CERN-TH/97-257

UB-ECM-PF 97/26

hep-th/9709180

September 1997

Duality in Quantum Field Theory

(and String Theory)
^{1}^{1}1Based on a lectures delivered by L. A.-G.
at The Workshop on Fundamental Particles and Interactions,
held in Vanderbilt University, and
at CERN-La Plata-Santiago de Compostela School of Physics,
both in May 1997.

Luis Álvarez-Gaumé and Frederic Zamora.

Theory Division, CERN,

1211 Geneva 23, Switzerland.

Departament d’Estructura i Constituents de la Materia,

Facultat de Física, Universitat de Barcelona,

Diagonal 647, E-08028 Barcelona, Spain.

ABSTRACT

These lectures give an introduction to duality in Quantum Field Theory. We discuss the phases of gauge theories and the implications of the electric-magnetic duality transformation to describe the mechanism of confinement. We review the exact results of supersymmetric QCD and the Seiberg-Witten solution of super Yang-Mills. Some of its extensions to String Theory are also briefly discussed.

CERN-TH/97-257

UB-ECM-PF 97/26

September 1997

###### Contents

- I The duality symmetry.
- II Dirac’s charge quantization.
- III A charge lattice and the group.
- IV The Higgs Phase
- V The Georgi-Glashow model and the Coulomb phase.
- VI The ’t Hooft-Polyakov monopoles
- VII The Confining phase.
- VIII The Higgs/confining phase.
- IX Supersymmetry
- X The uses of supersymmetry.
- XI SQCD.
- XII The vacuum structure of SQCD with .
- XIII The vacuum structure of SQCD with .
- XIV The vacuum structure of SQCD with .
- XV Seiberg’s duality.
- XVI supersymmetry.
- XVII super Yang-Mills theory in perturbation theory.
- XVIII The low energy effective Lagrangian.
- XIX BPS bound and duality.
- XX Singularities in the moduli space.
- XXI The physical interpretation of the singularities.
- XXII The Seiberg-Witten solution.
- XXIII Breaking to . Monopole condensation and confinement.
- XXIV Breaking to .
- XXV String Theory in perturbation theory.
- XXVI D-branes.
- XXVII Some final comments on nonperturbative String Theory.

## I The duality symmetry.

From a historical point of view we can say that many of the fundamental concepts of twentieth century Physics have Maxwell’s equations at its origin. In particular some of the symmetries that have led to our understanding of the fundamental interactions in terms of relativistic quantum field theories have their roots in the equations describing electromagnetism. As we will now describe, the most basic form of the duality symmetry also appears in the source free Maxwell equations:

(I.1) |

These equations are invariant under Lorentz transformations, and making all of Physics compatible with these symmetries led Einstein to formulate the Theory of Relativity. Other important symmetries of (I.1) are conformal and gauge invariance, which have later played important roles in our understanding of phase transitions and critical phenomena, and in the formulation of the fundamental interactions in terms of gauge theories. In these lectures however we will study the implications of yet another symmetry hidden in (I.1): duality. The simplest form of duality is the invariance of (I.1) under the interchange of electric and magnetic fields:

(I.2) |

In fact, the vacuum Maxwell equations (I.1)
admit a continuous transformation symmetry
^{2}^{2}2Notice that the duality transformations
are not a symmetry of the electromagnetic action. Concerning this
issue see [1].

(I.3) |

If we include ordinary electric sources the equations (1.1) become:

(I.4) |

In presence of matter, the duality symmetry is not valid. To keep it, magnetic sources have to be introduced:

(I.5) |

Now the duality symmetry is restored if at the same time we also rotate the electric and magnetic charges

(I.6) |

The complete physical meaning of the duality symmetry is still not clear, but a lot of work has been dedicated in recent years to understand the implications of this type of symmetry. We will focus mainly on the applications to Quantum Field Theory. In the final sections, we will briefly review some of the applications to String Theory, where duality make striking an profound predictions.

## Ii Dirac’s charge quantization.

From the classical point of view the inclusion of magnetic charges is not particularly problematic. Since the Maxwell equations, and the Lorentz equations of motion for electric and magnetic charges only involve the electric and magnetic field, the classical theory can accommodate any values for the electric and magnetic charges.

However, when we try to make a consistent quantum theory including monopoles, deep consequences are obtained. Dirac obtained his celebrated quantization condition precisely by studying the consistency conditions for a quantum theory in the presence of electric and magnetic charges [2]. We derive it here by the quantization of the angular momentum, since it allows to extend it to the case of dyons, i.e., particles that carry both electric and magnetic charges.

Consider a non-relativistic charge in the vicinity of a magnetic monopole of strength , situated at the origin. The charge experiences a force , where is the monopole field given by . The change in the orbital angular momentum of the electric charge under the effect of this force is given by

(II.1) |

Hence, the total conserved angular momentum of the system is

(II.2) |

The second term on the right hand side (henceforth denoted by ) is the contribution coming from the electromagnetic field. This term can be directly computed by using the fact that the momentum density of an electromagnetic field is given by its Poynting vector, , and hence its contribution to the angular momentum is given by

In components,

(II.3) | |||||

When the separation between the electric and magnetic charges is negligible compared to their distance from the boundary , the contribution of the first integral to vanishes by spherical symmetry. We are therefore left with

(II.4) |

Returning to equation (II.2), if we assume that orbital angular momentum is quantized. Then it follows that

(II.5) |

where is an integer. Equation (II.5) is the Dirac’s charge quantization condition. It implies that if there exists a magnetic monopole of charge somewhere in the universe, then all electric charges are quantized in units of . If we have a number of purely electric charges and purely magnetic charges , then any pair of them will satisfy a quantization condition:

(II.6) |

Thus, any electric charge is an integral multiple of . For a given , let these charges have as the highest common factor. Then, all the electric charges are multiples of . Similar considerations apply to the quantization of the magnetic charge.

Till now, we have only dealt with particles that carry either an electric or a magnetic charge. Consider now two dyons of charges and . For this system, we can repeat the calculation of by following the steps in (II.3) where now the electromagnetic fields are split as and . The answer is easily found to be

(II.7) |

The charge quantization condition is thus generalized to

(II.8) |

This is referred to as the Dirac-Schwinger-Zwanziger condition [3].

## Iii A charge lattice and the group.

In the previous section we derived the quantization of the electric charge of particles without magnetic charge, in terms of some smallest electric charge . For a dyon , this gives . Thus, the smallest magnetic charge the dyon can have is , with a positive integer dependent on the detailed theory considered. For two dyons of the same magnetic charge and electric charges and , the quantization condition implies , with a multiple of . Therefore, although the difference of electric charges is quantized, the individual charges are still arbitrary. It introduces a new parameter that contributes to the electric charge of any dyon with magnetic charge by

(III.1) |

Observe that the parameter gives the same electric charges that the parameter by shifting . Thus, we look at the parameter as an angular variable.

This arbitrariness in the electric charge of dyons through the parameter can be fixed if the theory is CP invariant. Under a CP transformation . If the theory is CP invariant, the existence of a state necessarily leads to the existence of . Applying the quantization condition to this pair, we get . This implies that or . If , the theory is not CP invariant. It indicates that the parameter is a source of CP violation. Later on we will identify with the instanton angle.

One can see that the general solution of the Dirac-Schwinger-Zwanziger condition (II.8) is

(III.2) | |||||

(III.3) |

with and integer numbers These equations can be expressed in terms of the complex number

(III.4) |

where

(III.5) |

Observe that this definition only includes intrinsic parameters of the theory, and that the imaginary part of is positive definite. This complex parameter will play an important role in supersymmetric gauge theories. Thus, physical states with electric and magnetic charges are located on a discrete two dimensional lattice with periods and , and are represented by the corresponding vector (see fig. 1).

Notice that the lattice of charges obtained from the quantization condition breaks the classical duality symmetry group that rotated the electric and magnetic charges (I.6). But another symmetry group arises at quantum level. Given a lattice as in figure 1 we can describe it in terms of different fundamental cells. Different choices correspond to transforming the electric and magnetic numbers by a two-by-two matrix:

(III.6) |

with satisfying . This transformation leaves invariant the Dirac-Schwinger-Zwanziger quantiztion condition (II.8). Hence the duality transformations are elements of the discrete group . Its action on the charge lattice can be implemented by modular transformations of the parameter

(III.7) |

This transformations preserve the sign of the imaginary part of , and are generated just by the action of two elements:

(III.8) | |||

(III.9) |

The effect of is to shift . Its action is well understood: it just maps the charge lattice to . As physics is -periodic in , it is a symmetry of the theory. Then, if the state is in the physical spectrum, the state , with any integer , is also a physical state.

The effect of is less trivial. If we take just for simplicity, the action is and sends the lattice vector to the lattice vector . So it interchanges the electric and magnetic roles. In terms of coupling constants, it represents the transformation , implying the exchange between the weak and strong coupling regimes. In this respect the duality symmetry could provide a new source of information on nonpertubative physics.

If we claim that the transformation is also a symmetry of the theory we have full symmetry. It implies the existence of any state in the physical spectrum, with and relatively to-prime, just from the knowledge that there are the physical states and . There are some examples of theories ‘duality invariant’, for instance the gauge theory with supersymmetry and the gauge theory with supersymmetry and four flavors [4].

A priori however there is no physical reason to impose -invariance, in contrast with -invariance. The stable physical spectrum may not be invariant. But if the theory still admits somehow magnetic monopoles, we could apply the -transformation as a change of variables of the theory, where a magnetic state is mapped to an electric state in terms of the dual variables. It could be convenient for several reasons: Maybe there are some physical phenomena where the magnetic monopoles become relevant degrees of freedom; this is the case for the mechanism of confinement, as we will see below. The other reason could be the difficulty in the computation of some dynamical effects in terms of the original electric variables because of the large value of the electric coupling . The -transformation sends to . In terms of the dual magnetic variables, the physics is weakly coupled.

Just by general arguments we have learned a good deal of information about the duality transformations. Next we have to see where such concepts appear in quantum field theory.

## Iv The Higgs Phase

### iv.1 The Higgs mechanism and mass gap.

We start considering that the relevant degrees of freedom at large distances of some theory in 3+1 dimensions are reduced to an Abelian Higgs model:

(IV.1) | |||||

where

(IV.2) |

and is the electric charge of the particle .

An important physical example of a theory described at large distances by the effective Lagrangian (IV.1) (in its nonrelativistic approximation) is a superconductor. Sound waves of a solid material causes complicated deviations from the ideal lattice of the material. Conducting electrons interact with the quantums of those sound waves, called phonons. For electrons near the Fermi surface, their interactions with the phonons create an attractive force. This force can be strong enough to cause bound states of two electrons with opposite spin, called Cooper pairs. The lowest state is a scalar particle with charge , which is represented by in (IV.1). To understand the basic features of a superconductor we only need to consider its relevant self-interactions and the interaction with the electromagnetic field resulting from its electric charge . This is the dynamics which is encoded in the effective Lagrangian (IV.1). The values of the parameters and depend of the temperature , and in general contribute to increase the energy of the system. To have an stable ground state, we require for any value of the temperature. But the function do not need to be negative for all . In fact, when the temperature drops below a critical value , the function becomes positive. In such situation, the ground state reaches its minimal energy when the Higgs particle condenses,

(IV.3) |

If we make perturbation theory around this minima,

(IV.4) |

with vanishing external electromagnetic fields, we find that there is a mass gap between the ground state and the first excited levels. There are particles of spin one with mass square

(IV.5) |

which corresponds to the inverse of the penetration depth of static electromagnetic fields in the superconductor. There are also spin zero particles with mass square

(IV.6) |

So perturbation theory already shows a quite different behavior of the Higgs theory from the Coulomb theory. There is only one real massive scalar field and the electromagnetic interaction becomes short-ranged, with the photon correlator being exponentially suppressed. This is a distinction that must survive nonperturbatively. But up to now, the above does not yet distinguish a Higgs theory from just any non-gauge theory with massive vector particles. There is yet another new phenomena in the Higgs mode which shows the spontaneous symmetry breaking of the gauge theory.

### iv.2 Vortex tubes and flux quantization.

We have seen that the Higgs condensation produces the electromagnetic interactions to be short-range. Ignoring boundary effects in the material, the electric and magnetic fields are zero inside the superconductor. This phenomena is called the Meissner effect.

If we turn on an external magnetic field beyond some critical value, one finds that small regions in the superconductor make a transition to a ‘non-superconducting’ state. Stable magnetic flux tubes are allowed along the material, with a transverse size of the order of the inverse of the mass gap. Their magnetic flux satisfy a quantization rule that can be understood only by a combination of the spontaneous symmetry breaking of the gauge symmetry and some topological arguments.

Parameterize the complex Higgs field by

(IV.7) |

and perform fluctuations around the configuration which minimizes the energy. i.e., we consider that almost everywhere, but at some points may be zero. At such points needs not be well defined and therefore in all the rest of space could be multivalued. For instance, if we take a closed contour around a zero of , then following around could give values that run from 0 to , with an integer number, instead of coming back to zero. These are exactly the field configurations that produce the quantized magnetic flux tubes [5].

Consider a two-dimensional plane, cut somewhere through a superconducting piece of material, with polar coordinates and work in the time-like . To have a finite energy per unit length static configuration we should demand that

(IV.8) |

for . Obviously, to keep the fields single valued, we must have

(IV.9) |

If , it is clear that at some point of the two-dimensional plane we should have that the continuous field vanishes. Such field configurations do not correspond to the ground state.

Solve the field equations with the boundary conditions
(IV.8) and (IV.9) fixed, and minimize
the energy. We find stable vortex tubes with non-trivial
magnetic flux through the two-dimensional plane.
To see this, perform a singular gauge transformation
^{3}^{3}3Singular in the sense of being not well defined
in all space.

(IV.10) |

with . We compute the magnetic flux in such a gauge and we find

(IV.11) |

It is important to realize that such field configurations, called Abrikosov vortices, are stable. The vortex tube cannot break since it cannot have an end point: as the magnetic flux is quantized, we would have be able to deform continuously the singular gauge transformation to zero, something obviously not possible for . Physically this is the statement that the magnetic flux is conserved, a consequence of the Maxwell equations. Mathematically it means that for the function belongs to a nontrivial homotopy class of the fundamental group .

The existence of these macroscopic stable objects can be used as another characterization of the Higgs phase. They should survive beyond perturbation theory.

### iv.3 Magnetic monopoles and permanent magnetic confinement.

The magnetic flux conservation in the Abelian Higgs model tells us that the theory does not include magnetic monopoles. But it is remarkable that the magnetic flux is precisely a multiple of the quantum of magnetic charge found by Dirac. If we imagine the effective gauge theory (IV.1) enriched somehow by magnetic monopoles, they would form end points of the vortex tubes. The energy per unit length, i.e., the string tension , of these flux tubes is of the order of the scale of the Higgs condensation,

(IV.12) |

It implies that the total energy of a system composed of a monopole and an anti-monopole, with a convenient magnetic flux tube attached between them, would be at least proportional to the separation length of the monopoles. In other words: magnetic monopoles in the Higgs phase are permanently confined.

## V The Georgi-Glashow model and the Coulomb phase.

The Georgi-Glashow model is a Yang-Mills-Higgs system which contains a Higgs multiplet transforming as a vector in the adjoint representation of the gauge group , and the gauge fields . Here, are the hermitian generators of satisfying . In the adjoint representation, we have and, for , . The field strength of and the covariant derivative on are defined by

(V.1) |

The minimal Lagrangian is then given by

(V.2) | |||||

where,

(V.3) |

The equations of motion following from this Lagrangian are

(V.4) |

The gauge field strength also satisfies the Bianchi identity

(V.5) |

Let us find the vacuum configurations in this theory. Introducing non-Abelian electric and magnetic fields, and , the energy density is written as

(V.6) | |||||

Note that , and it vanishes only if

(V.7) |

The first equation implies that in the vacuum, is pure gauge and the last two equations define the Higgs vacuum. The structure of the space of vacua is determined by which solves to such that . The space of Higgs vacua is therefore a two-sphere () of radius in field space. To formulate a perturbation theory, we have to choose one of these vacua and hence, break the gauge symmetry spontaneously The part of the symmetry which keeps this vacuum invariant, still survives and the corresponding unbroken generator is . The gauge boson associated with this generator is and the electric charge operator for this surviving is given by

(V.8) |

If the group is compact, this charge is quantized. The perturbative spectrum of the theory can be found by expanding around the chosen vacuum as

A convenient choice is . The perturbative spectrum (which becomes manifest after choosing an appropriate unitary gauge) consists of a massive Higgs of spin zero with a square mass

(V.9) |

a massless photon, corresponding to the gauge boson , and two charged massive W-bosons, and , with square mass

(V.10) |

This mass spectrum is realistic as long as we are at weak coupling, . At strong coupling, nonperturbative effects could change significatively eqs. (V.9) and (V.10). But the fact that there is an unbroken subgroup of the gauge symmetry ensures that there is some massless gauge boson, which a long range interaction. This is the characteristic of the Coulomb phase.

## Vi The ’t Hooft-Polyakov monopoles

Let us look for time-independent, finite energy solutions in the Georgi-Glashow model. Finiteness of energy requires that as , the energy density given by (V.6) must approach zero faster than . This means that as , our solution must go over to a Higgs vacuum defined by (V.7). In the following, we will first assume that such a finite energy solution exists and show that it can have a monopole charge related to its soliton number which is, in turn, determined by the associated Higgs vacuum. This result is proven without having to deal with any particular solution explicitly. Next, we will describe the ’t Hooft-Polyakov ansatz for explicitly constructing one such monopole solution, where we will also comment on the existence of Dyonic solutions. In the last two subsections we will derive the Bogomol’nyi bound and the Witten effect.

### vi.1 The Topological nature of the magnetic charge.

For convenience, in this subsection we will use the vector notation for the gauge group indices and not for the spatial indices.

Let denote the field in a Higgs vacuum. It then satisfies the equations

(VI.1) |

which can be solved for . The most general solution is given by

(VI.2) |

To see that this actually solves (VI.1), note that , so that

(VI.3) |

The first term on the right-hand side of Eq. (VI.2) is the particular solution, and is the general solution to the homogeneous equation. Using this solution, we can now compute the field strength tensor . The field strength corresponding to the unbroken part of the gauge group can be identified as

(VI.4) |

Using the equations of motion in the Higgs vacuum it follows that

This confirms that is a valid field strength tensor. The magnetic field is given by . Let us now consider a static, finite energy solution and a surface enclosing the core of the solution. We take to be far enough so that, on it, the solution is already in the Higgs vacuum. We can now use the magnetic field in the Higgs vacuum to calculate the magnetic charge associated with our solution:

(VI.5) |

It turns out that the expression on the right hand side is a topological quantity as we explain below: Since ; the manifold of Higgs vacua () has the topology of . The field defines a map from into . Since is also an , the map is characterized by its homotopy group . In other words, is characterized by an integer (the winding number) which counts the number of times it wraps around . In terms of the map , this integer is given by

(VI.6) |

Comparing this with the expression for magnetic charge, we get the important result

(VI.7) |

Hence, the winding number of the soliton determines its monopole charge. Note that the above equation differs from the Dirac quantization condition by a factor of . This is because the smallest electric charge which could exist in our model is for an spinorial representation of , the universal covering group of . Then, in this model .

### vi.2 The ’t Hooft-Polyakov ansatz.

Now we describe an ansatz proposed by ’t Hooft [6] and Polyakov [7] for constructing a monopole solution in the Georgi-Glashow model. For a spherically symmetric, parity-invariant, static solution of finite energy, they proposed:

(VI.8) |

For the non-trivial Higgs vacuum at , they chose . Note that this maps an at spatial infinity onto the vacuum manifold with a unit winding number. The asymptotic behavior of the functions and are determined by the Higgs vacuum as and regularity at . Explicitly, defining , we have: as and as . The mass of this solution can be parameterized as

For this ansatz, the equations of motion reduce to two coupled equations for and which have been solved exactly only in certain limits. For , one gets and which shows that the fields are non-singular at . For , we get and which leads to . Once again, defining , the magnetic field turns out to be . The associated monopole charge is , as expected from the unit winding number of the solution. It should be mentioned that ’t Hooft’s definition of the Abelian field strength tensor is slightly different but, at large distances, it reduces to the form given above.

Note that in the above monopole solution, the presence of the Dirac string is not obvious. To extract the Dirac string, we have to perform a singular gauge transformation on this solution which rotates the non-trivial Higgs vacuum into the trivial vacuum . In the process,the gauge field develops a Dirac string singularity which now serves as the source of the magnetic charge [6].

The ’t Hooft-Polyakov monopole carries one unit of magnetic charge and no electric charge. The Georgi-Glashow model also admits solutions which carry both magnetic as well as electric charges. An ansatz for constructing such a solution was proposed by Julia and Zee [8]. In this ansatz, and have exactly the same form as in the ’t Hooft-Polyakov ansatz, but is no longer zero: . This serves as the source for the electric charge of the dyon. It turns out that the dyon electric charge depends of a continuous parameter and, at the classical level, does not satisfy the quantization condition. However, semiclassical arguments show that, in CP invariant theories, and at the quantum level, the dyon electric charge is quantized as . This can be easily understood if we recognize that a monopole is not invariant under a gauge transformation which is, of course, a symmetry of the equations of motion. To deal with the associated zero-mode properly, the gauge degree of freedom should be regarded as a collective coordinate. Upon quantization, this collective coordinate leads to the existence of electrically charged states for the monopole with discrete charges. In the presence of a CP violating term in the Lagrangian, the situation is more subtle as we will discuss later. In the next subsection, we describe a limit in which the equations of motion can be solved exactly for the ’tHooft-Polyakov and the Julia-Zee ansatz. This is the limit in which the soliton mass saturates the Bogomol’nyi bound.

### vi.3 The Bogomol’nyi bound and the BPS states.

In this subsection, we derive the Bogomol’nyi bound [9] on the mass of a dyon in term of its electric and magnetic charges which are the sources for . Using the Bianchi identity (V.5) and the first equation in (V.4), we can write the charges as

(VI.9) |

Now, in the center of mass frame, the dyon mass is given by

(VI.10) |

where, is the energy momentum tensor. Using (VI.9) and some algebra we obtain

(VI.11) | |||||

where is an arbitrary angle. Since the terms in the first line are positive, we can write . This bound is maximized for . Thus we get the Bogomol’nyi bound on the dyon mass as

(VI.12) |

For the ’t Hooft-Polyakov solution, we have , and thus, . But and , so that

Here, is the fine structure constant and or , depending on whether the electron charge is or . Since is a small ( for electromagnetism), the above relation implies that the monopole is much heavier than the W-bosons associated with the symmetry breaking.

From (VI.11) it is clear that the bound is not saturated unless , so that . This is the Bogomol’nyi-Prasad-Sommerfield (BPS) limit of the theory [9, 10]. Note that in this limit, is no longer determined by the theory and, therefore, has to be imposed as a boundary condition on the Higgs field. Moreover, in this limit, the Higgs scalar becomes massless. Now, to saturate the bound we set

(VI.13) |

where, . In the BPS limit, one can use the ’t Hooft-Polyakov (or the Julia-Zee) ansatz either in (V.4), or in (VI.13) to obtain the exact monopole (or dyon) solutions [9, 10]. These solutions automatically saturate the Bogomol’nyi bound and are referred to as the BPS states. Also, note that in the BPS limit, all the perturbative excitations of the theory saturate this bound and, therefore, belong to the BPS spectrum. As we will see later, BPS states appears in a very natural way in theories with supersymmetry.

### vi.4 The parameter and the Witten effect.

In this section we will show that in the presence of a -term in the Lagrangian, the magnetic charge of a particle always contributes to its electric charge in the way given by formula (III.2) [11].

To study the effect of CP violation, we consider the Georgi-Glashow model with an additional -term as the only source of CP violation:

(VI.14) | |||||

Here, . The presence of the -term does not affect the equations of motion but changes the physics since the theory is no longer CP invariant. We want to construct the electric charge operator in this theory. The theory has an gauge symmetry but the electric charge is associated with an unbroken which keeps the Higgs vacuum invariant. Hence, we define an operator which implements a gauge rotation around the direction with gauge parameter . These transformations correspond to the electric charge. Under , a vector and the gauge fields transform as

Clearly, is kept invariant. At large distances where , the operator is a -rotation about and therefore . Elsewhere, the rotation angle is . However, by Gauss’ law, if the gauge transformation is at , it leaves the physical states invariant. Thus, it is only the large distance behavior of the transformation which matters and the eigenvalues of are quantized in integer units. Now, we use Noether’s formula to compute :

Since , only the gauge part (which also includes the -term) contributes:

Thus,

where, we have used (VI.9). Here, and are the electric and magnetic charge operators with eigenvalues and , respectively, and is quantized in integer units. This leads to the following formula for the electric charge:

For the ’t Hooft-Polyakov monopole, , , and therefore, . For a general dyonic solution we get

(VI.15) |

and we recover (III.2) and (III.3) for . In the presence of a -term, a magnetic monopole always carries an electric charge which is not an integral multiple of some basic unit. In section III we introduced the charge lattice of periods and . In this parameterization, the Bogomol’nyi bound (VI.12) takes the form

(VI.16) |

Notice that for a BPS state, equation (VI.16) implies that its mass is proportional to the distance of its lattice point from the origin.

## Vii The Confining phase.

### vii.1 The Abelian projection.

In non-Abelian gauge theories, gauge fixing is a subject full of interesting surprises (ghosts, phantom solitons, …) which often obscure the physical content of the theory [12].

’t Hooft gave a qualitative program to overcome these
difficulties and provided a scenario that explains confinement
in a gauge theory.
The idea is to perform the gauge fixing
procedure in two steps. In the first one
a unitary gauge is chosen for the non-Abelian degrees
of freedom. It reduces the non-Abelian gauge symmetry to the maximal
Abelian subgroup of the gauge group.
Here one gets particle gauge singularities
^{4}^{4}4We will discuss the physical meaning of them
later on.. This procedure is called the Abelian
projection [12].
In this way, the dynamics of the Yang-Mils
theory will be reduced to an Abelian gauge theory
with certain additional degrees of freedom.

We need a field that transforms without derivatives under gauge transformations. An example is a real field, in the adjoint representation of ,

(VII.1) |

Such a field can always be found; take for instance . We will use the field to implement the unitary gauge condition which will carry us to the Abelian projection of the gauge group. The gauge is fixed by requiring that be diagonal:

(VII.2) |

The eigenvalues of the matrix are gauge invariant. Generically they are all different, and the gauge condition (VII.2) leaves an Abelian gauge symmetry. It corresponds to the subgroup generated by the gauge transformations

(VII.3) |

There is also a discrete subgroup of transformations which still leave in diagonal form. It is the Weyl group of , which corresponds to permutations of the eigenvalues . We also fix the Weyl group with the convention .

At this stage, we have an Abelian gauge theory with photons, charged vector particles and some additional degrees of freedom that will appear presently.

### vii.2 The nature of the gauge singularities.

So far we assumed that the eigenvalues coincide nowhere. But there are some gauge field configurations that produce two consecutive eigenvalues to coincide at some spacetime points

(VII.4) |

These spacetime points are ‘singular’ points of the Abelian projection. The gauge subgroup corresponding to the block matrix with coinciding eigenvalues leaves invariant the gauge-fixing condition (VII.2).

Let us consider the vicinity of such a point. Prior to the complete gauge-fixing we may take to be

(VII.5) |

where and may safely be considered to be diagonalized because the other eigenvalues do not coincide. With respect to that subgroup of that corresponds to rotations among the th and st components, the three fields form an isovector. We may write the central block as

(VII.6) |

where are the Pauli matrices.

Consider static field configurations. The points of space where the two eigenvalues coincide correspond to the points that satisfy

(VII.7) |

These three equations define a single space point, and then the singularity is particle-like. Which is its physical interpretation?.

By analyticity we have that , and our gauge condition corresponds to rotating the isovector such that

(VII.8) |

From the previous sections, we know that the zero-point of at behaves as a magnetic charge with respect to the remaining rotations. We realize that those gauge field configurations that produce such a gauge ‘singularities’ correspond to magnetic monopoles.

The non-Abelian gauge theory is topologically such that it can be cast into a Abelian gauge theory, which will feature not only electrically charged particles but also magnetic monopoles.

### vii.3 The phases of the Yang-Mills vacuum.

We can now give a qualitative description of the possible phases of the Yang-Mills vacuum. It is only the dynamics which, as a function of the microscopic bare parameters, determines in which phase the Yang-Mills vacuum is actually realized.

Classically, the Yang-Mills Lagrangian is scale invariant. One can write down field configurations with magnetic charge and arbitrarily low energy. But quantum corrections are likely to violate their masslessness. If dynamics simply chooses to give a positive mass to the monopoles, we are in a Higgs or Coulomb phase. We must look for the magnetic vortex tubes to figure out if we are in a Higgs phase. It will be a signal that the ordinary Higgs mechanism has taken place in the Abelian gauge formulation of the Yang-Mills theory. The role of the dynamically generated Higgs field could be done by some scalar composite operator charged respect the gauge symmetries. There is also the possibility that no Higgs phenomenon occurs at all in the Abelian sector, or that some gauge symmetries are not spontaneously broken. In this case we are in the Coulomb phase, with some massless photons, or in a mixed Coulomb-Higgs phase.

There is a third possibility however. Maybe the quantum corrections give a formally negative mass squared for the monopole: a magnetically charged object condenses. We apply an ‘electric-magnetic dual transformation’ to write an effective Lagrangian which encodes the relevant magnetic degrees of freedom in the infrared limit. In such effective Lagrangian, the Higgs mechanism takes place in terms of dual variables. We are in a dual Higgs phase. We have electric flux tubes with finite energy per unit of length. There is a confining potential between electrically charged objects, like quarks.

In 1994, Seiberg and Witten gave a quantitative proof that such dynamical mechanism of color confinement takes place in super-QCD (SQCD) broken to [13], giving a non-trivial realization of ’t Hooft scenario. When SQCD is softly broken to the same mechanism of confinement persists [