, l$. In other words, duality requires that ${\partial\over\partial
s_j}{u_{j+l}\over j+l}$ should be independent of $j$. The proof of this statement relies on the following representation of the recursion relation : Differentiating  w.r.t. some $s_j$ we find: which makes the fact that ${\partial\over\partial s_j}u_n/n$ depends only on $n-j$ and not on $n$, $j$ separately manifest.
