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Home , Higgs phase, Seiberg duality, String duality, Super QCD, Type I string theory

CERN-TH/97-257 UB-ECM-PF 97/26 hep-th/9709180 September 1997

Duality in (and Theory) ∗

Luis Alvarez-Gaum´´ ea and Frederic Zamorab.

a Theory Division, CERN, 1211 Geneva 23, Switzerland.

b Departament d’Estructura i Constituents de la Materia, Facultat de F´ısica, 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 N = 1 supersymmetric QCD and the Seiberg-Witten solution of N = 2 super Yang-Mills. Some of its extensions to are also briefly discussed.

CERN-TH/97-257 UB-ECM-PF 97/26 September 1997

∗Based 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 , both in May 1997. Contents XIII The vacuum structure of SQCD with Nf = Nc.19 I The duality . 1 A A quantum modified . . . 19 B Patterns of spontaneous symmetry II Dirac’s charge quantization. 1 breaking and ’t Hooft’s match- ingconditions...... 20 III A charge lattice and the SL(2, Z) group. 2 XIV The vacuum structure of SQCD with IV The Higgs Phase 3 Nf = Nc +1.21 A The and gap. . 3 A The quantum moduli space...... 21 B Vortex tubes and flux quantization. . . 4 B S-confinement...... 21 C Magnetic monopoles and permanent magnetic confinement...... 5 XV Seiberg’s duality. 22 A ThedualSQCD...... 22 V The Georgi-Glashow model and the B Nc+1

VII The Confining phase. 9 XVII N =2SU(2) super Yang-Mills theory in A TheAbelianprojection...... 9 perturbation theory. 24 B The nature of the gauge singularities. . 9 ATheN= 2 Lagrangian...... 24 C The phases of the Yang-Mills vacuum. . 10 B Theflatdirection...... 25 D Oblique confinement...... 10 XVIII The low energy effective Lagrangian. 25 VIII The Higgs/confining phase. 10 XIX BPS bound and duality. 26 IX Supersymmetry 11 A The supersymmetry algebra and its XX Singularities in the moduli space. 27 masslessrepresentations...... 11 B and superfields...... 11 XXI The physical interpretation of the singu- C Supersymmetric Lagrangians...... 12 larities. 27 D R-symmetry...... 14 XXII The Seiberg-Witten solution. 28 X The uses of supersymmetry. 14 A The inputs...... 28 A Flat directions and super-Higgs mecha- B Thegeometricalpicture...... 28 nism...... 14 C The Physical connection with N =2 B Wilsonian effective actions and holo- super Yang-Mills...... 29 morphy...... 15 XXIII Breaking N =2to N =1. Monopole XI N =1SQCD. 16 condensation and confinement. 29 A Classical Lagrangian and symmetries. . 16 B The classical moduli space...... 17 XXIV Breaking N =2to N =0.30

XII The vacuum structure of SQCD with XXV String Theory in perturbation theory. 32 A The type IIA and type IIB string theories. 33 Nf

1 XXVII Some final comments on nonpertur- ∇·(E+iB)=(q+ig) , bative String Theory. 36 ∂ (E + iB)+i∇×(E+iB)=(j +ij ). (I.5) A D- and S-duality...... 36 ∂t e m B Aneleventhdimension...... 36 Now the duality symmetry is restored if at the same time we also rotate the electric and magnetic charges

iφ I. THE DUALITY SYMMETRY. (q + ig) → e (q + ig) . (I.6) The complete physical meaning of the duality symme- From a historical point of view we can say that many try is still not clear, but a lot of work has been dedicated of the fundamental concepts of twentieth century Physics in recent years to understand the implications of this type have Maxwell’s equations at its origin. In particular some of symmetry. We will focus mainly on the applications of the symmetries that have led to our understanding to Quantum Field Theory. In the final sections, we will of the fundamental interactions in terms of relativistic briefly review some of the applications to String Theory, quantum field theories have their roots in the equations where duality make striking an profound predictions. describing . As we will now describe, the most basic form of the duality symmetry also appears in the source free Maxwell equations: II. DIRAC’S CHARGE QUANTIZATION. ∇·(E+iB)=0, ∂ From the classical point of view the inclusion of mag- (E + iB)+i∇×(E+iB)=0. (I.1) netic charges is not particularly problematic. Since the ∂t Maxwell equations, and the Lorentz equations of motion These equations are invariant under Lorentz transforma- for electric and magnetic charges only involve the electric tions, and making all of Physics compatible with these and magnetic field, the classical theory can accommodate symmetries led Einstein to formulate the Theory of Rel- any values for the electric and magnetic charges. ativity. Other important symmetries of (I.1) are confor- However, when we try to make a consistent quantum mal and gauge invariance, which have later played im- theory including monopoles, deep consequences are ob- portant roles in our understanding of phase transitions tained. Dirac obtained his celebrated quantization condi- and critical phenomena, and in the formulation of the tion precisely by studying the consistency conditions for fundamental interactions in terms of gauge theories. In a quantum theory in the presence of electric and mag- these lectures however we will study the implications of netic charges [2]. We derive it here by the quantization yet another symmetry hidden in (I.1): duality. The sim- of the angular momentum, since it allows to extend it to plest form of duality is the invariance of (I.1) under the the case of dyons, i.e., particles that carry both electric interchange of electric and magnetic fields: and magnetic charges. Consider a non-relativistic charge q in the vicinity of a B → E , of strength g, situated at the origin. ¨ ˙ ~ ~ E →−B. (I.2) The charge q experiences a m~r = q~r × B,whereB is the monopole field given by B~ = g~r/4πr3. The change In fact, the vacuum Maxwell equations (I.1) admit a con- in the orbital angular momentum of the tinuous SO(2) transformation symmetry † under the effect of this force is given by

(E + i B) → eiφ(E + i B) . (I.3) d m~r × ~r˙ = m~r × ~¨r dt If we include ordinary electric sources the equations (1.1) qg  d qg ~r = ~r × ~r˙ × ~r = . (II.1) become: 4πr3 dt 4π r     ∇·(E+iB)=q, Hence, the total conserved angular momentum of the sys- ∂ tem is (E + iB)+i∇×(E+iB)=j . (I.4) ∂t e qg ~r J~ = ~r × m~r˙ − . (II.2) In presence of matter, the duality symmetry is not valid. 4π r To keep it, magnetic sources have to be introduced: The second term on the right hand side (henceforth de-

noted by J~em) is the contribution coming from the elec- tromagnetic field. This term can be directly computed

† by using the fact that the momentum density of an elec- Notice that the duality transformations are not a symmetry ~ ~ of the electromagnetic action. Concerning this issue see [1]. tromagnetic field is given by its Poynting vector, E × B,

2 and hence its contribution to the angular momentum is III. A CHARGE LATTICE AND THE SL(2, Z) given by GROUP.

g ~r J~ = d3x~r×(E~ ×B~)= d3x~r× E~ × . In the previous section we derived the quantization of em 4π r3 Z Z   the electric charge of particles without magnetic charge, in terms of some smallest electric charge q .Foradyon In components, 0 (qn,gn), this gives q0gn =2πn. Thus, the smallest mag- netic charge the dyon can have is g0 =2πm0/q0,with i g 3 j i J = d xE ∂j (ˆx ) m a positive integer dependent on the detailed theory em 4π 0 Z considered. For two dyons of the same magnetic charge g i g 3 i = xˆ E~ · ds~ − d x(∇·~ E~)ˆx . (II.3) g0 and electric charges q1 and q2, the quantization con- 4π 2 4π ZS Z dition implies q1 − q2 = nq0,withna multiple of m0. Therefore, although the difference of electric charges is When the separation between the electric and magnetic quantized, the individual charges are still arbitrary. It charges is negligible compared to their distance from the 2 ~ introduces a new parameter θ that contributes to the boundary S , the contribution of the first integral to Jem electric charge of any dyon with magnetic charge g by vanishes by spherical symmetry. We are therefore left 0 with θ q = q n + . (III.1) gq 0 e 2π J~ = − r.ˆ (II.4)   em 4π Observe that the parameter θ+2π gives the same electric Returning to equation (II.2), if we assume that orbital charges that the parameter θ by shifting ne → ne +1. angular momentum is quantized. Then it follows that Thus, we look at the parameter θ as an angular variable. This arbitrariness in the electric charge of dyons qg 1 = n, (II.5) through the θ parameter can be fixed if the theory is CP 4π 2 invariant. Under a CP transformation (q, g) → (−q, g). If the theory is CP invariant, the existence of a state where n is an integer. Equation (II.5) is the Dirac’s (q, g ) necessarily leads to the existence of ( q, g ). Ap- charge quantization condition. It implies that if there 0 − 0 plying the quantization condition to this pair, we get exists a magnetic monopole of charge g somewhere in the 2q = q integer. This implies that q = nq or , then all electric charges are quantized in units 0 × 0 q =(n+1)q.Ifθ=0,π, the theory is not CP invari- of 2π/g. If we have a number of purely electric charges 2 0 6 ant. It indicates that the θ parameter is a source of CP q and purely magnetic charges g , then any pair of them i j violation. Later on we will identify θ with the will satisfy a quantization condition: angle. One can see that the general solution of the Dirac- qigj =2πnij . (II.6) Schwinger-Zwanziger condition (II.8) is Thus, any electric charge is an integral multiple of 2π/g . j θ For a given gj , let these charges have n0j as the highest q = q n + n , (III.2) 0 e 2π m common factor. Then, all the electric charges are mul-   tiples of q0 = n0j2π/gj. Similar considerations apply to g = nmg0 , (III.3) the quantization of the magnetic charge. Till now, we have only dealt with particles that carry with ne and nm integer numbers These equations can be either an electric or a magnetic charge. Consider now two expressed in terms of the complex number dyons of charges (q1,g1)and(q2,). For this system, we q + ig = q0(ne + nmτ) , (III.4) can repeat the calculation of J~em by following the steps in (II.3) where now the electromagnetic fields are split as where E~ = E~1 + E~2 and B~ = B~1 + B~2. The answer is easily θ 2πim0 found to be τ ≡ + . (III.5) 2π q2 1 0 J~ = − (q g − q g )ˆr (II.7) em 4π 1 2 2 1 Observe that this definition only includes intrinsic pa- rameters of the theory, and that the imaginary part of τ The charge quantization condition is thus generalized to is positive definite. This complex parameter will play an important role in supersymmetric gauge theories. Thus, q1g2 − q2g1 1 = n12 (II.8) physical states with electric and magnetic charges (q, g) 4π 2 are located on a discrete two dimensional lattice with This is referred to as the Dirac-Schwinger-Zwanziger con- periods q0 and q0τ, and are represented by the corre- dition [3]. sponding vector (nm,ne) (see fig. 1).

3 g of coupling constants, it represents the transformation τ →−1/τ, implying the exchange between the weak and strong coupling regimes. In this respect the dual- ity symmetry could provide a new source of information on nonpertubative physics. q0τ If we claim that the S transformation is also a sym- metry of the theory we have full SL(2, Z) symmetry. It implies the existence of any state (nm,ne) in the physical spectrum, with nm and ne relatively to-prime, just from the knowledge that there are the physical states ±(0, 1) and ±(1, 0). There are some examples of theories ‘dual- q0 q ity invariant’, for instance the SU(2) with N = 4 supersymmetry and the SU(2) gauge theory with N = 2 supersymmetry and four flavors [4]. A priori however there is no physical reason to impose S-invariance, in contrast with T -invariance. The stable FIG. 1. The charge lattice with periods q0 and q0τ.The physical states are located on the points of the lattice. physical spectrum may not be SL(2, Z) invariant. But if the theory still admits somehow magnetic monopoles, Notice that the lattice of charges obtained from the we could apply the S-transformation as a change of vari- quantization condition breaks the classical duality sym- ables of the theory, where a magnetic state is mapped to metry group SO(2) that rotated the electric and mag- an electric state in terms of the dual variables. It could netic charges (I.6). But another symmetry group arises be convenient for several reasons: Maybe there are some at quantum level. Given a lattice as in figure 1 we can physical phenomena where the magnetic monopoles be- describe it in terms of different fundamental cells. Differ- come relevant degrees of freedom; this is the case for the ent choices correspond to transforming the electric and mechanism of confinement, as we will see below. The other reason could be the difficulty in the computation magnetic numbers (nm,ne)byatwo-by-twomatrix: of some dynamical effects in terms of the original electric 1 αβ− variables because of the large value of the electric cou- (n ,n ) (n ,n ) , (III.6) m e → m e γδ pling q0.TheS-transformation sends q0 to 1/q0.Interms   of the dual magnetic variables, the physics is weakly cou- with α, β, γ, δ ∈ Z satisfying αδ − βγ = 1. This transfor- pled. mation leaves invariant the Dirac-Schwinger-Zwanziger Just by general arguments we have learned a good deal quantiztion condition (II.8). Hence the duality transfor- of information about the duality transformations. Next mations are elements of the discrete group SL(2, Z). Its we have to see where such concepts appear in quantum action on the charge lattice can be implemented by mod- field theory. ular transformations of the parameter τ ατ + β τ → . (III.7) γτ + δ IV. THE HIGGS PHASE

This transformations preserve the sign of the imaginary A. The Higgs mechanism and mass gap. part of τ, and are generated just by the action of two elements: We start considering that the relevant degrees of free- dom at large distances of some theory in 3+1 dimensions T : τ → τ +1, (III.8) are reduced to an Abelian Higgs model: −1 S : τ → . (III.9) τ 1 L(φ∗,φ,A )=− F F µν +(D φ)∗(Dµφ) µ 4 µν µ The effect of T is to shift θ → θ +2π. Its action is λ 2 2 well understood: it just maps the charge lattice (nm,ne) − (φ∗φ−M ) , (IV.1) 2 to (nm,ne −nm).Asphysicsis2π-periodic in θ,itisa symmetry of the theory. Then, if the state (1, 0) is in the where physical spectrum, the state (1,ne), with any integer ne, is also a physical state. Fµν = ∂µAν − ∂ν Aµ , The effect of S is less trivial. If we take θ =0just Dµφ =(∂µ+iqAµ)φ, (IV.2) for simplicity, the S action is q0 → g0 and sends the lat- tice vector (nm,ne) to the lattice vector (−ne,nm). So and q is the electric charge of the particle φ. it interchanges the electric and magnetic roles. In terms

4 An important physical example of a theory described B. Vortex tubes and flux quantization. at large distances by the effective Lagrangian (IV.1) (in its nonrelativistic approximation) is a superconductor. We have seen that the Higgs condensation produces Sound waves of a solid material causes complicated de- the electromagnetic interactions to be short-range. Ig- viations from the ideal lattice of the material. Conduct- noring boundary effects in the material, the electric and ing electrons interact with the quantums of those sound magnetic fields are zero inside the superconductor. This waves, called phonons. For electrons near the Fermi sur- phenomena is called the Meissner effect. face, their interactions with the phonons create an at- If we turn on an external magnetic field H0 be- tractive force. This force can be strong enough to cause yond some critical value, one finds that small regions bound states of two electrons with opposite spin, called in the superconductor make a transition to a ‘non- Cooper pairs. The lowest state is a scalar particle with superconducting’ state. Stable magnetic flux tubes are charge q = −2e, which is represented by φ in (IV.1). allowed along the material, with a transverse size of the To understand the basic features of a superconductor we order of the inverse of the mass gap. Their magnetic flux only need to consider its relevant self-interactions and satisfy a quantization rule that can be understood only the interaction with the electromagnetic field resulting by a combination of the spontaneous symmetry break- from its electric charge q. This is the dynamics which ing of the U(1) gauge symmetry and some topological is encoded in the effective Lagrangian (IV.1). The val- arguments. 2 ues of the parameters λ and M depend of the tempera- Parameterize the complex Higgs field by ture T , and in general contribute to increase the energy of the system. To have an stable ground state, we re- φ(x)=ρ(x)eiχ(x), (IV.7) quire λ(T ) > 0 for any value of the temperature. But the function M 2(T ) do not need to be negative for all T . and perform fluctuations around the configuration which In fact, when the temperature T drops below a critical minimizes the energy. i.e., we consider that ρ(x) ' M 2 almost everywhere, but at some points ρ may be zero. At value Tc, the function M (T ) becomes positive. In such situation, the ground state reaches its minimal energy such points χ needs not be well defined and therefore in when the Higgs particle condenses, all the rest of space χ could be multivalued. For instance, if we take a closed contour C around a zero of ρ(x), then |hφi| = M. (IV.3) following χ around C could give values that run from 0 to 2πn,withnan integer number, instead of coming back If we make perturbation theory around this minima, to zero. These are exactly the field configurations that produce the quantized magnetic flux tubes [5]. φ(x)=M+ϕ(x), (IV.4) Consider a two-dimensional plane, cut somewhere through a superconducting piece of material, with polar with vanishing external electromagnetic fields, we find coordinates (r, θ) and work in the time-like A0 =0.To that there is a mass gap between the ground state and have a finite energy per unit length static configuration the first excited levels. There are particles of spin one we should demand that with mass square φ(x) → Meiχ(θ) , 2 2 MV =2qM , (IV.5) const A (x) → , (IV.8) i r which corresponds to the inverse of the penetration depth of static electromagnetic fields in the superconductor. for r →∞. Obviously, to keep the fields single valued, There are also spin zero particles with mass square we must have 2 2 χ(2π)=χ(0) + 2πn . (IV.9) MH =2λM . (IV.6) So perturbation theory already shows a quite different If n 6= 0, it is clear that at some point of the two- behavior of the Higgs theory from the Coulomb theory. dimensional plane we should have that the continuous There is only one real massive scalar field and the elec- field φ vanishes. Such field configurations do not corre- tromagnetic interaction becomes short-ranged, with the spond to the ground state. photon correlator being exponentially suppressed. This is Solve the field equations with the boundary conditions a distinction that must survive nonperturbatively. But (IV.8) and (IV.9) fixed, and minimize the energy. We up to now, the above does not yet distinguish a Higgs find stable vortex tubes with non-trivial magnetic flux theory from just any non-gauge theory with massive vec- through the two-dimensional plane. To see this, perform ‡ tor particles. There is yet another new phenomena in a singular gauge transformation the Higgs mode which shows the spontaneous symmetry breaking of the U(1) gauge theory.

‡Singular in the sense of being not well defined in all space.

5 iqΛ(x) φ(x) → e φ(x) , The field strength of Wµ and the covariant derivative on φa are defined by Aµ(x) → Aµ(x) − ∂µΛ(x) , (IV.10) with Λ = 2πnθ/q. We compute the magnetic flux in such Gµν = ∂µWν − ∂ν Wµ + ie[Wµ,Wν], a a abc b c a gauge and we find Dµφ =∂µφ −e Wµφ . (V.1) 2πn Φ= A dxµ =Λ(2π)−Λ(0) = . (IV.11) The minimal Lagrangian is then given by µ q I 1 L = − Ga Gaµν It is important to realize that such field configurations, 4 µν called Abrikosov vortices, are stable. The vortex tube 1 + DµφaD φa − V (φ) , (V.2) cannot break since it cannot have an end point: as the 2 µ magnetic flux is quantized, we would have be able to deform continuously the singular gauge transformation where,

Λ to zero, something obviously not possible for n 6=0. λ 2 Physically this is the statement that the magnetic flux V (φ)= φaφa−a2 . (V.3) 4 is conserved, a consequence of the Maxwell equations.  Mathematically it means that for n 6= 0 the function The equations of motion following from this Lagrangian χ(θ) belongs to a nontrivial homotopy class of the fun- are damental group Π (U(1)) = Z. 1 (D Gµν )a = eabc φb (Dµφ)c, The existence of these macroscopic stable objects can ν − µ a a 2 2 be used as another characterization of the Higgs phase. D Dµφ = −λφ (φ − a ) . (V.4) They should survive beyond perturbation theory. The gauge field strength also satisfies the Bianchi identity

D Gµνa =0. (V.5) C. Magnetic monopoles and permanent magnetic ν confinement. Let us find the vacuum configurations in this theory. e Introducing non-Abelian electric and magnetic fields, 0i i ij ij k The magnetic flux conservation in the Abelian Higgs Ga = −Ea and Ga = − kBa, the energy density is model tells us that the theory does not include magnetic written as monopoles. But it is remarkable that the magnetic flux 1 i 2 i 2 is precisely a multiple of the quantum of magnetic charge θ00 = (E ) +(B ) 2 a a 2π/q found by Dirac. If we imagine the effective gauge 0 2 i 2 theory (IV.1) enriched somehow by magnetic monopoles, + D φa) +(D φa) +V(φ). (V.6) they would form end points of the vortex tubes. The en- Note that θ00≥ 0, and it vanishes only if ergy per unit length, i.e., the string tension σ,ofthese µν flux tubes is of the order of the scale of the Higgs con- Ga =0,Dµφ=0,V(φ)=0. (V.7) densation, a The first equation implies that in the vacuum, Wµ is σ ∼ M 2. (IV.12) pure gauge and the last two equations define the Higgs vacuum. The structure of the space of vacua is deter- It implies that the total energy of a system composed of a mined by V (φ)=0whichsolvestoφa =φa such that monopole and an anti-monopole, with a convenient mag- vac |φvac| = a. The space of Higgs vacua is therefore a two- netic flux tube attached between them, would be at least sphere (S2)ofradiusain field space. To formulate a per- proportional to the separation length of the monopoles. turbation theory, we have to choose one of these vacua In other words: magnetic monopoles in the Higgs phase and hence, break the gauge symmetry spontaneously The are permanently confined. part of the symmetry which keeps this vacuum invariant, still survives and the corresponding unbroken generator c c is φvacT /a. The gauge associated with this gener- V. THE GEORGI-GLASHOW MODEL AND THE ator is A = φc W c/a and the electric charge operator COULOMB PHASE. µ vac µ for this surviving U(1) is given by

The Georgi-Glashow model is a Yang-Mills-Higgs sys- φc T c Q = e vac . (V.8) tem which contains a Higgs multiplet φa (a =1,2,3) a transforming as a vector in the adjoint representation If the group is compact, this charge is quantized. The of the gauge group SO(3), and the gauge fields W = µ perturbative spectrum of the theory can be found by ex- W aT a. Here, T a are the hermitian generators of SO(3) µ panding φa around the chosen vacuum as satisfying [T a,Tb]=if abcT c. In the adjoint representa- a a abc abc a a 0a tion, we have (T )bc = −ifbc and, for SO(3), f =  . φ = φvac + φ .

6 c c3 ~ A convenient choice is φvac = δ a. The perturbative To see that this actually solves (VI.1), note that ∂µφvac · spectrum (which becomes manifest after choosing an ap- φ~vac =0,sothat propriate unitary gauge) consists of a massive Higgs of spin zero with a square mass 1 (φ~ × ∂ φ~ ) × φ~ = ea2 vac µ vac vac M2 =2λa2, (V.9) 1 1 H ∂ φ~ a2 − φ~ (φ~ · ∂ φ ) = ∂ φ~ . (VI.3) ea2 µ vac vac vac µ vac e µ vac a massless photon, corresponding to the U(1) gauge bo-   3 1 2 son Aµ, and two charged massive W-, Aµ and Aµ, The first term on the right-hand side of Eq. (VI.2) is the with square mass particular solution, and φ~vacAµ is the general solution to the homogeneous equation. Using this solution, we can M2 = e2a2. (V.10) W now compute the field strength tensor G~ µν . The field strength F corresponding to the unbroken part of the Thismassspectrumisrealisticaslongasweareat µν gauge group can be identified as weak coupling, e2 ∼ λ  1. At strong coupling, nonper- turbative effects could change significatively eqs. (V.9) 1 F = φ~ · G~ = ∂ A − ∂ A and (V.10). But the fact that there is an unbroken sub- µν a vac µν µ ν ν µ group of the gauge symmetry ensures that there is some 1 + φ~ · (∂ φ~ × ∂ φ~ ) . (VI.4) massless gauge boson, which a long range interaction. a3e vac µ vac ν vac This is the characteristic of the Coulomb phase. Using the equations of motion in the Higgs vacuum it follows that VI. THE ’T HOOFT-POLYAKOV MONOPOLES µν µν ∂µF =0,∂µF=0. Let us look for time-independent, finite energy solu- This confirms that F is a valid U(1) field strength ten- tions in the Georgi-Glashow model. Finiteness of energy µν e sor. The magnetic field is given by Bi = − 1 ijkF .Let requires that as r , the energy density θ given by 2 jk →∞ 00 us now consider a static, finite energy solution and a sur- (V.6) must approach zero faster than 1/r3. This means face Σ enclosing the core of the solution. We take Σ to that as r , our solution must go over to a Higgs →∞ be far enough so that, on it, the solution is already in vacuum defined by (V.7). In the following, we will first the Higgs vacuum. We can now use the magnetic field assume that such a finite energy solution exists and show in the Higgs vacuum to calculate the magnetic charge g that it can have a monopole charge related to its soliton Σ associated with our solution: number which is, in turn, determined by the associated Higgs vacuum. This result is proven without having to i i gΣ = B ds deal with any particular solution explicitly. Next, we Σ will describe the ’t Hooft-Polyakov ansatz for explicitly Z 1 ~ j ~ k ~ i constructing one such monopole solution, where we will = − 3 ijk φvac · ∂ φvac × ∂ φvac ds . (VI.5) 2ea Σ also comment on the existence of Dyonic solutions. In Z   the last two subsections we will derive the Bogomol’nyi It turns out that the expression on the right hand side is bound and the Witten effect. a topological quantity as we explain below: Since φ2 = a; 2 the manifold of Higgs vacua (M0) has the topology of S . The field φ~vac defines a map from Σ into M0. Since Σ is A. The Topological nature of the magnetic charge. 2 also an S ,themapφvac :Σ→M0 is characterized by 2 its homotopy group π2(S ). In other words, φvac is char- For convenience, in this subsection we will use the vec- acterized by an integer ν (the winding number) which tor notation for the SO(3) gauge group indices and not counts the number of times it wraps Σ around M0.In for the spatial indices. terms of the map φvac, this integer is given by Let φ~vac denote the field φ~ in a Higgs vacuum. It then satisfies the equations 1 1 ~ j ~ k ~ i ν = 3 ijkφvac · ∂ φvac × ∂ φvac ds . (VI.6) 4πa Σ 2 2 Z φ~vac · φ~vac = a ,   ~ ~ ~ Comparing this with the expression for magnetic charge, ∂µφvac − e Wµ × φvac =0, (VI.1) we get the important result which can be solved for W~ . The most general solution −4πν µ g = . (VI.7) is given by Σ e 1 1 Hence, the winding number of the soliton determines its W~ = φ~ × ∂ φ~ + φ~ A . (VI.2) µ ea2 vac µ vac a vac µ monopole charge. Note that the above equation differs

7 from the Dirac quantization condition by a factor of 2. magnetic as well as electric charges. An ansatz for con- This is because the smallest electric charge which could structing such a solution was proposed by Julia and Zee a a exist in our model is e/2 for an spinorial representation [8]. In this ansatz, φ and Wi have exactly the same form a of SU(2), the universal covering group of SO(3). Then, as in the ’t Hooft-Polyakov ansatz, but W0 is no longer a a 2 in this model m0 =2. zero: W0 = x J(aer)/er . 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, B. The ’t Hooft-Polyakov ansatz. at the classical level, does not satisfy the quantization condition. However, semiclassical arguments show that, Now we describe an ansatz proposed by ’t Hooft [6] in CP invariant theories, and at the quantum level, the and Polyakov [7] for constructing a monopole solution in dyon electric charge is quantized as q = ne.Thiscan the Georgi-Glashow model. For a spherically symmet- be easily understood if we recognize that a monopole is ric, parity-invariant, static solution of finite energy, they not invariant under a gauge transformation which is, of proposed: course, a symmetry of the equations of motion. To deal with the associated zero-mode properly, the gauge degree xa φa = H(aer) , of freedom should be regarded as a collective coordinate. er2 Upon quantization, this collective coordinate leads to the xj existence of electrically charged states for the monopole W a = −a (1 − K(aer)) , i ij er2 with discrete charges. In the presence of a CP violating a W0 =0. (VI.8) term in the Lagrangian, the situation is more subtle as we will discuss later. In the next subsection, we describe For the non-trivial Higgs vacuum at r →∞, they chose a limit in which the equations of motion can be solved ex- c c c 2 φvac = ax /r = axˆ . Note that this maps an S at actly for the ’tHooft-Polyakov and the Julia-Zee ansatz. spatial infinity onto the vacuum manifold with a unit This is the limit in which the soliton mass saturates the winding number. The asymptotic behavior of the func- Bogomol’nyi bound. tions H(aer)andK(aer) are determined by the Higgs vacuum as r →∞and regularity at r = 0. Explicitly, defining ξ = aer,wehave:asξ→∞,H ∼ξ, K → 0 C. The Bogomol’nyi bound and the BPS states. and as ξ → 0,H ∼ξ, (K − 1) ∼ ξ.Themassofthis solution can be parameterized as In this subsection, we derive the Bogomol’nyi bound [9]

4πa 2 on the mass of a dyon in term of its electric and magnetic M = f (λ/e ) . µν µν e charges which are the sources for F = φ~ ·G~ /a.Using the Bianchi identity (V.5) and the first equation in (V.4), For this ansatz, the equations of motion reduce to two we can write the charges as coupled equations for K and H which have been solved exactly only in certain limits. For r → 0, one gets i 1 a i a 3 H ec r2 and K =1+ec r2 which shows that the g ≡ BidS = Bi (D φ) d x, → 1 2 2 a S∞ fields are non-singular at√ r =0.Forr→∞,we Z Z get H ξ + c exp( a 2λr)andK cξexp( ξ) i 1 a i a 3 → 3 − → 4 − q ≡ EidS = Ei (D φ) d x. (VI.9) a a j 2 2 a which leads to Wi ≈−ijx /er . Once again, defin- S∞ c c Z Z ing Fij = φ G /a, the magnetic field turns out to ij Now, in the center of mass frame, the dyon mass is given be Bi = −xi/er3. The associated monopole charge is by g = −4π/e, as expected from the unit winding number of the solution. It should be mentioned that ’t Hooft’s 3 3 1 a 2 a 2 definition of the Abelian field strength tensor is slightly M≡ d xθ00 = d x (E ) +(B ) 2 k k different but, at large distances, it reduces to the form Z Z  a 2 a 2 given above. +(Dkφ) +(D0φ ) +V(φ) , (VI.10) Note that in the above monopole solution, the presence   of the is not obvious. To extract the Dirac where, θµν is the energy momentum tensor. Using (VI.9) string, we have to perform a singular gauge transforma- and some algebra we obtain tion on this solution which rotates the non-trivial Higgs 1 vacuum φc = axˆc into the trivial vacuum φc = aδc3. M = d3x (Ea − D φa sin θ)2 vac vac 2 k k In the process,the gauge field develops a Dirac string sin- Z  a a 2 a 2 gularity which now serves as the source of the magnetic +(Bk−Dkφcos θ) +(D0φ ) charge [6]. + V(φ)) + a(q sin θ + g cos θ) , (VI.11) The ’t Hooft-Polyakov monopole carries one unit of  magnetic charge and no electric charge. The Georgi- where θ is an arbitrary angle. Since the terms in the first Glashow model also admits solutions which carry both line are positive, we can write M≥(qsin θ + g cos θ).

8 This bound is maximized for tan θ = q/g.Thusweget physics since the theory is no longer CP invariant. We the Bogomol’nyi bound on the dyon mass as want to construct the electric charge operator in this the- ory. The theory has an SO(3) gauge symmetry but the M≥a g2+q2. (VI.12) electric charge is associated with an unbroken U(1) which keeps the Higgs vacuum invariant. Hence, we define an For the ’t Hooft-Polyakovp solution, we have q =0,and operator N which implements a gauge rotation around thus, ag.Butg =4π/e and = ae = aq,so M≥ | | || MW ˆ a a that the φ direction with gauge parameter Λ = φ /a.These transformations correspond to the electric charge. Under 4π 4π 4π ν a a M≥a = M = M = M . N, a vector v and the gauge fields Aµ transform as e e2 W q2 W α W a 1 abc b c a 1 a Here, α is the fine structure constant and ν =1or1/4, δv =  φ v ,δAµ= Dµφ . depending on whether the electron charge is q or q/2. a ea Since α is a small (∼ 1/137 for electromagnetism), the Clearly, φa is kept invariant. At large distances where above relation implies that the monopole is much heavier |φ| = a, the operator e2πiN is a 2π-rotation about φˆ than the W-bosons associated with the symmetry break- and therefore exp (2πiN) = 1. Elsewhere, the rotation ing. angle is 2π|φ|/a. However, by Gauss’ law, if the gauge From (VI.11) it is clear that the bound is not satu- transformation is 1 at ∞, it leaves the physical states rated unless λ → 0, so that V (φ) = 0. This is the invariant. Thus, it is only the large distance behavior of Bogomol’nyi-Prasad-Sommerfield (BPS) limit of the the- the transformation which matters and the eigenvalues of 2 2 ory [9,10]. Note that in this limit, φvac = a is no longer N are quantized in integer units. Now, we use Noether’s determined by the theory and, therefore, has to be im- formula to compute N: posed as a boundary condition on the Higgs field. More- over, in this limit, the Higgs scalar becomes massless. δL δL N = d3x δAa + δφa . Now, to saturate the bound we set δ∂ Aa i δ∂ φa Z  0 i 0  a D0φ =0, ~ a a Since δφ = 0, only the gauge part (which also includes Ek =(Dkφ) sin θ, the θ-term) contributes: a a Bk =(Dkφ) cos θ, (VI.13) δ a aµν aoi ai where, tan θ = q/g. In the BPS limit, one can use the ’t a Fµν F =4F = −4E , δ∂0A Hooft-Polyakov (or the Julia-Zee) ansatz either in (V.4), i δ  or in (VI.13) to obtain the exact monopole (or dyon) ˜a aµν ijk a ai a Fµν F =2 Fjk = −4B . solutions [9,10]. These solutions automatically saturate δ∂0Ai the Bogomol’nyi bound and are referred to as the BPS   Thus, states. Also, note that in the BPS limit, all the per- turbative excitations of the theory saturate this bound 1 3 ~ ~i θe 3 ~ ~i and, therefore, belong to the BPS spectrum. As we will N = d xDiφ · E − d xDiφ·B ae 8π2a see later, BPS states appears in a very natural way in Z Z 1 θe theories with N = 2 supersymmetry. = Q − Q , e e 8π2 m

where, we have used (VI.9). Here, Qe and Qm are the D. The θ parameter and the Witten effect. electric and magnetic charge operators with eigenvalues q and g, respectively, and N is quantized in integer units. In this section we will show that in the presence of a θ- This leads to the following formula for the electric charge: term in the Lagrangian, the magnetic charge of a particle always contributes to its electric charge in the way given θe2 q = ne + g. by formula (III.2) [11]. 8π2 To study the effect of CP violation, we consider the Georgi-Glashow model with an additional θ-term as the For the ’t Hooft-Polyakov monopole, n =1,g=−4π/e, only source of CP violation: and therefore, q = e(1 − θ/2π). For a general dyonic solution we get 1 1 L = − F a F aµν + (D φa)2 − λ(φ2 − a2)2 4 µν 2 µ 4π θe g = n ,q=ne+n . (VI.15) θe2 e m e 2π m + F a F˜aµν . (VI.14) 32π2 µν and we recover (III.2) and (III.3) for q0 = e.Inthe ˜aµν 1 µνρσ a presence of a θ-term, a magnetic monopole always carries Here, F = 2  Fρσ. The presence of the θ-term does not affect the equations of motion but changes the an electric charge which is not an integral multiple of

9 some basic unit. In section III we introduced the charge There is also a discrete subgroup of transformations lattice of periods e and eτ. In this parameterization, the which still leave X in diagonal form. It is the Weyl Bogomol’nyi bound (VI.12) takes the form group of SU(N), which corresponds to permutations of √ the eigenvalues λi. We also fix the Weyl group with the M≥ 2|ae(ne + nmτ)| . (VI.16) convention λ1 >λ2 >···λN. At this stage, we have an Abelian U(1) gauge theory Notice that for a BPS state, equation (VI.16) implies with N − 1 photons, N(N − 1) charged vector particles that its mass is proportional to the distance of its lattice and some additional degrees of freedom that will appear point from the origin. presently.

VII. THE CONFINING PHASE. B. The nature of the gauge singularities.

A. The Abelian projection. So far we assumed that the eigenvalues λi coincide nowhere. But there are some gauge field configurations In non-Abelian gauge theories, gauge fixing is a subject that produce two consecutive eigenvalues to coincide at full of interesting surprises (ghosts, phantom solitons, ...) some spacetime points which often obscure the physical content of the theory [12]. λi = λi+1 = λ, for certain i. (VII.4) ’t Hooft gave a qualitative program to overcome these difficulties and provided a scenario that explains confine- These spacetime points are ‘singular’ points of the ment in a gauge theory. The idea is to perform the gauge Abelian projection. The SU(2) gauge subgroup corre- fixing procedure in two steps. In the first one a unitary sponding to the 2×2 block matrix with coinciding eigen- gauge is chosen for the non-Abelian degrees of freedom. values leaves invariant the gauge-fixing condition (VII.2). It reduces the non-Abelian gauge symmetry to the maxi- Let us consider the vicinity of such a point. Prior to mal Abelian subgroup of the gauge group. Here one gets the complete gauge-fixing we may take X to be particle gauge singularities §. This procedure is called D 000 the Abelian projection [12]. In this way, the dynamics 1 0λ+−i 0 of the Yang-Mils theory will be reduced to an Abelian X = 312 ,(VII.5)  0  + i λ −  0  gauge theory with certain additional degrees of freedom. 1 2 3 00 0D We need a field that transforms without derivatives  2   under gauge transformations. An example is a real field, where D1 and D2 may safely be considered to be diag- X in the adjoint representation of SU(N), onalized because the other eigenvalues do not coincide. With respect to that SU(2) subgroup of SU(N)thatcor- X ΩXΩ−1. (VII.1) → responds to rotations among the ith and i +1stcompo- Such a field can always be found; take for instance nents, the three fields a(x) form an isovector. We may a a write the central block as X = G12. We will use the field X to implement the uni- tary gauge condition which will carry us to the Abelian λI +  σa, (VII.6) projection of the SU(N) gauge group. The gauge is fixed 2 a by requiring that X be diagonal: where σa are the Pauli matrices. Consider static field configurations. The points of λ1 0 . space where the two eigenvalues coincide correspond to X = .. . (VII.2)   the points x0 that satisfy 0 λN a    (x0)=0. (VII.7) The eigenvalues of the matrix X are gauge invariant. Generically they are all different, and the gauge condi- These three equations define a single space point, and tion (VII.2) leaves an Abelian U(1)N−1 gauge symmetry. then the singularity is particle-like. Which is its physical It corresponds to the subgroup generated by the gauge interpretation?. a a transformations By analyticity we have that  ∼ (x − x0) ,andour a iω gauge condition corresponds to rotating the isovector  e 1 0 N . such that Ω= .. , ωi =0. (VII.3)   iωN i=1 0 0 e X    = 0 . (VII.8) |3| ! From the previous sections, we know that the zero-point § a We will discuss the physical meaning of them later on. of  at x0 behaves as a magnetic charge with respect

10 to the remaining U(1) ⊂ SU(2) rotations. We realize θ, the physical electric charge of a particle with electric that those gauge field configurations that produce such a (resp. magnetic) number ne (resp. nm)is: gauge ‘singularities’ correspond to magnetic monopoles. The non-Abelian SU(N) gauge theory is topologically θ q=(ne+ nm)e. (VII.9) such that it can be cast into a U(1)N−1 Abelian gauge 2π theory, which will feature not only electrically charged Dyons with large electric charges may have larger self- particles but also magnetic monopoles. energies contributing positively to their mass squared. If the state (ne,nm) condenses at θ ' 0, it is likely that the state (n 1,n ) condenses at θ 2π. It suggests C. The phases of the Yang-Mills vacuum. e − m ' that there is a phase transition around θ ' π. Such first order phase transitions has been observed in softly We can now give a qualitative description of the pos- broken N =2SQCDtoN= 0 [16]. sible phases of the Yang-Mills vacuum. It is only the ’t Hooft proposed a new condensation mode at θ ' π dynamics which, as a function of the microscopic bare [12]. He imagined the possibility that a bound state of the parameters, determines in which phase the Yang-Mills dyons (ne,nm)and(ne−1,nm), with zero electric charge vacuum is actually realized. at θ = π, could be formed. Its smaller electric charge Classically, the Yang-Mills Lagrangian is scale invari- could favor its condensation, leading to what he called ant. One can write down field configurations with mag- an oblique confinement mode. These oblique modes have netic charge and arbitrarily low energy. But quantum also been observed in softly broken N = 2 SQCD with corrections are likely to violate their masslessness. If matter [14,15]. 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 VIII. THE HIGGS/CONFINING PHASE. in a Higgs phase. It will be a signal that the ordinary Higgs mechanism has taken place in the Abelian gauge In the previous section we have characterized the con- formulation of the Yang-Mills theory. The role of the dy- fining phase as the dual of the Higgs phase, i.e., the phys- namically generated Higgs field could be done by some ical states are gauge singlets made by the electric degrees scalar composite operator charged respect the U(1)N−1 of freedom bound by stable electric flux tubes. A good gauge symmetries. There is also the possibility that no gauge invariant order parameter measuring such behav- Higgs phenomenon occurs at all in the Abelian sector, or ior is the Wilson loop [17]: that some U(1) gauge symmetries are not spontaneously broken. In this case we are in the Coulomb phase, with µ some massless photons, or in a mixed Coulomb-Higgs W (C) = Tr exp ig dx Aµ . (VIII.1) phase.  IC  There is a third possibility however. Maybe the quan- For SU(N) Yang-Mills in the confining phase, for con- tum corrections give a formally negative mass squared for tours C, the Wilson loop obeys the area law, the monopole: a magnetically charged object condenses. We apply an ‘electric-magnetic dual transformation’ to hW (C)i∼exp(−σ · (area)), (VIII.2) write an effective Lagrangian which encodes the relevant magnetic degrees of freedom in the infrared limit. In such with σ the string tension of the electric flux tube. effective Lagrangian, the Higgs mechanism takes place in But dynamical matter fields in the fundamental rep- terms of dual variables. We are in a dual Higgs phase. resentation immediately create a problem in identify- We have electric flux tubes with finite energy per unit of ing the confining phase of the theory through the Wil- length. There is a confining potential between electrically son loop. The criterion used for confinement in the charged objects, like . pure gauge theory, the energy between static sources, no In 1994, Seiberg and Witten gave a quantitative proof longer works. Even if the energy starts increasing as the that such dynamical mechanism of color confinement sources separate, it eventually becomes favorable to pro- takes place in N = 2 super-QCD (SQCD) broken to duce a particle-antiparticle pair out of the vacuum. This N = 1 [13], giving a non-trivial realization of ’t Hooft pair shields the gauge charge of the sources, and the en- scenario. When N = 2 SQCD is softly broken to N =0 ergy stops growing. So even in a theory that ‘looks’ very the same mechanism of confinement persists [14,15]. confining our signal fails, and the perimeter law replaces (VIII.2),

D. Oblique confinement. hW (C)i =∼ exp(−Λ · (perimeter)) (VIII.3)

For simplicity let us consider an SU(2) gauge group. If some scalar field is in the fundamental representation We have seen that for a non-zero CP violating parameter of the gauge group, there is no distinction at all between the confinement phases and the Higgs phase. Using the

11 scalar field in the fundamental representation one can This is a Clifford algebra with 2 fermionic generators build gauge invariant interpolating operators for all pos- and has a 2-dimensional representation. From the point sible physical states. As the of view of the angular momentum algebra, a is a ris- of the Higgs field in the fundamental representation con- ing operator and a† is a lowering operator for the helic- tinuously changes from large values to smaller ones, the ity of massless states. We choose the vacuum such that spectrum of all physical states, and all other measurable J3|Ωλ >= λ|Ωλ > and a|Ωλ >=0.Then quantities, changes smoothly [18]. There is no gauge in- † 1 † variant operator which can distinguish between the Higgs J3(a |Ωλ >)=(λ− )(a |Ωλ >). (IX.2) or confining phases. We are in a Higgs/confining phase. 2 In supersymmetric gauge theories, it is common to The irreducible representations are not necessarily have scalar fields in the fundamental representation of CPT invariant. Therefore, if we want to assign physical the gauge group, the scalar quarks. In such situation, states to these representations, we have to supplement when the theory is not in the Coulomb phase, we will see them with their CPT conjugates |−λ>CPT .Ifarep- that the theory is presented in a Higgs/confining phase. resentation is CPT self-conjugate, it is left unchanged. We could take the phase description which is more ap- Thus, from a Clifford vacuum with helicity λ =1/2we propriate for the theory. For instance, if the theory is in obtain the N = 1 supermultiplet: the weak coupling region, it is better to realize it in the Higgs phase; if the theory in the strong coupling region, {|1/2>, |−1/2> } CPT (IX.3) it is better to think it in a confining phase. {|0>, |0 >CPT }   which contains a Weyl spinor ψ and a complex scalar φ. It is called the scalar multiplet. IX. SUPERSYMMETRY The other relevant representation of a renormalizable quantum field theory is the vector multiplet. It is con- A. The supersymmetry algebra and its massless structed from a Clifford vacuum with helicity λ =1: representations. {|1>, |−1> } CPT . (IX.4) {|1/2>, |−1/2>CPT } The N = 1 supersymmetry algebra is written as [19]   µ It contains a vector Aµ and a Weyl spinor λ. {Qα, Qα˙ } =2σαα˙Pµ

{Qα,Qβ}=0,{Qα˙ ,Qβ˙}=0. (IX.1) B. Superspace and superfields. Here, Q and Q are the supersymmetry generators and transform as spin 1/2 operators, α, α˙ =1,2. Moreover, the supersymmetry generators commute with the mo- To make supersymmetry linearly realized it is conve- mentum operator P and hence, with P 2. Therefore, nient to use the superspace formalism and superfields µ [20]. Superspace is obtained by adding four spinor de- all states in a given representation of the algebra have α µ the same mass. For a theory to be supersymmetric, it is grees of freedom θ , θα˙ to the spacetime coordinates x . necessary that its particle content form a representation Under the supersymmetry transformations implemented α α˙ of the above algebra. The irreducible representations of by the operator ξ Qα + ξα˙ Q with transformation pa- (IX.1) can be obtained using Wigner’s method. rameters ξ and ξ, the superspace coordinates transform For massless states, we can always go to a frame where as µ P = E(1, 0, 0, 1). Then the supersymmetry algebra be- µ 0µ µ µ µ comes x → x = x + iθσ ξ − iξσ θ, θ→θ0 =θ+ξ, 00 Q , Q = . 0 { α α˙ } 04E θ→θ =θ+ξ. (IX.5)   In a unitary theory the norm of a state is always posi- These transformations can easily be obtained by the following representation of the supercharges acting on tive. Since Qα and Qα˙ are conjugate to each other, and (x, θ): {Q1, Q1˙ } = 0, it follows that Q1|phys >= Q1˙ |phys >=0. As for the other generators, it is convenient to re-scale ∂ α˙ them as Q = − iσµ θ ∂ , α ∂θα αα˙ µ 1 1 † ∂ µ a = √ Q2 ,a=√Q2˙. Q = + iθασ ∂ . (IX.6) 2 E 2E α˙ − α˙ αα˙ µ ∂θ Then, the supersymmetry algebra takes the form µ These satisfy {Qα, Qα˙ } =2iσαα˙ ∂µ. Moreover, using the {a, a†} =1, {a, a} =0, {a†,a†}=0. chain rule, it is easy to see that ∂/∂xµ is invariant under

12 (IX.5) but not ∂/∂θ and ∂/∂θ. Therefore, we introduce It can be verified that Wα is a chiral superfield. Since it is the super-covariant derivatives gauge invariant, it can be computed in the Wess-Zumino gauge, ∂ µ Dα = α + iσαα˙ ∂µ , ∂θ i µ ν Wα = −iλα(y)+θαD− (σ σ θ)αFµν ∂ µ α 2 Dα˙ = − α˙ − iσαα˙ θ ∂µ . (IX.7) 2 µ ∂θ + θ (σ ∂µλ)α , (IX.12) µ They satisfy {Dα, Dα˙ } = −2iσαα˙ ∂µ and anti-commute where, Fµν = ∂µAν − ∂ν Aµ. with Q and Q. In the non-Abelian case, V belongs to the adjoint A The quantum fields transform as components of a su- representation of the gauge group: V = VAT ,where, perfield defined on superspace, F (x, θ, θ). Since the θ- TA† =TA. The gauge transformations are now imple- variables are anti-commuting, the Taylor expansion of mented by F (x, θ, θ)in(θ, θ) is finite, indicating that the supersym- † metry representations are finite dimensional. The coeffi- e−2V → e−iΛ e−2V eiΛ , cients of the expansion are the component fields. A To have irreducible representations we must impose where Λ = ΛAT is a chiral superfield. The non-Abelian supersymmetric invariant constraints on the superfields. gauge field strength is defined by The scalar multiplet (IX.3) is represented by a chiral 1 2 scalar superfield, Φ, satisfying the chiral constraint W = D e2V D e−2V α 8 α D Φ=0. (IX.8) α˙ and transforms as µ µ µ µ Note that for y = x + iθσ θ,wehaveDα˙y = iΛ iΛ β W W 0 = e− W e . 0, Dα˙θ =0.Therefore, any function of (y,θ)isa α → α α chiral superfield. It can be shown that this also is a In components, in the WZ gauge it takes the form necessary condition. Hence, any chiral superfield can be expanded as a a a i µ ν a √ Wα = −iλα + θαD − (σ σ θ)αFµν Φ(y,θ)=φ(y)+ 2θψ(y)+θθF(y) . (IX.9) 2 2 µ a + θ σ Dµλ , (IX.13) Here, ψ and φ are the fermionic and scalar components respectively and F is an auxiliary field linear and homo- where, geneous. Similarly, an anti-chiral superfield is defined by † a a a abc b c DαΦ = 0 and can be expanded as Fµν = ∂µAν − ∂ν Aµ + f AµAν , a a c √ D λ = ∂ λ + f abcAb λ . Φ†(y†, θ)=φ†(y†)+ 2θψ(y†)+θθF †(y†) , (IX.10) µ µ µ where, yµ† = xµ − iθσµθ. Now we are ready to construct supersymmetric La- The vector multiplet (IX.4) is represented off-shell by grangians in terms of superfields. a real scalar superfield V = V †. (IX.11) C. Supersymmetric Lagrangians. In local quantum field theories, spin one massless par- Clearly, any function of superfields is, by itself, a super- ticles carry gauge symmetries [21]. These symmetries field. Under supersymmetry, the superfield transforms commute with the supersymmetry transformations. For as δF =(ξQ + ξQ)F, from which the transformation a vector superfield, many of its component fields can of the component fields can be obtained. Note that the be gauged away using the Abelian gauge transformation 2 2 V → V +Λ+Λ†,whereΛ(Λ†) are chiral (anti-chiral) coefficient of the θ θ component is the field component superfields. In the Wess-Zumino gauge [19], it becomes of highest dimension in the multiplet. Then, its vari- ation under supersymmetry is always a total derivative 2 1 2 V = −θσµθA + iθ2θλ − iθ θλ + θ2θ D. of other components. Thus, ignoring surface terms, the µ 2 spacetime integral of this component is invariant under supersymmetry. This tells us that a supersymmetric La- 2 1 µ 2 2 3 In this gauge, V = 2 AµA θ θ and V = 0. The Wess- grangian density may be constructed as the highest di- Zumino gauge breaks supersymmetry, but not the gauge mension component of an appropriate superfield. symmetry of the Abelian gauge field Aµ. The Abelian Let us first consider the product of a chiral and an superfield gauge field strength is defined by anti-chiral superfield Φ†Φ. This is a general superfield 1 2 1 and its highest component can be computed using (IX.9) W = − D D V, W =− D2D V. α 4 α α˙ 4 α˙ as

13 † 1 † 1 † 1 † µ where, τ = θ/2π +4πi/g2. Φ Φ | 2 = − φ 2φ − 2φ φ + ∂µφ ∂ φ θ2θ 4 4 2 We now include matter fields by the introduction of the i µ i µ † chiral superfield Φ in a given representation of the gauge − ψσ ∂µψ + ∂µψσ ψ + F F. (IX.14) a 2 2 group in which the generators are the matrices Tij .The † Dropping some total derivatives we get the free field La- kinetic energy term Φ Φ is invariant under global gauge transformations Φ0 = e−iΛΦ. In the local case, to insure grangian for a massless scalar and a massless 0 with an auxiliary field. that Φ remains a chiral superfield, Λ has to be a chiral superfield. The supersymmetric gauge invariant kinetic The product of chiral superfields is a chiral superfield. † −2V In general, any arbitrary function of chiral superfields is energy term is then given by Φ e Φ. We are now in a chiral superfield: a position to write down the full N=1 supersymmetric gauge invariant Lagrangian as √ W(Φi)=W(φi+ 2θψi + θθFi) 1 2 α ∂W√ L = Im τTr d θW Wα = W(φi)+ 2θψi 8π ∂φi  Z  2 2 † −2V 2 2 ∂W 1 ∂2W + d θd θ (Φ e Φ) + d θ W + d θ W . (IX.18) + θθ Fi − ψiψj . (IX.15) ∂φi 2 ∂φiφj Z Z Z   Note that since each term is separately invariant, the W is referred to as the superpotential. Moreover, the relative normalization between the scalar part and the space of the chiral fields Φ may have a non-trivial met- Yang-Mills part is not fixed by N = 1 supersymmetry. ric gij in which case the scalar kinetic term, for exam- In fact, under loop effects, by virtue of the perturbative ij † µ ple, takes the form g ∂µφi ∂ φj , with appropriate mod- non- theorem [23], only the term with ifications for other terms. In such cases, the free field the complete superspace integral d2θd2θ gets an overall Lagrangian above has to be replaced by a non-linear σ- renormalization factor Z(µ, g(µ)), with µ the renormal- model [22]. Thus, the most general N = 1 supersymmet- ization scale and g(µ) the renormalizedR gauge coupling ric Lagrangian for the scalar multiplet is given by constant. Observe the unique dependence on Re(τ)in Z, breaking the holomorphic τ-dependence of the La- L = d4θK(Φ, Φ†)+ d2θW(Φ) + d2θW(Φ†) . grangian L. But quantities as the superpotential W are renormalization group invariant under perturbation the- Z Z Z ory [23] (we will see dynamically generated superpoten- Note that the θ-integrals pick up the highest component tials by nonperturbative effects). 2 2 of the superfield and in our conventions, d θθ =1 In terms of component fields, the Lagrangian (IX.18) 2 2 and d θ θ = 1. In terms of the non-holomorphicR becomes function K(φ, φ†), the metric in field space is given by R 1 a aµν θ a aµν gij = ∂2K/∂φ ∂φ†, i.e., the target space for chiral su- L = − F F + F F i j 4g2 µν 32π2 µν perfields is always a K¨ahler space. For this reason, the i a 1 function K(Φ, Φ†) is referred to as the K¨ahler potential. − λaσµD λ + DaDa e g2 µ 2g2 Remember that the super-field strength Wα is a chiral a b a a † µ aµ a a † a superfield spinor. Using the normalization Tr(T T )= +(∂µφ−iAµT φ) (∂ φ − iA T φ) − D φ T φ 1 ab 2δ ,wehavethat µ a a † − i ψσ (∂µψ − iAµT ψ)+F F √ 2 α a µ a 1 a a † a a ∂W 1 ∂ W Tr(W Wα ) |θθ = −iλ σ Dµλ + D D + −i 2φ T λ ψ+ F − ψψ + h.c. . 2 ∂φ 2 ∂φ∂φ 1 i   − F aµν F a + µνρσ F a F a . (IX.16) (IX.19) 4 µν 8 µν ρσ Here, W denotes the scalar component of the superpo- The first three terms are real and the last one is pure tential. The auxiliary fields F and Da can be eliminated imaginary. It means that we can include the gauge cou- by using their equations of motion: pling constant and the θ parameter in the Lagrangian in acompactform ∂W F = , (IX.20) ∂φ 1 L= Im τ Tr d2 θWαW a 2 † a 4π α D = g (φ T φ) . (IX.21)  Z  1 θ =− Fa F aµν + F a F˜aµν The terms involving these fields, thus, give rise to the 4g2 µν 32π2 µν scalar potential 1 1 a + ( DaDa − iλaσµD λ ) , (IX.17) 1 g2 2 µ V = |F |2 + DaDa . (IX.22) 2g2

14 a † a Using the supersymmetry algebra (IX.1) it is not difficult D = φf Tf φf . (X.1) 0 to see that the hamiltonian P = H is a positive semi- f X definite operator, hHi≥0, and that the ground state has a zero energy if and only if it is supersymmetric invariant. The solutions of D = 0 usually lead to the concept of flat At the level of local fields, the equation (IX.22) means directions. They play an important role in the analysis that the supersymmetric ground state configuration is of SUSY theories. These flat directions may be lifted by such that F -terms in the Lagrangian, as for instance mass terms. As an illustrative example of flat directions and some F = Da =0. (IX.23) of its consequences, consider the SU(2) gauge group, one chiral superfield Q in the fundamental representa- tion of SU(2) and another chiral superfield Q˜ in the anti-fundamental representation of SU(2). This is su- D. R-symmetry. persymmetric QCD (SQCD) with one massless flavor. In this particular case, the equation (X.1) becomes

The supercharges Qα and Qα˙ are complex spinors. In a † a a † the supersymmetry algebra (IX.1) there is a U(1) sym- D = q σ q − qσ˜ q˜ . (X.2) metry associated to the phase of the supercharges: The equations Da = 0 have the general solution (up to gauge and global symmetry transformations) Q → Q0 = eiβ Q 0 −iβ a Q → Q = e Q. (IX.24) q =˜q†= ,aarbitrary . (X.3) 0 This symmetry is called the R-symmetry. It plays an im-   portant role in the study of supersymmetric gauge theo- The scalar superpartners of the fermionic quarks, (q, q), ries. called squarks, play the role of Higgs fields. As these are In terms of superspace, the R-symmetry is introduced in the fundamental representation of the gauge group, through the superfield generator (θQ + θQ). Then, it SU(2) is completely broken by the super-Higgs mecha- rotates the phase of the superspace components θ and nism (for a 6= 0). It is just the supersymmetric gen- θ in the opposite way as Q and Q.ItgivesdifferentR- eralization of the familiar Higgs mechanism: three real charges for the component fields of a superfield. Consider scalars are eaten by the , in the adjoint represen- that the chiral superfield Φ has R-charge n, tation, and three Weyl spinor combinations of the spinors are eaten by the to form a massive Dirac Φ(x, θ) → Φ0(x, θ)=einβΦ(x, e−iβ θ) . (IX.25) spinor in the adjoint of SU(2). and ac- quire the classical square mass In terms of its component fields we have that: 2 2 2 Mg =2g0|a| , (X.4) φ → φ0 = einβ φ, ψ →ψ0 =ei(n−1)βψ, where g0 is the bare gauge coupling. We see that the the- F →F0 =ei(n−2)βF. ory is in the Higgs/confining phase. But there is not mass gap; it remains a massless superfield. Its corresponding Since d2(e−iβ θ)=e2iβ d2θ, we derive that the superpo- massless scalar must move along some flat direction of tential has R-charge two, the classical potential. This flat direction is given by the arbitrary value of the real number |a|. This degen- W(Φ) →W(Φ0,θ)=e2iβ W(Φ,e−iβ θ) , (IX.26) eracy is not unphysical, as in the spontaneous breaking of a symmetry. When we move along the supersymmet- and that the K¨ahler potential is R-neutral. ric flat direction the physical observables change, as for instance the gluon mass (X.4). Different values of |a| cor- respond to physically inequivalent vacua. The space they X. THE USES OF SUPERSYMMETRY. expand is called the moduli space. It would be nice to have a gauge invariant parameterization of such an addi- A. Flat directions and super-Higgs mechanism tional parameter of the gauge invariant vacuum. It can only come from the vacuum expectation value of some We have seen that the fields configuration of the super- gauge invariant operator, since it is an independent new classical parameter which does not appear in the bare symmetric ground state are those corresponding to zero energy. To find them we solve (IX.23). Consider a super- Lagrangian. The simplest choice is to take the following gauge invariant chiral superfield: symmetric gauge theory with gauge group G, and matter superfields Φi in the representation R(f)ofG. The clas- M = QQ.˜ (X.5) sical equations of motion of the Da (a =1, ..., dim G) auxiliary fields give Classically, its vacuum expectation value is

15 hMi = |a|2, (X.6) B. Wilsonian effective actions and holomorphy. a gauge invariant statement and a good parameterization of the flat direction. The concept of Wilsonian effective action is simple. There is one consequence of the flat directions in su- Any physical process has a typical scale. The idea of the persymmetric gauge theories that, when combined with Wilsonian effective action is to give the Lagrangian of the property of holomorphy, will be important to obtain some physical processes at its corresponding characteris- exact results in supersymmetric theories. SQCD depends tic scale µ: 2 of the complex coupling τ(µ)=θ(µ)/2π+4πi/g (µ)at (µ) i scale µ. The angle θ(µ) measures the strength of CP vi- L (x)= g(µ)Oi(x, µ) . (X.9) i olation at scale µ. By , the theory X is weakly coupled at scales higher than the dynamically Oi(x, µ) are some relevant local composite operators of generated scale Λ , which is defined by | | the effective fields ϕa(p, µ). These are the effective de- 2πiτ(µ0) grees of freedom at scale µ, with momentum modes p Λ ≡ µ0e b0 , (X.7) running from zero to µ. There could be some symme- tries in the operators that our physical system could where µ0 is the ultraviolet cut-off where the bare parame- Oi ter τ0 = τ(µ0) is defined, and b0 is the one-loop coefficient realize in some way, broken or unbroken. The constants i of the beta function, g (µ) measure the strength of the interaction Oi of ϕa at scale µ. ∂g 2 2 4 µ (µ)=g −b0(g /16π )+O(g ) . (X.8) Behind some macroscopic physical processes, there is ∂µ usually a microscopic theory, with a bare Lagrangian (µ ) The complex parameter Λ is renormalization group in- L 0 (x) defined at scale µ0. The microscopic theory has variant in the scheme of the Wilsonian effective actions, also its characteristic scale µ0, much higher than the low where holomorphy is not lost (see below). Observe also energy scale µ. Also its corresponding microscopic de- that the bare instanton angle θ0 plays the role of the grees of freedom, φj (p, µ0), may be completely different complex phase of Λb0 . than the macroscopic ones ϕa(p, µ). The bare Lagrangian At scales µ ≤Mg all the gluons decouple and the rel- encodes the dynamics at scales below the ultraviolet cut- evant degrees of freedom are those of the ‘meson’ M.Its off µ0. The effective Lagrangian (X.9) is completely de- (µ ) self-interactions are completely determined by the ‘mi- termined by the microscopical Lagrangian L 0 (x). It croscopic’ degrees of freedom of the super-gluons and is obtained by integrating out the momentum modes p super-quarks. We must perform a matching condition between µ and µ0. It gives the values of the effective i for the physics at some scale of order Mg; the renormal- couplings in terms of the bare couplings g0(µ0), ization group will secure the physical equivalence at the i i i g (µ)=g(µ;µ0,g0(µ0)) . (X.10) other energies. If Mg  Λ, this matching takes place at weak coupling, where perturbation theory in the gauge In the macroscopic theory there is no reference to the coupling g is reliable, and we can trust the semiclassi- scale µ0. Physics is independent of the ultraviolet cut-off cal arguments, like those leading to formulae (X.4) and µ0: (X.6). So far we have shown the existence of a flat direction ∂gi at the classical level. When quantum corrections are in- =0. (X.11) ∂µ0 cluded, the flat direction may disappear and a definite i value of hM i is selected. For the Wilsonian effective The µ0-dependence on the bare couplings g0(µ0) cancel description in terms of the relevant degrees of freedom the explicit µ0-dependence in (X.10). This is the action M, this is only possible if a superpotential W(M)isdy- of the renormalization group. It allows to perform the namically generated for M. By the perturbative non- continuum limit µ0 →∞without changing the low en- renormalization theorem, this superpotential can only ergy physics. be generated by nonperturbative effects, since classically In supersymmetric theories, there are some operators there was no superpotential for the massless gauge singlet Oi(z), depending only on z =(x, θ), the chiral super- M because of the masslessness of the quark multiplet. space coordinate, not on θ. Clearly, their field con- If we turn on a bare mass for the quarks, m, the flat tent can only be made of chiral superfields. Those direction is lifted at classical level and a determined value of most relevant physical importance are the superpo- of mass dependent function hMi is selected. But the tential W(Φi,τ0,mf), and the gauge kinetic operator α advantage of the flat direction to carry hMi→∞to be τ(µ/µ0,τ0)W Wα. We say that the superpotential W at weak coupling is not completely lost. This limit can and the effective gauge coupling τ are holomorphic func- now be performed by sending the free parameter m to tions, with the chiral superfields Φi, the dimensionless the appropriate limit, as far as we are able to know the quotient µ/µ0 and the bare parameters τ0 and mf play- mass dependence of the vacuum expectation value of the ing the role of the complex variables. The K¨ahler poten- † meson superfield M. Here holomorphy is very relevant. tial K(Φ , Φ) is a real function of the variables Φi, but

16 α as far as supersymmetry is not broken and the theory ( i lnZi) W Wα, giving a non-holomorphic contribu- is not on some Coulomb phase, the vacuum structure is tion to the effective coupling τ.ForN= 2 theories, P determined by the superpotential in the limit µ → 0. Zi = 1 and holomorphy is not lost for τ [26,27]. We know that complex analysis is substantially more powerful than real analysis. For instance, there are a lot of real functions f(x)thatatx→0andx→∞go like f(x) → x. But there is only one holomorphic function XI. N =1SQCD. f(z)(∂zf(z) = 0) with those properties: f(z)=z.The holomorphic constraint is so strong that sometimes the A. Classical Lagrangian and symmetries. symmetries of the theory, together with some consistency conditions, are enough to determine the unique possible We now analyze N = 1 SQCD with gauge group form of the functions W and τ [24]. SU(N )andN flavors ∗∗ . The field content is the An illustrative example is the saturation at one-loop c f following: There is a spinor chiral superfield Wα in the of the holomorphic gauge coupling τ(µ/µ0,τ0)atanyor- 2 adjoint of SU(Nc), which contains the gluons Aµ and the der of perturbation theory. Since τ0 = θ0/2π + i4π/g0, gluinos λ. The matter content is given by 2Nf scalar chi- physical periodicity in θ0 implies ral superfields Qf and Q˜f , f,f =1, ..., Nf ,intheNc and ∞ µ µ Nc representations of SU(Nc) respectively. The renor- τ( ,τ )=τ + c e2πniτ0 , (X.12) µ 0 0 n µ malizable bare Lagrangian ise the following: 0 n=0 0 X   1 where the sum is restricted to n ≥ 0toensureawell L = Im τ d2θWαW SQCD 8π 0 α defined weak coupling limit g0 → 0. The unique term  Z  compatible with perturbation theory is the n =0term. + d4θ Q†e−2VQ +Q˜ e2VQ˜† Terms with n>0 corresponds to instanton contributions. f f f f The function c (t)mustsatisfyc(tt)=c (t )+c (t ) Z   0 0 1 2 0 1 0 2 2 ˜ and hence it must be a logarithm. Hence + dθmfQfQf +h.c. , (XI.1) Z  µ ib µ τ ,τ =τ + 0 ln , (X.13) with τ = θ /2π + i4π/g2 and m the bare couplings. In pert µ 0 0 2π µ 0 0 0 f  0  0 the massless limit the global symmetry of the classical Lagrangian is SU(Nf)L × SU(Nf)R × U(1)B × U(1)A × with b0 the one-loop coefficient of the beta function. We can use the definition (X.7) of the dynamically generated U(1)R.ForNc=2therepresentations2and 2 are equiv- scale Λ to absorb the bare inside the alent, and the global symmetry group is enlarged. In logarithm general we consider Nc > 2. The U(1)A and U(1)R sym- metries are anomalous and are broken by instanton ef- µ ib0 µ fects. But we can perform a linear combination of U(1)A τpert = ln , (X.14) Λ 2π Λ and U(1)R,callitU(1)AF , that is anomaly free. We   have the following table of representations for the global showing explicitly the independence of the effective gauge symmetries of SQCD: coupling in the ultraviolet cut-off µ0. We would like to comment that the one-loop saturation of the perturbative beta function and the renormalization SU(Nf)L SU(Nf)R U(1)B U(1)AF group invariance of the scale Λ can be lost by the effect of the Konishi anomaly [25,26]. In general, after the in- Wα 1 1 0 1 tegration of the modes µ

17 2 Consider now that the fermionic quarks ψ have charge Rψ SU(Nc−Nf). By the super-Higgs mechanism, Nc −(Nc− 2 2 under an U(1)AF transformation. In the one-instanton Nf ) =2NcNf −Nf chiral superfields are eaten by the sector, λ has 2N zero modes, and one for each Q and 2 c f vector superfields. This leaves 2Nf Nc −(2Nf Nc −Nf )= Q˜ . In total we have 2N +2N R =0toavoidthe 2 f c f ψ Nf chiral superfields. They can be described by the me- anomalies. We derive that Rψ = −Nc/Nf . Since this is son operators the charge of the , the superfields (Qf , Q˜f )have Mfg ≡ Q˜fQg. (XI.5) RAF charge 1 − Nc/Nf =(Nf −Nc)/Nf . which provide a gauge invariant description of the clas- sical moduli space. B. The classical moduli space. b) Nf ≥ Nc: In this case the general solution of (XI.3) is:

The classical equations of motion of the auxiliary fields v1 0 ··· 0 ··· 0 are . .  0 v2 . . qf = . . , (XI.6) F qf = −mf q˜f =0, .. .  . F = m q =0,  v ··· 0 q˜f − f f  Nc    a † a a † D = qfT qf −q˜fT q˜f =0. (XI.3) v˜1 0 ··· 0 ··· 0 f . . X  0˜v . . q˜† =  2 , (XI.7) f . . If there is a massive flavor mf 6= 0, then we must have .. .  . qf =˜qf = 0. As we want to go to the infrared limit to  v˜ 0  Nc ···  analyze the vacuum structure, the interesting case is the   situation of Nf massless flavors. If some quark has a non- with the parameters vi,˜vi (i=1, ..., Nc) subject to the zero mass m, its physical effects can be decoupled at very constraint 2 2 low energy, by taking into account the appropriate phys- |vi| −|v˜i| = constant independent of i. (XI.8) ical matching conditions at the decoupling scale m (see below). If all quarks are massive, in the infrared limit Now the gauge group is completely higgsed. The gauge invariant parameterization of the classical moduli space we only have a pure SU(Nc) supersymmetric gauge the- 2 must be done by 2Nf Nc −(Nc −1) chiral superfields. For ory. The Witten index of pure SU(Nc) super Yang-Mills 2 F instance, if Nf = Nc, we need Nc + 1 superfields. The is tr(−1) = Nc [29]. We know that supersymmetry is 2 not broken dynamically in this theory, and that there are meson operators Mfg provide Nc . The remaining degree of freedom comes from the -like operators Nc equivalent vacua. The 2Nc gaugino zero modes break f1···fN the U(1)R symmetry to Z2Nc by the instantons. Those B =  f Q ···Q , f1 fNf Nc vacua corresponds to the spontaneously broken dis- f ···f B˜ = 1 Nf Q˜ ···Q˜ , (XI.9) crete symmetry Z2Nc to Z2 by the gaugino condensate f1 fNf hλλi6=0. with the color indices also contracted by the -tensor. If there are some massless super-quarks, they can These are two superfields, but there is a holomorphic have non-trivial physical effects on the vacuum structure. constraint Consider that we have Nf massless flavors. We can look ˜ at the qf andq ˜f scalar quarks as Nc × Nf matrices. Us- detM − BB =0. (XI.10) ing SU(Nc)×SU(Nf) transformations, the qf matrix can 2 For Nf = Nc + 1, we need 2Nc(Nc +1)−(Nc −1) = be rotated into a simple form. There are two cases to be 2 Nc +2Nc + 1 independent chiral superfields. We can distinguished: construct the baryon operators: a) N

18 XII. THE VACUUM STRUCTURE OF SQCD we assign the charge 2 − 2(Nf − Nc)/Nf =2Nc/Nf to WITH NF

A. The Afleck-Dine-Seiberg’s superpotential. of ΛNf ,Nc and m. To implement the same action under SU(Nf)L × SU(Nf)R rotations, we must have First we consider the case of massless flavors. At hMi = f(detm, Λ )m−1. (XII.3) the classical level there are flat directions parameter- Nf ,Nc ized by the free vacuum expectation values of the me- The dependence in detm of the function f is determined son fields Mfg. They belong to the representation by the RAF charge. Then, the ΛNf ,Nc dependence is (Nf , Nf , 0, 2(Nf −Nc)/Nf ) of the global symmetry group worked out by dimensional analysis. The result is SU(Nf)L×SU(Nf)R×U(1)B ×U(1)AF . If nonperturba- 1 tive effects generate a Wilsonian effective superpotential 3N N Nc hMi =(const) Λ c− fdet m m−1 . (XII.4) W, it must depend in a holomorphic way of the light chi- Nf,Nc ral superfields Mfg and the bare coupling constant τ0.   The renormalization group invariance of the Wilsonian The Nc roots give Nc vacua. Observe that this is an exact effective action demands that the dependence on the bare result, and valid also for Nf ≥ Nc. There is only an di- coupling constant τ0 of W enters thought the dynami- mensionless constant (in general Nf and Nc dependent) to be determined. It would be nice to be able to carry cally generated scale ΛNf ,Nc . The invariance of W under SU(Nf)L × SU(Nf)R rotations reduces the dependence its computation in the weak coupling limit, since holo- in the mesons fields to the combination detM.Thereis morphy would allow to extend (XII.4) also to the strong coupling region. only one holomorphic function W = W(detM,ΛNf,Nc ), with RAF charge two that can be built from the variables The result (XII.4) suggest the existence of an effec- tive superpotential out of which (XII.4) can be obtained. detM and ΛNf ,Nc , which have RAF charge 2(Nf − Nc) and zero, respectively. It is the Afleck-Dine-Seiberg’s su- Holomorphy and symmetries tell us that the possible su- perpotential [30,31] perpotential would have to be

1 1 (Nc−N ) (Nc−Nf ) ΛN ,N f ΛNf ,Nc f c W = c , (XII.1) W(M,ΛNf ,Nc ,m)= · Nf ,Ng detM detM    −1 (Nc−Nf ) ΛNf ,Nc where cNf ,Nc are some undetermined dimensionless con- f t =tr(mM) . (XII.5) stants. If c = 0, (XII.1) corresponds to an exact detM Nf ,Nc 6   ! nonperturbative dynamically generated Wilsonian super- potential. It has catastrophic consequences, the the- In the limit of weak coupling, ΛNf ,Nc → 0, we know that ory has no vacuum. If we try to minimize the energy f(t)=cNf,Nc + t. But we can play at the same time derived from the superpotential (XII.1) we find that with the free values of m to reach any desired value of |hdetMi| → ∞. t. This fixes the function f(t) and the superpotential

W(M,ΛNf,Nc ,m)tobe

1 B. Massive flavors. ΛN ,N (Nc−Nf ) W(M,Λ ,m)=c f c Nf ,Nc Nf,Nc detM   When we add mass terms for all the flavors we expect + tr (mM). (XII.6) to find some physical vacua. In fact, by Witten index, we should find Nc of them. To verify this, let us try to As a consistency check, when we solve the equations compute hMfgi taking advantage of its holomorphy and ∂W/∂M = 0, we obtain the previously determined vac- symmetries. uum expectation values (XII.4). A bare mass matrix mfg 6= 0 breaks explicitly the Finally, we have to check the non-vanishing of cNf ,Nc . SU(Nf) × SU(Nf)R × U(1)AF global symmetry of the We take advantage of the decoupling theorem to obtain bare Lagrangian (XI.1). In terms of the meson operator further information about the constants cNf ,Nc .Letus the mass term is add a mass term m only for the Nf flavor,

Wtree =tr(mM). (XII.2) 1 ΛN ,N (Nc−Nf ) W(M,Λ ,m)= f c Nf ,Nc detM We see that, under an L and R rotation of SU(Nf)L and   SU(Nf)R respectively, we can recover the SU(Nf)L × + mMNf Nf . (XII.7) SU(Nf)R invariance if we require m to transform as m → L−1mR. In the same way, as the superpotential Solving for the equations: has R-charge two, the U(1)AF invariance is recovered if

19 ∂W (M,Λ ,m)=0, from the one-instanton sector. A direct instanton calcu- Nf ,Nc †† ∂MfNf lation reveals that the constant c2,1 6= 0 [31] . ∂W For Nf

Nc−Nf 1 from the classical one. This is the case of Nf = Nc. cN ,N Nc−Nf +1 ΛN 1,N (Nc−Nf +1) f c f − c , (XII.11) Nc − Nf detMˆ     XIII. THE VACUUM STRUCTURE OF SQCD and we obtain the relation WITH NF = NC .

Nc−Nf Nc−Nf+1 cN ,N cN 1,N f c = f − c . A. A quantum modified moduli space. N − N N − N +1  c f   c f  (XII.12) For Nf = Nc , the classical moduli space is spanned by the gauge singlet operators Mfg, B and B˜ subject to the Similarly, we can try to obtain another relation between constraint detM − BB˜ = 0. At quantum level, instanton the constants cNf ,Nc for different numbers of colors. To effects could change the classical constraint to this end we give a large expectation value to MNf Nf with respect the expectation values of Mˆ . Then below the detM − BB˜ =Λ2Nc, (XIII.1) scale hMNf Nf i we have SQCD with Nc − 1 colors and 2 2Nc −8π/g +iθ Nf − 1 flavors. Following the same strategy as before we since Λ ∼ e corresponds to the one- instanton factor, it has the right dimensions, and the find that cNf −1,Nc−1 = cNc,Nf .ItmeansthatcNc,Nf = operators (Q , Q˜ )haveR charge zero. cNf −Nc , which together with the relation (XII.12) gives f f AF To check if the quantum correction (XIII.1) really takes

cNf ,Nc =(Nc−Nf)c1,2. (XII.13) place, add a mass term for the quarks. The unique pos- sible holomorphic term with RAF charge two that can be We just have to compute the dimensionless constant c1,2 generated with the variables (Mfg,B,B,˜ Λ,m)is of the gauge group SU(2) with one flavor. In this case, or for the general case of Nf = Nc −1, the gauge group is completely higgsed and there are not infrared divergences in the instanton computation. In the weak coupling limit †† the unique surviving nonperturbative contributions come In the DR scheme c2,1 = 1 [32]. If we do not say the contrary, we will work on such a scheme.

20 W =trmM . (XIII.2) with SU(Nf)V the diagonal part of SU(Nf)×SU(Nf)R. To check it, the unbroken symmetries must satisfy the ’t Imagine now that the Nc-flavor is much heavier, with Hooft’s anomaly matching conditions [37]. bare mass m, than the Nc − 1 other ones, with bare mass With respect to the unbroken symmetries the quan- matrixm ˆ . The degree of freedom MNcNc is given by the tum numbers of the elementary and composite massless ˜ constraint. Locate at B = B = MfNc =0.Byequation fermions, at high and low energy respectively, are (XII.4) we know that the (Nc − 1) × (Nc − 1) matrix Mˆ is determined to be SU(Nf)V U(1)B U(1)AF 1 N Mˆ = Λ2Nc+1 detm ˆ c mˆ −1, (XIII.3) Nc−1,Nc λ 1 0 1   ψq Nf 1 −1 which has a non-zero determinant. It indicates that the ψ N −1 −1 constraint (XIII.1) is really generated at quantum level q˜ f [36]. As a final check, consider the simplest situation 2 ψM Nf − 1 0 −1 of Nc − 1 massless flavors. When we use the constraint ˆ ψB 1 Nf −1 (XIII.1) to express MNcNc as function of detM we obtain ψB˜ 1 −Nf −1 mΛ2Nc W = , (XIII.4) detMˆ 2 Observe there are only Nf − 1 independent meson fields, the Afleck-Dine-Seiberg’s superpotential for N = N 1 arranged in the adjoint of SU(Nf)V , since the constraint f c − 2 massless flavors. (XIII.1) eliminates one of them. There are Nf −1 gluinos Far from the origin of the moduli field space we are and Nf extra components for each quark ψq and anti- at weak coupling and the quantum moduli space given quark ψq˜ because of the gauge group SU(Nc). The by the constraint (XIII.1) looks like the classical mod- anomaly coefficients are: uli space (XI.10). But far from the origin of order Λ, the one-instanton sector is sufficiently strong to change triangles high energy low energy significatively the vacuum structure. Observe that the ˜ classically allowed point M = B = B = 0 is not a point 2 2 SU(Nf) × U(1)AF −2Nf T (Nf ) −T (Nf − 1) of the quantum moduli space and the gluons never be- 3 2 2 2 U(1)AF −2Nf +(Nf −1) −(Nf − 1) − 2 come massless. 2 2 2 2 U(1)B × U(1)AF −Nf − Nf −2Nf 2 2 2 tr U(1)AF −2Nf + Nf − 1 −(Nf − 1) − 2 B. Patterns of spontaneous symmetry breaking and ’t Hooft’s anomaly matching conditions. The constants T (R) are defined by tr(T aT a)= T(R)δab,withTa in the representation R of the group Our global symmetries are SU(N ) × SU(N ) × f L f R SU(N). For the fundamental representation, T (N)= U(1)B × U(1)AF . Since for Nf = Nc the super-quarks 2 are neutral with respect to the non-anomalous symme- 1/2. For the adjoint representation, T (N − 1)=N. try U(1) , it is never spontaneously broken. The other The coefficient of tr U(1)AF corresponds to the gravita- AF tional anomaly. One can check that all the anomalies symmetries present different patterns of symmetry break- ing depending on which point of the moduli space the match perfectly, supporting the spontaneous symmetry breaking pattern of (XIII.6). vacuum is located ‡‡. For instance, the point The quantum moduli space of Nf = Nc allows another particular point with a quite different breaking pattern. 2 ˜ It is: M =Λ 1Nf,B=B=0, (XIII.5)

˜ Nc suggests the spontaneous symmetry breaking M =0,B=−B=Λ . (XIII.7) At this point, only the vectorial baryon symmetry is bro- SU(Nf)L × SU(Nf)R × U(1)B × U(1)AF ken, all the chiral symmetries SU(Nf)L × SU(Nf)R × −→ SU(Nf)V × U(1)B × U(1)AF , (XIII.6) U(1)AF remain unbroken. We check this pattern with the help of the ’t Hooft’s anomaly matching conditions again. In this case we have the quantum numbers:

‡‡Different patterns of symmetry breaking have also been observed in softly broken N = 2 SQCD [15].

21 SU(Nf)L SU(Nf)R U(1)AF B. S-confinement.

λ 1 1 fg 1 In the massless limit m → 0, (XIV.1) and (XIV.2) ψq Nf 1 −1 are satisfied at the quantum level. It means that the ψq˜ 1 Nf −1 origin of field space, M = B = B = 0, is an allowed point of the quantum moduli space. On such a point, ψ N N 1 M f f − there is no spontaneous symmetrye breaking at all. We ψB 1 1 −1 use the ’t Hooft’s anomaly matching conditions to check ψB˜ 1 1 −1 it. The quantum numbers of the massless fermions at high and low energy are: and the anomaly coefficients are:

SU(Nf)L SU(Nf)R U(1)B U(1)AF triangles high energy low energy λ 1 1 0 1 3 1 SU(N ) N C N C ψq Nf 1 1 − 1 f L f 3 f 3 Nf 3 1 SU(Nf)R NfC3 NfC3 ψq˜ 1 Nf −1 − 1 2 Nf SU(Nf) × U(1)AF −Nf T (Nf ) −Nf T (Nf ) 3 2 2 2 U(1)AF −2Nf + Nf − 1 −Nf − 1 2 ψM Nf Nf 0 − 1 Nf 1 ψB Nf 1 Nf − 1 − Nf a b c abc a 1 where C3 is defined by tr(T {T ,T })=C3d ,withT ψ ˜ 1 Nf 1 − Nf − B Nf in the fundamental representation of SU(Nf). Because of the constraint (XIII.1) there is only one independent baryonic degree of freedom. The anomaly coefficients and the anomaly coefficients are: match perfectly.

triangles high energy low energy XIV. THE VACUUM STRUCTURE OF SQCD 3 WITH NF = NC +1. SU(Nf) NcC3 NfC3 + C3 2 Nc 2 SU(Nf) NcT(Nf)(− ) Nf T (Nf )( − 1) Nf Nf A. The quantum moduli space. 1 ×U(1)AF +T (Nf )(− ) Nf 2 Nc 2 1 U(1) × U(1)AF 2NcNf (− ) 2Nf N (− ) B Nf c Nf First we consider if the classical constraints: 3 2 2 2 3 U(1)AF (Nc − 1) Nf ( N − 1) g f f M B = M B˜ =0, (XIV.1) Nc 3 1 3 fg fg +2Nf Nc(− ) +2Nf (− ) Nf Nf −1 fg f g 2 2 2 detM(M ) −B B =0, (XIV.2) tr U(1)AF (N − 1) N ( − 1) c f Nf Nc 1 +2Nf Nc(− ) +2Nf (− ) are modified quantum mechanically. For N = N +1 Nf Nf e f c the quark multiplets (Qf , Qf )haveRAF charge equal to 1/N . The mass matrix breaks the U(1) symmetry f AF with complete agreement. Hence, at the origin of field with a charge of 2 − 2/N =2N/N . It is exactly the fe c f space we have massless mesons and , and the charge U(1) of equation (XIV.2). On the other hand, AF full global symmetry is manifest. It is a singular point, the instanton factor Λ2Nc−1 supplies the right dimen- with the number of massless degrees of freedom larger sionality. Then, there is the possibility that the classical than the dimensionality of the space of vacua. As we constraint (XIV.2) is modified by nonperturbative con- move along the moduli space away from the origin, the tributions to ‘extra’ fields become massive and the massless fluctua- detM(M −1)fg −BfBg =Λ2Nc−1mfg. (XIV.3) tions match with the dimensionality of the moduli space. As we are in a Higgs/confining phase, there should be a On the other hand, one can see that the classical con- e smooth connection of the dynamics at the origin of field straints (XIV.1) do not admit modification. Then if space with the one away from it. This dynamics must be M 6=0wehaveBf =Bg = 0. Using (XII.4), we ob- given by some nonperturbative superpotential of mesons tain and baryons. A theory with the previous characteristics

−1 fg e 2Nc−1 fg is called s-confining. detM(M ) =Λ m , (XIV.4) There is a unique effective superpotential yielding all and the quantum modification (XIV.3) really takes place the constraints [36], [36].

22 1 g f This mass scale relates the intrinsic scales Λ and Λofthe W = (B Mgf B − detM) , (XIV.5) 2Nf −3 Λ SU(Nc)andSU(Nc) gauge theories through the equation it satisfies: e e 3N −N 3N −N N −N N i) Invariance under all the symmetries. Λ c f Λ e c f =(−1) f c µ f . (XV.3) ii) The equations of motion ∂W/∂M = ∂W/∂B = We see that an stronglye coupled SU(Nc) gauge theory ∂W/∂B = 0 give the constraints (XIV.1, XIV.2). e corresponds to a weakly coupled SU(N ) gauge theory, iii) At the origin all the fields are massless. c f f in analogy with the electric-magnetic duality. From this iv) Addinge the bare term tr (mM)+bfB +bfB we analogy, we call the SU(Nc) gauge theorye the electric recover the Nf Nc+1, i.e., as being in a Higgs/confining phase with the vacuum structure determined by meson and baryons operators satisfying the corresponding clas- SU(Nf)L SU(Nf)R U(1)B U(1)AF sical constraints, to the case of Nf >Nc+1 (it is not pos- sible to modify the classical constraints for Nf >Nc+1), λ 1 1 0 1 we obtain inconsistencies. It is not possible to generate Nc Nc ψd Nf 1 N a superpotential yielding to the constraints, and the ’t e Nc f Nc Nc Hooft’s anomaly matching conditions are not satisfied. ψd˜ 1 Nf − N Nc ef It indicates that for Nf >Nc+ 1 the Higgs/confining Nc ψm Nf Nf e0 1 − 2 description of SQCD at large distances in terms of just e Nf M, B and B is no longer valid. e For N >N + 1, Seiberg conjectured [38] that the f c with λ the magnetic gluinos. One can check that both infrared limit of SQCD with N flavors admits a dual e f theories give the same anomalies. description in terms of an N = 1 super Yang-Mills gauge It can be verified that applying duality again we obtain theory with N = N N number of colors, N flavors e c f − c f the original theory. Df and Df in the fundamental and anti-fundamental 2 representationse of SU(Nf − Nc) respectively, and Nf (m) (m) e B. Nc +1

f1···f f f C. 3Nc/2

In the baryone operatorse e the eSUe(Nc) color indices of (Df , Df ) are contracted with the N antisymmetric ten- As in QCD, the N = 1 SQCD has a Banks-Zaks fixed c point [39] for N ,N ,whenN/N =3 with sor. The scale µ is introduced becausee the dimension c f →∞ f c −   1. We still have asymptotic freedom and under the of the bare operator M (m), derived from (XV.1), is one. e gf e renormalization group transformations the theory flows

23 from the ultraviolet free fixed point to an infrared fixed familiar R-symmetry of supersymmetric theories that ro- (I) point with a non-zero finite value of the gauge coupling tate the global phase of the supercharges Qα .Withre- constant. If there is an interacting superconformal gauge (I) spect the SU(2)R group, the supercharges Qα are in the theory the scaling dimensions of some gauge invariant doublet representation 2. operators should be non-trivial. As in massless N = 1 supersymmetric representations, The superconformal invariance includes an R- half of the supercharges are realized as vanishing opera- symmetry, from which the scaling dimensions of the op- (I) tors: Q = 0. We normalize the other two supercharges, erators satisfy the lower bound 2 1 3 a(I) = √ Q(I) , (XVI.2) D ≥ |R| (XV.4) 1 1 2 2 E with equality for chiral and anti-chiral operators. The which are an SU(2)R doublet. The massless N = 2 vec- R-current is in the same supermultiplet as the energy- tor multiplet is a representation constructed from the momentum tensor, whose trace anomaly is zero on the Clifford vacuum |1 >, which has helicity λ =1andis fixed point. It implies that there the R-symmetry must an SU(2)R singlet. From it we obtain two fermionic (I) (I) † be the anomaly-free U(1)AF symmetry. It gives the scal- states, |1/2 > =(a )|1>, and a scalar boson |0 >= ing dimensions of the following chiral operators: (a(1))†(a(2))†|1 >.AfterCPT doubling we obtain the N = 2 vector multiplet: 3 N −N D(M)= R (M)=3 f c, (XV.5) AF {|1>, |−1>CPT } 2 Nf 3Nc(Nf −Nc)  1 (1) 1 (1) 1 (2) 1 (2)  D(B)=D(B)= . (XV.6) {|2 > , |− 2 >CPT }{|2>, |− 2 >CPT } 2 Nf    {|0>, |0 > }  Unitarity restrictse the scaling dimensions of the gauge  CPT    invariant operators to be D ≥ 1. If D =1,thecor- (XVI.3) responding operator O satisfies the free equation of mo- tion ∂2O =0.IfD>1, there are non-trivial interactions In terms of local fields we have: a vector Aµ (the gauge between the operators. bosons of some gauge group G, since we consider mass- For the range 3Nc/2 1. Seiberg conjectured the exis- blet; and a complex scalar φ, playing the role of the Higgs, tence of such a non-trivial fixed point for any value of a singlet of SU(2)R but in the adjoint of the gauge group e G. These fields arrange as 3Nc/2 ,whichisan

(I) µ I SU(2)R singlet. The action of the two grassmanian oper- {Q , Q ˙ } =2(σ ) ˙Pµδ , I α β(J) αβ J ators aα seems to produce the same particle content that {Q(I),Q(J)}= 0 (XVI.1) the N = 1 chiral multiplet, but |1/2 >= |1/2, R > is α β usually in some non-trivial representation R of a gauge with I,J =1,2. The algebra (XVI.1) has a new symme- group G.AsR→Runder a CPT transformation, it try. We can perform unitary rotations of the two super- to make the CPT doubling, and the N =2hy- permultiplet is built from two N = 1 chiral multiplets in charges Q(I) that do leave the anti-commutator relations α complex conjugate gauge group representations: (XVI.1) invariant. We have an U(2)R = U(1)R ×SU(2)R symmetry. The Abelian factor U(1)R corresponds to the

24 1 1 √ {|2,R>, |− 2,R>CPT } M= 2|Z|, the operators in (XVI.11) are trivially real- ized and the algebra resembles the massless case. The di-  (1) (1) (2) (2)  {|0,R> , |0, R >CPT }{|0,R> , |0, R >CPT } mension of the representation is greatly reduced. For ex-   ample, a reduced massive N = 2 multiplet has the same  1,R>, 1,R>   {|−2 |2 CPT }  number of states as a massless N = 2 multiplet. Thus     the representations of the N = 2 algebra with a central   (XVI.5) charge√ can be classified as either long multiplets√ (when M > 2|Z|) or short multiplets (when M = 2|Z|). Which represents the local fields From (XVI.11) it is clear that the BPS states [9,10] (which saturate the bound) are annihilated by half of the ψq supersymmetry generators and thus belong to reduced . representations of the supersymmetry algebra. An im-  q q†  (XVI.6) portant consequence of this is that, for BPS states, the .   relationship between their charges and is dictated  ψ   q e  by supersymmetry and does not receive perturbative or   nonperturbative corrections in the quantum theory. This with the complex scalar fields (q, q†) in a doublet rep- is so because a modification of this relation implies that e resentation of SU(2)R.IntermsofN=1superfields the states no longer belong to a short multiplet. On the we have one chiral superfield Q =(eq, ψq) in gauge repre- other hand, quantum corrections are not expected to gen- erate the extra degrees of freedom needed to convert a sentation R and another chiral superfield Q =(q,ψq)in gauge representation R.Allthefieldinthehypermulti- short multiplet into a long multiplet. Since there is no e e other possibility, we√ conclude that for short multiplets plet have spin ≤ 1/2. Because of the CPT doubling,e e the matter content of extended supersymmetry (N>1) is the relation M = 2|Z| is not modified either perturba- always in vectorial representations of the gauge group. tively or nonperturbatively.

XVII. N =2SU(2) SUPER YANG-MILLS THEORY B. The central charge and massive short IN PERTURBATION THEORY. representations.

A. The N =2Lagrangian. As shown by Haag, Lapuszanski and Sohnius [40], the N = 2 supersymmetry algebra admits a central exten- sion: The N = 2 superspace has two independent chiral spinors θ(I), I =1,2. The N = 2 vector multiplet can be √ a b ab writtenintermsofN= 2 superspace by the N =2su- {Qα,Qβ}=2 2αβ Z, √ perfield Ψ(x, θ(I)) subject to the superspace constraints {Qαa˙ , Qβb˙ } =2 2α˙β˙abZ. (XVI.7) [41]:

Since Z commutes with all the generators, we can fix it (I) Ψ=0, to be the eigenvalue for the given representation. Now, ∇·α (K) (L) let us define: ∇(I)∇(J)Ψ=IKJL∇ ∇ Ψ. (XVII.1) 1 1 2 † where ∇(I)α = D(I)α +Γ(I)α is the generalized super- aα = {Qα + αβ(Qβ) } , (XVI.8) 2 (I) covariant derivative of the variable θ ,withΓ(I)α the 1 1 2 † superconnection. The N = 1 superfields are connected bα = {Qα − αβ(Qβ) } . (XVI.9) 2 to the N = 2 vector superfield through the equations:

Then, in the rest frame, the N = 2 supersymmetry alge- (1) (1) Ψ| (2) =Φ(x, θ , θ ) , bra reduces to θ(2)=θ =0 √ (1) 1 √ Ψ (2) = i 2W (x, θ , θ ) . (XVII.2) † ∇(2)α |θ(2)=θ =0 α {aα,aβ}=δαβ(M + 2Z) , (XVI.10) √ † It results that the renormalizable N = 2 super Yang- {bα,b }=δαβ(M− 2Z), (XVI.11) β Mills Lagrangian is with all other anti-commutators vanishing. Since all 1 physical states have positive definite norm, it follows that L = Im τ d2θ(1)d2θ(2) ΨaΨa (XVII.3) 8π for massless states, the central charge is trivially realized  Z  (i.e.,, Z = 0), as we used before. For massive states, √ with our old friend τ = θ/2π+i4π/g2.IntermsofN=1 this leads to a bound on the mass M≥ 2|Z|.When superspace, using (XVII.1) and (XVII.2), with θ ≡ θ(1), the Lagrangian is

25 1 2 α 1 2 2 2V field through the potential derived from the Lagrangian L = Im τ d θW W + dθd θ Φ†e− Φ . 8π α g2 (XVII.4),  Z  Z (XVII.4) 1 V (φ, φ†)= [φ†,φ]2 . (XVII.7) 2g2 It looks like N =1SU(2) gauge theory with an ad- 2 joint chiral superfield Φ. The point is that the 1/g The supersymmetric minimum is obtained by the solu- normalization in front of the kinetic term of Φ gives tion of N = 2 supersymmetry. In fact, when we perform the remaining superspace integral in (XVII.4), we obtain a [φ†,φ]=0, (XVII.8) Lagrangian that looks like a Georgi-Glashow model with a complex Higgs triplet and the addition of a Dirac spinor whose solution, up to gauge transformations, is φ = aσ3, (2) (λ(1), λ ) in the adjoint also. This Lagrangian does with a an arbitrary complex number. This is our flat not have all the gauge invariant renormalizable terms. direction. Along it, the SU(2) gauge group is spon- ± N = 2 supersymmetry restricts the possible terms and taneously broken to the U(1) subgroup. The Ψ = √1 (Ψ1 iΨ2) superfield components have U(1) electric gives relations between their couplings, such that at the 2 ± end there are only the parameters g2 and θ. charge Qe = ±g, respectively, and they have the classical If we apply perturbation theory to the Lagrangian squared mass

(XVII.3) we only have to perform a one loop renormaliza- 2 2 tion. This is an indication that in N = 2 supersymmetry, MW =2|a| . (XVII.9) holomorphy is not lost by radiative corrections. The rea- The Ψ3 superfield component remains massless. We son is the following: We expained that the multi-loop A≡ know that the Lagrangian (XVII.3) admits semi-classical renormalization of the coupling τ came from the gener- dyons with electric charge Q = n g+θ/2π and magnetic ation of non-holomorphic factors Z(µ/µ ,g)infrontof e e 0 charge Q =(4π/g), i.e., the points (1,n ) in the charge the complete N = 1 superspace integrals. At the level of m e lattice. They have the classical squared mass the Lagrangian (XVII.4), consider the bare coupling τ0 at scale µ and integrate out the modes between µ and 2 2 2 0 0 M (1,n )=2|a| |n +τ| . (XVII.10) µ. If we consider only the renormalizable terms, N =1 e e supersymmetry gives us Physical masses are gauge invariant. We can use the gauge invariant parametrization of the moduli space in 1 L = Im τ(µ/Λ) d2θW αW terms of the chiral superfield ren 8π α  Z  µ 1 U =trΦ2, (XVII.11) + Z ,g d2θd2θ Φ†e−2V Φ (XVII.5) µ 0 g2(µ)  0  Λ Z and traslate the a-dependence in previous formulae by 2 where an u-dependence through the relation u =trhφi.The classical relation is just u = a2/2. µ 2i µ ∞ Λ 4n Then, semi-classical analysis gives A as the unique τ( )= ln + c (XVII.6) Λ π Λ n µ light degree of freedom. Only at u =0thefullSU(2) n=0 X   gauge symmetry is restored. How is this picture modified by the nonperturbative corrections?. The Seiberg-Witten is the renormalized coupling constant at scale µ.Weused §§ the one-loop beta function of N =2SU(2) gauge theory solution answers this question [13] . b0 = 4 and the renormalization group invariant scale Λ ≡ µ exp(iπτ /2). The dimensionless constants c are the 0 0 n XVIII. THE LOW ENERGY EFFECTIVE coefficients of the n-instanton contribution (Λ/µ)4n = LAGRANGIAN. exp(−8πn/g2(µ)+iθ(µ)n). If we compare with the N = 2 renormalizable La- The N = 2 vector superfield is invariant under the grangian (XVII.4) we derive that Z(µ/µ0,g0) = 1. Then, A there is no Konishi anomaly and the one-loop renormal- unbroken U(1) gauge transformations. At a scale of the 1/2 ization of τ is all there is in perturbation theory. order of the MW mass, i.e., of the order or |u| ,the most general N = 2 Wilsonian Lagrangian, with two derivatives and four fermions terms, that can be con- B. The flat direction. structed from the light degrees of freedom in A is

Unlike N = 1 super Yang-Mills, N = 2 super Yang- Mills theory includes a complex scalar φ in the adjoint of the gauge group. This scalar plays the role of a Higgs §§Some additional reviews on the Seiberg-Witten solution are [42].

26 1 L = Im d2θ(1)d2θ(2) F(A) (XVIII.1) XIX. BPS BOUND AND DUALITY. eff 4π Z  The N = 2 supersymmetry algebra gives the mass with F a holomorphic function of A, called the pre- potential. We stress that the unique inputs to equa- bound tion (XVIII.1) are N = 2 supersymmetry and that A √ M≥ 2|Z|, (XIX.1) is a vector multiplet. We derive an immediate conse- quence of the general form of the effective Lagrangian with Z the central charge. The origin of the central (XVIII.1): N = 2 supersymmetry prevents the genera- charge is easy to understand: the supersymmetry charges tion of a superpotential for the N = 1 chiral superfield Q and Q are space integrals of local expressions in the of A. It means that the previously derived flat direction, 2 fields (the time component of the super-currents). In parametrized by the arbitrary value u =trhφi,isnot calculating their anti-commutators, one encounters sur- lifted by nonperturbative corrections. face terms which are normally neglected. However, in In terms of N = 1 superspace we have the presence of electric and magnetic charges, these sur- face terms are non-zero and give rise to a central charge. 1 2 1 α Leff = Im d θ τ(A)W Wα When one calculates the central charge that arises from 4π 2 Z  the classical Lagrangian (XVII.3) one obtains [43] + d2θd2θK(A, A) , (XVIII.2) Z = ae(n + mτ) , (XIX.2) Z where √ so that M≥ 2|Z|coincides with the Bogomol’nyi ∂2F bound (VI.12). τ(A)= (A), (XVIII.3) ∂A2 But the equation (XIX.2) is a classical result. The ∂F effective Lagrangian (XVIII.1) includes all the nonper- K(A, A)=Im A , (XVIII.4) turbative quantum corrections of the higher modes. To ∂A   get their contribution to the BPS bound, we just have and A is the N = 1 chiral multiplet of A. to compute the central charge that is derived from the The Wilsonian Lagrangian (XVIII.2) is an Abelian effective Lagrangian (XVIII.1). The result is gauge theory defined at some scale of order MW ∼ |u|1/2. Interaction terms come out after the expansion Z(nm,ne)=nea+nmaD, (XIX.3) A = a + Aˆ,withathe vacuum expectation value of the for a supermultiplet located in the charge lattice at Higgs field, and Aˆ the quantum fluctuations of the chiral (n ,n ). We have defined the a function superfield. The matching at scale |u|1/2 with the high en- m e D ergy SU(2) theory is performed by the renormalization ∂F group: aD ≡ (a). (XIX.4) ∂a i u ∞ Λ2 2n τ(u)= ln + c . (XVIII.5) This function plays a crucial role in duality. Observe that π Λ2 n u αβ n=0 under the SL(2, Z) transformation M = of the X   γδ Observe that the phase of the dimensionless quotient charge lattice,   2 u/Λ plays the role of the bare θ0 angle. If we are able to −1 know the relation between the u and a variables, i.e.,the (nm,ne)→(nm,ne)M , (XIX.5) function u(a), we can replace it into (XVIII.5) to obtain τ(a). Integrating twice in the variable a we obtain the the invariance of the central charge demands prepotential a a D → M D . (XIX.6) i a2 ∞ Λ 4k a a F(a)= a2ln + a2 F . (XVIII.6)     2π Λ2 k a n=1 X   Its action on the effective gauge coupling τ = ∂aD/∂a is If we look at the terms of the Lagrangian (XVIII.2) pro- ατ + β portional to the dimensionless constant Fn, they corre- τ → . (XIX.7) spond to the effective interaction terms created by the γτ + δ n-instanton contribution, as expected. For a →∞,the The S-transformation, that interchanges electric with instanton contributions go to zero. This is an expected magnetic charges, makes result, since at a →∞the matching takes place at weak coupling due to asymptotic freedom. In this region per- aD → a, turbation theory is applicable and we can believe the a→−a . (XIX.8) semi-classical relation, u ∼ a2/2. D

27 Then, aD is the dual scalar photon, that couples locally τ → τ − 2 . (XX.1) with the monopole (1, 0) through the dual gauge coupling Its associated monodromy is τD = −1/τ. From (XVIII.3) and (XVIII.4), we see that Imτ(a)is −12 the K¨ahler metric of the K¨ahler potential K(a, a), M = =PT−2 . (XX.2) ∞ 0−1   d2s = [Imτ(a)]dada. (XIX.9) which acts on the variables (aD,a)as Physical constraints demands the metric be positive def- inite, Imτ>0. However, if τ(a) is globally defined the aD →−aD +2a, (XX.3) metric cannot be positive definite as the harmonic func- a →−a. (XX.4) tion Imτ(a) cannot have a minimum. This indicates that the above description of the metric in terms of the vari- As it should be, the monodromy is a symmetry of the −2 able a must be valid only locally. In the weak coupling theory. T just shifts the θ parameter by −4π,andP region, |u||Λ|,whereτ(a)∼(2i/π)ln(a/Λ), we have is the action of the Weyl subgroup of the SU(2) gauge that Imτ(a) > 0, but for a ∼ Λ, when the theory is at group. Then, the monodromy at infinity M∞ leaves the strong coupling and the nonperturbative effects become a variable invariant (up to a gauge transformation). important, the perturbative result does not give the cor- The monodromy at infinity means there must be some rect physical answer. Two things should happen: the singularity in the u plane. How many singularities?. We instanton corrections must secure the positivity of the know that the anomalous U(1)R symmetry is broken by metric and physics must be described in terms of a new instantons, and that there is an unbroken Z8 subgroup 0 because the one-instanton sector has eight fermionic zero local variable a . Which is this new local variable? If we 2 do not want to change the physics, the change of vari- modes. The U =trΦ operator has R-charge four. It ables must be an isometry of the K¨ahler metric (XIX.9). means that the u →−usymmetry is spontaneously bro- ken, leading to equivalent physical vacua. Then, if u0 is a In terms of the variables (aD,a)theK¨ahler metric is singular point, −u0 must be also another singular point. i Let us assume that there is only one singularity. If d2s =Im(da da)=− (da da − dada ) , (XIX.10) D 2 D D this were the situation, the monodromy group would be Abelian, generated only by the monodromy at infinity. a The complete isometry group of (XIX.10) is D → From the monodromy invariance of the variable a under a   M∞, we would have that a is a good variable to describe a p M D + with M ∈ SL(2, R)andp, q ∈ R. the physics of the whole moduli space. This is in contra- a q     diction with the holomorphy of τ(a). But the invariance of the central charge puts p = q =0 Seiberg and Witten made the assumption that there ∗∗∗ and the Dirac quantization condition restricts M ∈ are only two singularities, which they normalized to be SL(2, Z). We arrive to an important result: in some 2 2 u1 =Λ and u2 = −Λ . This assumption leads to a region of the moduli space we have to perform an electric- unique and elegant solution that passes many tests. magnetic duality transformation.

XXI. THE PHYSICAL INTERPRETATION OF XX. SINGULARITIES IN THE MODULI SPACE. THE SINGULARITIES.

As Imτ cannot be globally defined on the u plane, there The most natural physical interpretation of singulari- must be some singularities ui indicating the multivalued- ties in the u plane is that some additional massless par- ness of τ(u). If we perform a loop arround a singularity ticles appear at the singular point u = u0. ui, there is a non-trivial monodromy action Mi on τ(u). The particles will arrange in some N =2supermul- This action should be an isometry of the K¨ahler metric, tiplet and will be labeled by some quantum numbers if we do not want to change the physics. It implies that (nm,ne). If the massless particle is purely electric, the the monodromies M are elements of the SL(2, Z) group. i Bogomol’nyi bound implies a(u0) = 0. It would mean In fact, we have found already one non-trivial mon- that the W-bosons become massless at u0 and the whole odromy because of the perturbative contributions. The SU(2) gauge symmetry is restored there. It would im- multivalued logarithmic dependence of τ gives the mon- ply the existence of a non-Abelian infrared fixed point 2 odromy. For u ∼∞,τ∼(i/π)ln(u/Λ ). In that region, with trφ2 = 0. By conformal invariance, the scaling di- 2πi h i6 the loop u → e u applied on τ(u)gives mension of the operator trφ2 at this infrared fixed point would have to be zero, i.e., it would have to be the iden- tity operator. It is not possible since trφ2 is odd under a global symmetry. ∗∗∗In N = 2 SQCD with massive matter, the central charge Then, the particles that become massless at the sin- allows to have p, q 6= 0 [44]. gular point u0 are arranged in an N = 2 supermultiplet

28 2 2 of spin ≤ 1/2. The possibilities are severely restricted have that θeff (−Λ )=2πRe(τ(−Λ )) = 2π,andbythe by the structure of N = 2 supersymmetry: the multi- Witten effect gives the same physical electric charge to plet must be an hypermultiplet that saturates the BPS the massless states at u = ±Λ2. bound. As we have derived that we should have a 6=0 Seiberg and Witten took the simplest solution: a for all the points of the moduli space, the singular BPS purely magnetic monopole (1, 0) ††† becomes massless at state must have a non-zero magnetic charge. u =Λ2. With our chosen monodromy base point, the Near its associated singularity, the light N =2hyper- state with quantum numbers (1, −1) has vanishing mass multiplet is a relevant degree of freedom to be considered at u = −Λ2. in the low energy Lagrangian. The coupling to the mass- less photon of the unbroken U(1) gauge symmetry has to be local. Therefore, we apply a duality transformation XXII. THE SEIBERG-WITTEN SOLUTION. to describe the relevant degree of freedom (nm,ne)asa purely electric state (0, 1), A. The inputs. (0, 1) = (n ,n )N−1 , (XXI.1) m e After this long preparation, we can present the solu- with N the appropiate SL(2, Z) transformation. The tion of the model. The moduli space is the compactified u-plane punctured at u =Λ2, Λ2, . These singular dual variables are the good local variables near the u0 − ∞ singularity. It implies that the monodromy matrix must points generate the monodromies: leave invariant the singular state (n ,n ). This con- m e 10 straint plus the U(1) β-function give the monodromy ma- M 2 = , Λ −21 trix   −12 2 2 1+2nmne 2ne M−Λ= , M(nm,ne)= 2 . (XXI.2) −23 −2nm 1−2nenm     −12 M = , (XXII.1) In fact, in terms of the local variables, ∞ 0−1   a0 a which act on the holomorphic function τ(u)bythecor- D = N D , (XXI.3) a0 a responding modular transformations. Physically, the     function τ(u) is the effective coupling at the vacuum u the monodromy matrix is just T 2. This result can be and its asymptotic behavior near the punctured points understood as follows: The renormalizable part of the u =Λ2,−Λ2,∞,isknown. low energy Lagrangian is just√ N = 2√ QED with one light 0 hypermultiplet with mass 2|a | = 2|nmaD + nea|.It has a trivial infrared fixed point, and the theory is weakly B. The geometrical picture. coupled at large distances. Perturbation theory gives i A torus is a two dimensional compact Riemann surface τ 0 '− lna0 . (XXI.4) of genus one. Topologically it can be described by a two π dimensional lattice with complex periods ω and ωD.The On the other hand, by the monodormy invariance of a0, construction is the following: a point z in the complex 0 we have a (u) ' c0(u − u0), this gives the monodromy plane is identifyed with the points z +ω and z +ωD (with 2 matrix T : τ 0 → τ 0 +2. the convention Im(ωD/ω) > 0), to get the topology of a With all the monodromies taken in the counter clock- torus. Then, the SL(2, Z) transformations wise direction, and the monodromy base point chosen in ω ω the negative imaginary part of the complex u plane, we D → M D (XXII.2) have the topological constraint ω ω    

M−Λ2 MΛ2 = M∞ . (XXI.5) leave invariant the torus. If we rescale the lattice with 1/ω, the torus is characterized just by the modulus If we use the expression (XXI.2) for the monodromies −2 ωD M±Λ2 and that M∞ = PT , (XXI.5) implies that the τ ≡ , magnetic charge of the singular states must be ±1. Then, ω they exist semi-classically and are continuousy connected with the weak coupling region. Moreover, if the state 2 (1,ne) becomes massless at u =Λ, then (XXI.5) gives 2 ††† the massless state (1,ne−1) at u = −Λ . It is consistent Observe that by Witten effect, the shift θ → θ +2πn with the action of the spontaneously broken symmetry transforms (1, 0) → (1,n). There is a complete democracy u →−u, since by the expression of τ(u) in (XVIII.5) we between the semi-classical stable dyons.

29 up to SL(2, Z) transformations, ∂a /∂u dx/y τ(u)= D = β = τ , (XXII.6) ∂a/∂u dx/y u ατ + β Hα τ ∼ . γτ + δ leads to the formulae: H Algebraically the torus can be described by a complex a = λ(u) , (XXII.7) elliptic curve D Iβ 2 y =4(x−e1)(x − e2)(x − e3) . (XXII.3) a = λ(u) , (XXII.8) Iα ThetoricstructurearisesbecauseofthetwoRiemman sheets in the x plane joined through the two branch cuts where λ(u) is an Abelian differential with the property that going from e1 to e2 and e3 to infinity (see fig. 2). ∂λ dx = f(u) + dg . (XXII.9) ∂u y x-plane β β Then, the solution of the problem is reduced to finding e3 the family of elliptic curves (XXII.5) and the holomorphic function f(u). The conditions at the begining of this e3 e1 e2 section fix a unique solution. The family of elliptic curves e1 is determined by the monodromy group generated by the monodromy matrices. The matrices (XXII.1) generate α the group Γ(2), the subgroup of SL(2, Z) consisting of α matrices congruent to the identity modulo 2. It gives the elliptic curves e2 FIG. 2. The elliptic curve (XXII.3) gives the topology of a 2 2 4 torus. y =(x −Λ )(x − u) . (XXII.10) Finally, the function f(u) is determined by the asymp- The lattice periods are obtained by integrating the totic behavior of (a ,a) at the singular points. The an- Abelian differential of first kind dx/y along the two ho- √ D mologically non-trivial one-cycles α and β, with intersec- swer is f = − 2/4π. tion number β · α =1, dx XXIII. BREAKING N =2TO N =1. ω = , D y MONOPOLE CONDENSATION AND Iβ CONFINEMENT. dx ω = . (XXII.4) y Iα In this section we will exhibit an explicit realization They have the property that Imτ>0. of the confinement mechanism envisaged by Mandelstam [45] and ’t Hooft’s through the condensation of light monopoles. C. The Physical connection with N =2super In the N = 2 model, we have found points in the mod- Yang-Mills. uli space where the relevant light degrees of freedom are magnetic particles. Since we have the exact solution of the low energy N = 2 model, it would be nice to answer The breakthough of Seiberg and Witten for the solu- in which phase the dynamics of the model, or controlable tion of the model was the identification of the complex deformations of it, locates the vacuum. effective coupling τ(u) at a given vacuum u with the mod- For the N = 2 model we already know from section ulus of a u-dependent torus. At any point u of the moduli XVIII that N = 2 supersymmetry does not allow the space, they associated an elliptic curve generation of a superpotential just for the N =1chiral 3 multiplet of the N = 2 vector multiplet. It means that 2 y =4 (x − ei(u)) , (XXII.5) the theory is always in an Abelian Coulomb phase. The i=1 exact solution of the model allowed us to know which Y are all the instanton corrections to the low energy La- with its lattice periods given by (XXII.4). grangian. Remarkably enough, the instanton series ad- The identification of the physical coupling τ(u)= mits a resumation in terms of magnetic variables. ∂aD/∂a with the modulus τu = ωD(u)/ω(u) of the el- To go out of the Coulomb branch, we need a super- liptic curve (XXII.5), potential for the chiral superfield Φ. In [13] an explicit

30 mass term for the chiral superfield was added in the bare XXIV. BREAKING N =2TO N =0. Lagrangian, 2 Wtree = m tr Φ . (XXIII.1) When the N = 2 theory is broken to the N =1theory through the decoupling of the chiral superfield Φ in the It breaks N =2toN= 1 supersymmetry. At adjoint, we have seen that the mechanism of confinement low energy, we will have an effective superpotential takes place because of the condensation of a magnetic W(m, M, M,AD). Once again, holomorphy of the su- monopole. The natural question is if this results can be perpotential and selection rules from the symmeries will extended to non supersymmetric gauge theories. fix the exactf form of W.IntermsofN= 1 superspace, The N =1,2 results were based on the use of holo- only the subgroup U(1)J ⊂ SU(2)R is manifestly a sym- morphy; the question is whether the properties connected metry. It is a non-anomalous R-symmetry (rotates the with holomorphy can be extended to the N =0case.The complex phases of θ(I), I =1,2, in opposite directions.). answer is positive provided supersymmetry is broken via The corresponding charge of Φ is zero. As superpoten- soft breaking terms. tials should have charge two, from (XXIII.1) we derive The method is to promote some couplings in the su- that the parameter m 6= 0 breaks the U(1)J symmetry persymmetric Lagrangian to the quality of frozen super- by two units. On the other hand, the N =1chiralsu- fields, called spurion superfields. We could think they perfields M and M are in an N = 2 hypermultiplet and correspond to some heavy degrees of freedom which at therefore, both have charge one. Imposing that W is a low energies have been decoupled. Their trace is only regular function atf m = MM = 0, we find that it is of through their vacuum expectation values appearing in the form W = mf1(AD)+MMf2(AD). For m → 0, the Lagrangian and are parametrized by the spurion su- the effective superpotentialf flows to the tree level super- perfields [46]. √ In the N = 2 theory we will promote some couplings to potential (XXIII.1) plus thef term 2ADMM.Asthe the status of spurion superfields. The property of holo- functions f1 and f2 are independent of m, we obtain the morphy in the prepotential will be secured if the intro- exact result f √ duced spurions are N = 2 vector superfields [14,15] ‡‡‡. W = 2ADMM +mU(AD) . (XXIII.2) In the bare Lagrangian of the N =2SU(2) gauge the- We found what we were looking for: an exact effective ory (XVII.3), there is only one parameter: τ0.TheN=2 superpotential withf a term which depends only of the softly broken theory is obtained by the bare prepotential N = 1 chiral composite operator U. It presumely will 1 a a remove the flat direction. The N =2toN= 1 breaking F0 = SA A , (XXIV.1) makes no loger valid the hiden N = 2 holomorphy in π the K¨ahler potential K(A, A). But as long as there is where S is an dimensionless N = 2 vector multiplet an unbroken supersymmetry, the vacuum configuration whose scalar component gives the bare coupling constant, corresponds to the solution of the equations π s = 2 τ0. The factor of proporcionality is related with dW =0, (XXIII.3) the one loop coefficient of the beta function, such that Λ=µ0exp(is). Inspired by String Theory, we call S 2 2 D = |M| −|M| =0. (XXIII.4) the spurion. The source of soft breaking comes from the non vanishing auxiliary fields, F and D ,inthe From the exact solution we know that du/daD 6=0at 0 0 f dilaton spurion S. aD= 0. Thus (up to gauge transformations) The tree level mass terms arising from the softly bro- √ 1/2 M = M = −mu0(0)/ 2 , ken bare Lagrangian (XXIV.1) are the following: the W- bosons get a mass term by the usual Higgs mechanism, aD =0.   (XXIII.5) with the mass square equal to 2 a 2; the photon of the un- f | | Expanding around this vacuum we find: broken U(1) remains massless; the gauginos get a mass 2 2 2 −1 i) There is a mass gap of the order (mΛ)1/2. square M1/2 =(|F0| +D0/2)(4Ims) ; all the scalar ii) The objects that condense are magnetic monopoles. components, except the real part of φ3 which do not have 2 2 There are electric flux tubes with a non-zero string ten- a bare mass term, get a square mass M0 =4M1/2. sion of the order of the mass gap, that confines the elec- At low energy, i.e., at scales of the order |u|1/2 ∼ Λ, the tric charges of the U(1) gauge group. Wilsonian effective Lagrangian up to two derivatives and The spontaneously broken symmetry u →−ucarries four fermions terms is given by the effective prepotential the theory to the ‘dyon region’, with the local variable F(a, Λ) found in the N = 2 model, but with the differ- aD − a. The perturbing superpotential there, mU(aD − ence that the bare coupling constant is replaced by the a), also produces the condensation of the ‘dyon’ with physical electric charge zero at the point aD − a =0. Then, we have two physically equivalent vacua, related by an spontaneously broken symmetry, in agreement with ‡‡‡ the Witten index of N =1SU(2) gauge theory. Soft breaking of N = 1 SQCD has been studied in [47].

31 dilaton spurion, i.e.,Λ→µ0exp(iS). Then, the prepo- But for moderate values of the supersymmetry break- tential depends on two vector multiplets and the effective ing parameter, the effective Lagrangian (XXIV.2) gives Lagrangian becomes the large distance physics of a non-supersymmetric gauge

2 theory at strong coupling. If we minimize the effective 1 ∂F i 1 ∂ F L = Im d4θ A + d2θ WiWj potential (XXIV.4) with respect to the monopoles, we 4π ∂Ai 2 ∂Ai∂Aj obtain the energy of the vacuum u Z Z  + LHM . (XXIV.2) 2 b01(u) 2 i V (u)= b (u) F with A =(S, A)andLHM the N = 2 Lagrangian that eff 00 − | 0| b11(u) includes the monopole hypermultiplet. Observe that the   2 dilaton spurion do not enter in the Lagrangian of the − ρ4(u) , (XXIV.5) hypermultiplets, in agreement with the N = 2 non- b11(u) renormalization theorem of [27]. The low energy cou- where ρ(u) is a positive function that gives the monopole plings are determined by the 2 2matrix × condensate at u ∂2F τij (a, s)= . (XXIV.3) 2 2 2 |b01|f0 2 ∂ai∂aj |m| = |m| = ρ (u)= √ − b11|a| > 0 (XXIV.6) 2 The supersymmetry breaking generates a non-trivial ef- √ 2 fective potential for the scalar fields, or m = me= ρ(u)=0if|b01|f0 < 2b11|a| . b2 1 V = b − 01 |F |2 + D2 eff 00 b 0 2 0 e  11   b01 √ 2 2 + 2(F0mm + F 0mm)+D0(|m| −|m| ) b11 h i 1 2 2 2 2 2 2 + (|m| + |me| ) +2|ea| (|m| +|m| ),e(XXIV.4) 2b11 −1 0.008 wherewehavedefinede bij =(4π) Imτij . em and m are the scalar components of the chiral superfields M and 0.006 1 0.004 M of the monopole hypermultiplet, respectively.e Ob- serve that the first line of (XXIV.4) is independent of 0.002 0.5 the monopole degrees of freedom. To be sure that such 0 f 0 0 quantity gives the right amount of energy at any point 0.5 of the moduli space, where different local descriptions of the physics are necessary, it must be duality invariant. 1 -0.5 This is the case for any SL(2, Z) transformation. 1.5 The auxiliary fields of the dilaton spurion are in the 2 -1 adjoint representation of the group SU(2) and have R FIG. 3. The monopole condensate ρ2, at the monopole re- U(1)R charge two. We can consider the situation of 2 gion u ∼ Λ ,forf0 =Λ/10. D0 =0,F0 = f0 > 0 without any loss of generality, since it is related with the case of D 6= 0 and complex 0 Notice that b diverges logarithmically at the singu- F just by the appropiate SU(2) rotation. 11 0 R larities u = Λ2, but the corresponding local variable a We have to be careful with the validity of our approx- ± vanishes linearly at u = Λ2. It implies that b a 2 0 imations. Because of supersymmetry, the expansion in ± 11| | → for u Λ2. It can be shown that the Seiberg-Witten derivatives is linked with the expansion in fermions and →± solution gives b Λ/8π for u Λ. It means that the expansion in auxiliary fields. The exact solution of 01 ∼ ∼ the monopole condenses at the monopole region (see Seiberg and Witten is only for the first terms in the fig. 3), since from the expression of the effective potential derivative expansion of the effective Lagrangian, in par- (XXIV.5), such condensation is energetically favoured. If ticular up to two derivatives. At the level of the softly we look at the dyon region, we find that b 0for broken effective Lagrangian, the exact solution of Seiberg 01 → u Λ2. Numerically, there is a very small dyon con- and Witten only gives us the terms at most quadratic in →− densate without any associated minimum in the effective the supersymmetry breaking parameter f . The expan- 0 potential in that region. On the other hand, there is sion is performed in the dimensionless parameter f /Λ. 0 a clear absolute minimum in the monopole region (see Our ignorance on the higher derivative terms of the effec- fig. 4). The different behaviors of the broken theory un- tive Lagrangian is traslated into our ignorance the terms 4 der the transformation u →−uis an expected result if of O((f0/Λ) ). Hence our results are reliable for small we take into account that f0 6= 0 breaks explicitly the values of f0/Λ, and this is far from the supersymmetry U(1)R symmetry. decoupling limit f0/Λ →∞.

32 of Quantum Field Theory, it corresponds to an effective low energy interaction, with lp the natural length scale at which the effects of quantum become impor- tant. The natural suspicion is that there is new physics at such short distances, which smears out the interaction. The idea of String Theory is to replace the point par- ticle description of the interactions by one-dimensional 0 objects, strings with size of the order of the Planck’s length l 10−33cm (see fig. 5). Such simple change 0.4 p ∼ -0.0005 has profound consequences on the physical behavior of 0.2 the theory, as we will briefly review below. It is still not -0.001 0 clear whether the stringy solution to 0.6 should work. Because Planck’s length scale is so small, 0.8 -0.2 up to now String Theory is only constructed from internal 1 1.2 consistency. But it is at the moment the best candidate -0.4 we have. Let us quickly review some of the major impli- 1.4 cations of String Theory, derived already at perturbative level. FIG. 4. The effective potential Veff (u) (XXIV.5), at the 2 monopole region u ∼ Λ ,forf0 =Λ/10.

The softly broken theory selects a unique minimum at the monopole region, with a non vanishing expectation value for the monopole. The theory confines and has a 1/2 mass gap or order (f0Λ) .

XXV. STRING THEORY IN PERTURBATION FIG. 5. The point particle graviton interchange is replaced THEORY. by the smeared string interaction.

String Theory is a multifaceted subject. In the sixties The first important consequence of String Theory is strings were first introduced to model the dynamics of the existence of vibrating modes of the string. They dynamics. In section VII we described the con- correspond to the physical particle spectrum. For phe- fining phase as the dual Higgs phase, where magnetic nomenology the relevant part comes from the massless degrees of freedom condense. The topology of the gauge modes, since the massive modes are excited at energies −1 group allows the existence of electric vortex tubes, end- of the order of the Planck’s mass lp . At low energies all ing on quark-antiquark bound states. The transverse size the massive modes decouple and we end with an effective of the electric tubes is of the order of the compton wave Quantum Field Theory for the massless modes. In the length of the ‘massive’ W-bosons. At large distances, massless spectrum of the closed string, there is a parti- these electric tubes can be considered as open strings cle of spin two. It is the graviton. Then String Theory with a quark and an anti-quark at their end points. This includes gravity. If we know how to make a consistent is the QCD string, with an string tension of the or- and phenomenologically satisfactory quantum theory of der of the characteristic length square of the , strings, we have quantized gravity. α0 ∼ (1GeV)−2. Up to now, String Theory is only well understood at But the major interest in String Theory comes from the perturbative level. The field theory diagrams are re- being a good candidate for quantum gravity [48]. The placed by two dimensional Riemann surfaces, with the macroscopic gravitational force includes an intrinsic con- loop expansion being performed by an expansion in the stant, GN , with dimensions of length square genus of the surfaces. It is a formulation of first quanti- zation, where the path integral is weighed by the area of 2 −33 2 GN = lp =(1.6×10 cm) . (XXV.1) the Riemann surface and the external states are included by the insertion of the appropiate vertex operators (see In a physical process with an energy scale E for the fun- fig. 6). The perturbative string coupling constant is de- damental constituents of matter, the strength of the grav- termined by the vacuum expectation value of a massless itational interaction is given by the dimensionless cou- real scalar field, called the dilaton, through the relation G E2 pling N to the graviton. This interaction can be g =exphsi. The thickening of Feynman diagrams into neglected when the graviton probes length scales much s 2 ‘surface’ diagrams improves considerably the ultraviolet larger than the Planck’s size, GN E  1. The interac- behavior of the theory. String Theory is ultraviolet finite. tion is also non-renormalizable. From the point of view

33 mensions.

++ +. . . . A. The type IIA and type IIB string theories. FIG. 6. The preturbative loop expansion in String Theory is equivalent to expand in the number of genus of the Riemann A type II string theory is constructed from closed surfaces. superstrings with N = 2 spacetime . The spectrum is obtained as a tensor product of a left- The third important consequence is the introduction and right-moving world-sheet sectors of the closed string. of supersymmetry. For the bosonic string, the lowest Working in the light-cone gauge, the massless states of vibrating mode correponds to a . It indicates each sector are in the representation 8v ⊕ 8± of the little that we are performing perturbation theory arround an group SO(8). The representations 8v and 8± are the vec- unestable minimum. Supersymmetry gives a very eco- tor representation and the irreducible chiral spinor rep- nomical solution to this problem. In a supersymmetric resentations of SO(8), respectively. theory the hamiltonian operator is positive semi-definite The type IIA string theory corresponds to the choice and the ground state has always zero energy. It is also of opposite chiralities for the spinorial representations in very appealing from the point of view of the cosmologi- the left- and right-moving sectors, cal constant problem. Furthermore, supersymmetry also introduces fermionic degrees of freedom in the physical Type IIA : (8v ⊕ 8+) ⊗ (8v ⊕ 8−) . (XXV.2) spectrum. If nature really chooses to be supersymmet- ric at sort distances, the big question is: How is su- The bosonic massless spectrum is divided between the persymmetry dynamically broken? The satisfactory an- NS-NS fields: swer must include the observed low energy phenomena of 8 ⊗ 8 = 1 ⊕ 28 ⊕ 35 , (XXV.3) the standard model and the vanishing of the cosmologi- v v cal constant. As a last comment on supersymmetry we which corresponds to the dilaton s, the antisymmetric will say that the Green-Schwarz formulation of the su- tensor Bµν and the gravitation field gµν , respectively, perstring action demands invariance under a world-sheet and the R-R fields: local fermionic symmetry, called κ-symmetry. It is only possible to construct κ-symmetric world-sheet actions if 8+ ⊗ 8− = 8v ⊕ 56, (XXV.4) the number of spacetime symmetries is N ≤ 2(inten spacetime dimensions). which correspond to the light-cone degrees of freedom of The fourth important consequence is the prediction on the antisymmetric tensors Aµ and Aµνρ, respectively. As the number of dimensions of the target space where the the chiral spinors have opposite chiralities, in the vertex perturbative string propagates. Lorentz invariance on operators of the R-R fields only even forms appear, F2 the target space or conformal invariance on the world- and . The physical state conditions on the massless sheet fixes the number of spacetime dimensions (twenty- states give the following equations on these even forms: six for bosonic strings and ten for superstrings). As our dF =0 d?F =0, (XXV.5) low energy world is four dimensional, String Theory in- corporates the Kaluza-Klein idea in a natural way. But with ?F the Poincare dual (10 − n)-form of the n-form again the one-dimensional nature of the string gives a Fn. These are the Bianchi identity and the equation of quite different behavior of String Theory with respect to motion for a field strength. Their relation with the R-R field theory. The dimensional reduction of a field the- fields is then Fn = dAn−1. The Abelian field strengths ory in D spacetime dimensions is another field theory in Fn are gauge invariant, and since these are the fields that D−1 dimensions. The effect of a non-zero finite radius R appear in the vertex operators, the fundamental strings for the compactified dimension is just a tower of Kaluza- do not carry RR charges. Klein states with masses n/R. But in String Theory, the The fermionic massless spectrum is given by the NS− string can wind m times around the . R and R − NS fields: This process gives a contribution to the momentum of 0 the string proportional to the compact radius, mR/α . 8v ⊗ 8− = 8+ ⊕ 56− , These quantum states become light for R 0. The di- → 8+ ⊗ 8v = 8− ⊕ 56+ . (XXV.6) mensional reduction of a String Theory in D dimensions is another String Theory in D dimensions. This is T The 8± states are the two dilatini. The 56± states are the duality [49]. two gravitini, with a spinor and a vector index. Observe The fifth important consequence comes from the can- that the fermions have opposite chiralities, which prevent cellation of spacetime anomalies (gauge, gravitational the type IIA theory from gravitational anomalies. and mixed anomalies). It gives only the following five The Type IIB String Theory corresponds to the choice anomaly-free superstring theories in ten spacetime di- of the same chirality for the spinor representations of the left- and right-moving sector,

34 Type IIB : (8v ⊕ 8+) ⊗ (8v ⊕ 8+) . (XXV.7) For the physical massless states, the supersymmetric right-moving sector gives the factor 8v ⊗ 8+,whichto- TheNS-NSfieldsarethesameasforthetypeIIAstring. gether with the lattice points of length squared two of The difference comes from the R-R fields: the left-moving sector, give an N = 1 vector multiplet in the adjoint representation of the gauge group SO(32) or 8+ ⊗ 8+ = 1 + ⊕28 ⊕ 35+ . (XXV.8) E8 × E8. There is also a T -duality symmetry relating the two They correspond, respectively, to the forms A0, A2 and heterotic strings. A4 (self-dual). For the massless fermions there are two dilatini and two gravitini, but now all of them have the same chiral- ity. In spite of it, the theory does not have gravitational XXVI. D-BRANES. anomalies [50]. Under spacetime compactifications, the type IIA and Perturbation theory is not the whole history. In the the type IIB string theories are unified by the T -duality field theory sections we have learned how much the non- symmetry. It is an exact symmetry of the theory already perturbative effects could change the perturbative pic- at the perturbative level and maps a type IIA string with ture of a theory. In particular, there are nonperturbative a compact dimension of radius R toatypeIIBstringwith stable field configurations (solitons) that can become the radius α0/R. relevant degrees of freedom in some regime. In that situa- tion it is convenient to perform a duality transformation to have an effective description of the theory in terms B. The . of these solitonic degrees of freedom as the fundamental objects. It is constructed from unoriented open and closed su- What about the nonperturbative effects in String The- perstrings, leading only N = 1 spacetime supersymme- ory?. Does String Theory incorporate nonperturbative try. The massless states are: excitations (string solitons)?. Are there also strong-weak coupling duality transformations in String Theory?. Be- Open : 8v ⊗ 8+ (XXV.9) fore the role of D-branes in String Theory were appreci- ated, the answers to these three questions were not clear. Closed sym. :[(8v⊕8+)⊗(8v⊕8+)]sym = For instance, it was known, by the study of large orders =[1⊕28 ⊕ 35]bosonic ⊕ [8− ⊕ 56−]fermionic . (XXV.10) of string perturbation theory, that the nonperturbative The massless sector of the spectrum that comes from the effects in string theory had to be stronger than in field theory, in the sense of being of the order of exp(−1/gs) unoriented open superstring (XXV.9) gives N =1su- 2 per Yang-Mills theory, with a gauge group SO(N )or instead of order exp(−1/gs) [51], but it was not known c which were the nature of such nonperturbative effects. USp(Nc) introduced by Chan-Paton factors at the ends of the open superstring. The sector coming from the With respect the existence of nonperturbative objects, unoriented closed string (XXV.10) gives N =1super- the unique evidence came form solitonic solutions of the gravity. Cancellation of spacetime anomalies restricts the equations of motion which are the low en- gauge group to SO(32). ergy limits of string theories. These objects were in gen- eral extended membranes in p + 1 dimensions, called p- branes [52]. In relation to the utility of the duality transformation C. The SO(32) and E8 × E8 heterotic strings. in String Theory, there is strong evidence of some string dualities [53]. There is for instance the SL(2, Z)self- The heterotic string is constructed from a right- duality conjecture of the type IIB theory [54]. Under an moving closed superstring and a left-moving closed S-transformation,the string coupling value g is mapped bosonic string. cancellation de- s to the value 1/g , and the NS-NS field B is mapped mands twenty-six bosonic target space coordinates in the s µν to the R-R field A . Then, self-duality of type IIB de- left-moving sector. The additional sixteen left-moving µν I 16 mands the existence of an string with a tension scaling coordinates XL, I =1, ..., 16, are compactified on a T −1 as gs and non-zero RR charge. torus, defined by a sixteen-dimensional lattice, Λ16,with I some basis vectors {ei }, i =1, ..., 16. The left-moving I momenta pL live on the dual lattice Λ16. The mass oper- A. Dirichlet boundary conditions. 16 I I ator gives an even lattice ( I=1 ei ei =2foranyi). The modular invariance of the one-loop diagramse restricts the In open string theory, it is possible to impose two dif- P lattice to be self-dual (Λ16 =Λ16). There are only two ferent boundary conditions at the ends of the open string: even self-dual sixteen-dimensional lattices. They corre- Neuman : ∂ Xµ =0. (XXVI.1) spond to the root latticese of the Lie groups SO(32)/Z2 ⊥ µ and E8 × E8. Dirichlet : ∂tX =0. (XXVI.2)

35 An extended topological defect with p + 1 dimensions B. BPS states with RR charges. is described by the following boundary conditions on the open strings:

0,1,···p p+1,···9 ∂⊥X = ∂tX =0. (XXVI.3)

We call it a D p- (for Dirichlet [55]), an extended (p+1)-dimensional object (located at Xp+1,···9 =const) with the end points of open strings attached to it. The Dirichlet boundary conditions are not Lorentz in- variant. There is a momentum flux going from the ends of open strings to the D-branes to which they are attached. In fact, the quantum fluctuations of the open string end- points in the longitudinal directions of the D-brane live on the world-volume of the D-brane. The quantum fluc- tuations of the open string endpoints in the transverse directions of the D-brane, makes the D-brane fluctuate locally. It is a dynamical object, characterized by a ten- sion Tp and a RR charge µp.Ifµp6= 0, the world-volume of a p-brane will couple to the R-R (p +1)-formAp+1. FIG. 7. Two parallel D-branes with the one-loop vacuum Far from the D-brane, we have closed superstrings, but fluctuation of an open string attached between them. By the world-sheet boundaries (XXVI.3) relates the right- modular invariance, it also corresponds to a tree level inter- moving supercharges to the left-moving ones, and only change of a closed string. a linear combination of both is a good symmetry of the given configuration. In presence of the D-brane, half of To check if really the D-branes are the nonperturba- the supersymmetries are broken. The D-brane is a BPS tive string solitons required by , Polchinski state. In fact, in [56] it was shown that the D-brane computed explicitly the tension and RR charge of a D tension arises from the disk and therefore that it scales p-brane [57]. He first computed the one-loop amplitude −1 as gs . This is the same coupling constant dependence of an open string attached to two parallel D p-branes. as for BPS solitonic branes carrying RR charges [52]. The resulting Casimir force between the D-branes was The Dirichlet boundary condition becomes the Neu- zero, supporting its BPS nature. By modular invariance, man boundary condition in terms of the T -dual coordi- it can also be interpreted as the amplitude for the inter- nates, and vice versa. It implies that if we T -dualize change of a closed string between the D-branes (see fig. a direction longitudinal to the world- volume of the D 7). In the large separation limit, only the massless closed p-brane, it becomes a (p − 1)-brane. Equally, if the T - modes contribute. These are the NS-NS fields (graviton dualized direction is transverse to the D p-brane, we ob- and dilaton) and the R-R (p +1)form. Onthespace tain a D (p + 1)-brane. Consider a 9-brane in a type between the D-branes these fields follow the low energy IIB background. The 9-brane fills the spacetime and the type II action (type IIA for p even and type IIB for p endpoints of the open strings attached to it are free to odd). On the D p-branes, the coupling to the NS-NS and move in all the directions. It is a type I theory, with R-R fields is only N = 1 supersymmetry. Now T -dualize one direc- p+1 −s 1/2 tion of the target space. We obtain an 8-brane in a type Sp = Tp d ξe |detGab| + µp Ap+1 . p brane IIA background. If we proceed further, we obtain that Z Z − a type IIB background can hold p =9,7,5,3,1,−1p- (XXVI.4) branes. A D (−1)-brane is a D-instanton, a localized From (XXVI.4) we see that the actual D-brane action spacetime point. For a type IIA background we obtain includes a dilaton factor τp = Tp/gs,withgs the coupling p =8,6,4,2,0p-branes. constant of the closed string theory. Comparing the field theory calculation with the contribution of the massless closed modes in the string theory computation, one can obtain the values of Tp and µp. The result is [57] 2 2 2 0 3−p µp =2Tp =(4π α) . (XXVI.5) Observe that the R-R charge is really non-zero. In fact, if one checks (the generalization of) the Dirac’s quanti- zation condition for the charge µp and its dual charge µ(6−p), one obtains that µpµ(6−p) =2π.Theysatisfy the minimal quantization condition. It means that the D-branes carry the minimal allowed RR charges.

36 XXVII. SOME FINAL COMMENTS ON is characteristic of the appearance of an additional di- NONPERTURBATIVE STRING THEORY. mension. Type IIA theory at strong coupling feels an eleventh dimension of some size 2πR, with the 0-branes A. D-instantons and S-duality. playing the role of the Kaluza-Klein states [60]. If we compactify 11D supergravity [61] on a circle of The answers to the three questions at the beginning radius R and compare its action with the 10D type IIA of the previous section can now be more concrete, since supergravity action, we obtain the relation some nonperturbative objects in String Theory has been R ∼ g2/3 . (XXVII.1) identified: the D-branes. s Consider a D p-brane wrapped around a non-trivial This eleventh dimension is invisible in perturbation (p + 1)-cycle. This configuration is topologically stable. theory, where we perform an expansion near gs =0. Its action is TpVp+1/gs,withVp+1 the volume of the non- trivial (p + 1) cycle. It contributes in amplitudes with This has been a lightning review of some aspects of du- factors e−TpVp+1/gs , a generalized instanton effect. Now ality in String Theory. We hope it will serve to whet the we understand why the nonperturbative effects in String appetite of the reader and encourage her/him to learn Theory are stronger than in field theory, it is related to more about the subject and to eventually contribute to the peculiar nature of the string solitons. some of the outstanding open problems. More informa- The D-branes also give the necessary ingredient for the tion can be found from the references [62]. SL(2, Z) self-duality of the type IIB string theory. This 0 −1 theory allows D 1-branes, with a mass τ1 ∼ (2πα gs) in the string metric and non-zero RR charge. Also, one can Acknowledgments see that on the D 1-brane there are the same fluctuations We have benefited from valuable conversations with of a fundamental IIB string [58]. Then, it is the required many colleages. We would like to thank in particular object for the S-duality transformation of the type IIB E. Alvarez,´ J.M. F. Barb´on, J. Distler, D. Espriu, C. string. In fact, at strong coupling the D 1-string becomes G´omez, J. Gomis, K. Kounnas, J. Labastida, W. Lerche, light and it is natural to formulate the type IIB theory M. Mari˜no, J.M. Pons and E. Verlinde for discussions. in terms of weakly coupled D 1-branes. F. Z. would like to thank the Theory Division at CERN There is another S-duality relation in String Theory. for its hospitality. The work of F. Z. is supported by a Observe that the type I theory and the SO(32) heterotic fellowship from Ministerio de Educaci´on y Ciencia. theory have the same low energy limit. It could be that they correspond to the same theory but for different val- ues of the string coupling constant. Again D-branes help to make this picture clearer. Consider a D 1-brane in a type I background with open strings attached to it, but also with open strings with one end point attached to a 9-brane. We call them 1 − 9 strings. The 9-brane fills [1] S. Deser and C. Teitelboim, Phys. Rev. D13 (1976), the spacetime, and the 1 − 9 strings, having one Chan- 1592; Paton index, are vectors of SO(32). One can see that S. Deser, A. Gomberoff, M. Henneaux and C. Teitelboim, the world-sheet theory of the D 1-brane is precisely that Phys. Lett. B400 (1997), 80. of the SO(32) heterotic string [59]. Having a tension −1 [2] P.A.M. Dirac, Proc. Roy. Soc. A133 (1931), 60. that scales as gs , one can argue that this D heterotic [3]J.Schwinger,Phys.Rev.144 (1966) 1087; 173 (1968) string sets the lightest scale in the theory when gs  1. 1536; The strong coupling behavior of the type I string can be D. Zwanziger, Phys. Rev. 176 (1968) 1480. modeled by the weak coupling behavior of the heterotic [4] C. Vafa and E. Witten, Nucl. Phys. B431 (1994), 3; string. F. Ferrari, hep-th/9702166. [5] H.B. Nielsen and P. Olesen, Nucl. Phys. B61(1973), 45- 61. B. An eleventh dimension. [6]G.’tHooft,Nucl.Phys.B79 (1974) 276. [7]A.M.Polyakov,JETPLett.20 (1974) 194. Type IIA allows the existence of 0-branes that cou- [8] B. Julia and A. Zee, Phys. Rev. D11 (1975) 2227. [9] E. B. Bogomol’nyi, Sov. J. Nucl. Phys. 24 (1976) 449. ple to the R-R one-form A1. The 0-brane mass is 0 −1/2 [10] M. K. Prasad and C. M. Sommerfield, Phys. Rev. Lett. τ0 ∼ (α ) /gs in the string metric. At strong cou- 35 (1975) 760. pling in the type IIA theory, gs  1, this mass is the lightest scale of the theory. In fact, n 0-branes can form [11] E. Witten, Phys. Lett. 86B (1979) 283. a BPS bound state with mass nτ . This tower of states [12] G. ’t Hooft, Nucl. Phys. B190[FS3](181), 455, and 1981 0 Cargese Summer School Lecture Notes on Fundamental becoming a continuum of light states at strong coupling Interactions, in ‘Under the Spell of the Gauge Principle’, World Scientific, 1994.

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