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Duality in Quantum Field Theory (And String Theory) ∗ CERN-TH/97-257 UB-ECM-PF 97/26 hep-th/9709180 September 1997 Duality in Quantum Field Theory (and String 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 String Theory 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 Physics, both in May 1997. Contents XIII The vacuum structure of SQCD with Nf = Nc.19 I The duality symmetry. 1 A A quantum modified moduli space. 19 B Patterns of spontaneous symmetry II Dirac’s charge quantization. 1 breaking and ’t Hooft’s anomaly 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 Higgs mechanism and mass 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 <Nf ≤3Nc/2. An infrared free Coulomb phase. 5 non-Abelian Coulomb phase. 22 C3Nc/2<Nf <3Nc.Aninteracting VI The ’t Hooft-Polyakov monopoles 6 non-Abelian Coulomb phase. 22 A The Topological nature of the magnetic charge................... 6 XVI N =2supersymmetry. 23 B The ’t Hooft-Polyakov ansatz. 7 A The supersymmetry algebra and its C The Bogomol’nyi bound and the BPS masslessrepresentations......... 23 states................... 7 B The central charge and massive short DTheθparameter and the Witten effect. 8 representations.............. 24 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 Superspace 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 <Nc.18 A The Afleck-Dine-Seiberg’s superpotential. 18 B TheTypeIstringtheory......... 34 B Massiveflavors.............. 18 CTheSO(32) and E8 ×E8 heterotic strings. 34 XXVI D-branes. 34 A Dirichlet boundary conditions. 34 B BPS states with RR charges. 35 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-instantons 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 electromagnetism. 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 , magnetic monopole of strength g, situated at the origin. ¨ ˙ ~ ~ E →−B. (I.2) The charge q experiences a force 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 electric charge 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.
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