Electromagnetic Duality for Children JM Figueroa-O'Farrill [email protected] Version of 8 October 1998 Contents I The Simplest Example: SO(3) 11 1 Classical Electromagnetic Duality 12 1.1 The Dirac Monopole ....................... 12 1.1.1 And in the beginning there was Maxwell... 12 1.1.2 The Dirac quantisation condition . 14 1.1.3 Dyons and the Zwanziger{Schwinger quantisation con- dition ........................... 16 1.2 The 't Hooft{Polyakov Monopole . 18 1.2.1 The bosonic part of the Georgi{Glashow model . 18 1.2.2 Finite-energy solutions: the 't Hooft{Polyakov Ansatz . 20 1.2.3 The topological origin of the magnetic charge . 24 1.3 BPS-monopoles .......................... 26 1.3.1 Estimating the mass of a monopole: the Bogomol'nyi bound ........................... 27 1.3.2 Saturating the bound: the BPS-monopole . 28 1.4 Duality conjectures ........................ 30 1.4.1 The Montonen{Olive conjecture . 30 1.4.2 The Witten e®ect ..................... 31 1.4.3 SL(2; Z) duality ...................... 33 2 Supersymmetry 39 2.1 The super-Poincar¶ealgebra in four dimensions . 40 2.1.1 Some notational remarks about spinors . 40 2.1.2 The Coleman{Mandula and Haag{ÃLopusza¶nski{Sohnius theorems .......................... 42 2.2 Unitary representations of the supersymmetry algebra . 44 2.2.1 Wigner's method and the little group . 44 2.2.2 Massless representations . 45 2.2.3 Massive representations . 47 No central charges .................... 48 Adding central charges . 49 1 [email protected] draft version of 8/10/1998 2.3 N=2 Supersymmetric Yang-Mills . 51 2.3.1 N=1 d=6 supersymmetric Yang{Mills . 52 2.3.2 From N=1 in d=6 to N=2 in d=4 . 57 2.3.3 Higgsed N=2 supersymmetric Yang{Mills . 59 2.3.4 N=2 avatar of the BPS-monopole . 60 2.3.5 The supersymmetry bound is the Bogomol'nyi bound . 62 The supersymmetry algebra in six dimensions . 63 The supersymmetry algebra in four dimensions . 64 2.4 N=4 Supersymmetric Yang-Mills . 66 2.4.1 N=1 d=10 supersymmetric Yang{Mills . 67 2.4.2 Reduction to d=4: N=4 supersymmetric Yang{Mills . 69 2.4.3 Monopoles and gauge bosons in N=4 supersymmetric Yang{Mills ........................ 74 2.4.4 The mass bound for N=4 super Yang{Mills . 76 The supersymmetry algebra in ten dimensions . 77 The supersymmetry algebra in four dimensions . 78 3 Collective Coordinates 80 3.1 The metric on the moduli space . 81 3.1.1 The physical con¯guration space . 81 3.1.2 The metric on the physical con¯guration space . 82 3.1.3 The metric on the moduli space . 84 3.1.4 The 1-monopole moduli space . 85 3.1.5 The quantisation of the e®ective action . 87 3.1.6 Some general properties of the monopole moduli space 88 3.2 dim Mk = 4k ........................... 89 3.2.1 The dimension as an index . 90 3.2.2 Computing the index of D . 92 3.2.3 Computing the current Ji(x; x) . 96 3.3 A quick motivation of hyperkÄahlergeometry . 99 3.3.1 Riemannian geometry . 100 3.3.2 KÄahlergeometry . 103 3.3.3 Ricci flatness . 107 3.3.4 HyperkÄahlergeometry . 109 3.4 Mk is hyperkÄahler . 111 3.4.1 Symplectic quotients . 111 3.4.2 KÄahlerquotients . 113 3.4.3 HyperkÄahlerquotients . 116 3.4.4 Mk as a hyperkÄahlerquotient . 116 3.4.5 Another proof that Mk is hyperkÄahler . 119 2 [email protected] draft version of 8/10/1998 4 The E®ective Action for N=2 Supersymmetric Yang{Mills 122 4.1 Fermionic collective coordinates . 123 4.1.1 Computing the index . 124 4.1.2 Using supersymmetry . 125 4.2 The e®ective action . 127 4.3 N=4 supersymmetry of the e®ective action . 130 4.3.1 N=4 supersymmetry in R4: a toy model . 131 4.3.2 N=4 supersymmetry in hyperkÄahlermanifolds . 132 4.3.3 N=4 supersymmetry of Le® . 134 4.4 A brief review of harmonic theory . 136 4.4.1 Harmonic theory for riemannian manifolds . 136 4.4.2 Harmonic theory for KÄahlermanifolds . 139 4.4.3 Explicit formulas for @¹ and @¹¤ . 142 4.5 Quantisation of the e®ective action . 144 4.5.1 Canonical analysis . 144 4.5.2 The quantisation of the e®ective hamiltonian . 146 5 The E®ective Action for N=4 Supersymmetric Yang{Mills 149 5.1 Fermionic collective coordinates . 150 II Arbitrary Gauge Groups 153 6 Monopoles for Arbitrary Gauge Groups 154 6.1 Topologically stable solutions . 154 6.1.1 Some elements of homotopy . 156 The fundamental group . 157 Higher homotopy groups . 162 6.1.2 Homotopy classi¯cation of ¯nite-energy con¯gurations . 166 Adding topological charges . 171 6.2 The Dirac quantisation condition . 174 6.3 Some facts about compact Lie groups and Lie algebras . 177 6.3.1 Compact Lie groups . 177 6.3.2 The Weyl group . 180 6.3.3 Root systems and simple Lie algebras . 182 Reconstructing the group . 184 The centre of G~ . 186 An example: A3 = D3 . 188 Another example: D4 . 189 All the connected compact simple Lie groups . 190 6.3.4 Some simple examples . 192 3 [email protected] draft version of 8/10/1998 The simple root system A2 . 192 The simple root systems B2 = C2 . 194 The simple root system G2 . 195 6.4 The magnetic dual of a compact Lie group . 195 6.4.1 Some lattices and dual groups . 197 H abelian . 198 H simple . 198 An example: Spin(8) and its quotients . 200 Another example: Spin(12) and its quotients . 202 4 List of Exercises 1.1 Angular momentum due to the electromagnetic ¯eld . 17 1.2 The Zwanziger{Schwinger quantisation condition . 17 1.3 Dyonic spectrum in CP non-violating theories . 17 1.4 The spectrum of the model .................... 19 1.5 Boundary conditions on H and K . 22 1.6 The equations of motion for H and K . 22 1.7 Asymptotic form of the electromagnetic ¯eld . 23 1.8 Gauge ¯eld-strength in the Higgs vacuum . 25 1.9 Additivity of the magnetic charge g§ . 26 1.10 Dirac quantisation condition revisited . 26 1.11 The Bogomol'nyi equation implies (1.10) . 29 1.12 The BPS-monopole ........................ 29 1.13 The mass density at the origin is ¯nite . 29 1.14 (P )SL(2; Z) and its action on the upper half-plane . 34 1.15 SL(2; Z)-invariance of the mass formula . 35 1.16 Orbifold points in the fundamental domain D . 36 1.17 Properties of the mass matrix A(¿) . 37 2.1 The little groups for positive-energy particles . 45 2.2 Helicity content of massless multiplets . 46 2.3 Highest spin in the multiplet ................... 48 2.4 Massive N=2 multiplets with s=0 and s=1=2 . 48 2.5 Short N=2 multiplets with s=0 and s=1=2 . 50 2.6 Short N=4 multiplets with s=0 . 51 2.7 N=1 supersymmetric Yang{Mills . 52 2.8 Supersymmetry variation of L . 54 2.9 A Fierz rearrangement ...................... 55 2.10 Some ¡-matrix identities ..................... 56 2.11 ... and the proof of supersymmetry invariance . 56 2.12 The supersymmetry multiplet . 56 2.13 L in a Majorana basis ...................... 57 5 [email protected] draft version of 8/10/1998 2.14 Explicit N=2 supersymmetry transformations . 58 2.15 The SO(2) Noether current ................... 59 2.16 The perturbative spectrum of the model . 59 2.17 Supersymmetric BPS-monopoles . 61 2.18 More γ-matrix identities ..................... 61 2.19 Some euclidean γ-matrices .................... 61 2.20 BPS-monopoles break one half of the supersymmetry . 62 2.21 Supersymmetric variation of the supercurrent . 63 2.22 The symmetric gauge-invariant energy momentum tensor . 64 2.23 The topological current ...................... 64 2.24 The \momenta" in the extra dimensions . 64 2.25 The e®ective electromagnetic ¯eld strength . 65 2.26 The space components of the topological charge . 66 2.27 The Majorana condition ..................... 67 2.28 Properties of Majorana and Weyl fermions . 68 2.29 Varying the lagrangian density . 68 2.30 A ten-dimensional Fierz identity . 69 2.31 Some more ¡-matrix identities . 69 2.32 so(4) »= so(3) £ so(3) explicitly . 70 2.33 Explicit realisation for ®I and ¯J . 71 2.34 The fundamental representation of su(4) . 71 2.35 The fermionic terms in the lagrangian . 73 2.36 N=4 supersymmetry transformations . 73 2.37 su(4) invariance .......................... 73 2.38 The perturbative spectrum after higgsing . 75 2.39 The supersymmetry algebra in ten dimensions . 77 2.40 Another topological current ................... 78 2.41 A self-dual 5-form ......................... 78 2.42 Momentum and topological charge in this background . 79 3.1 The BPS-monopole as an instanton . 82 3.2 D2 has no normalisable zero modes . 85 3.3 The electric charge ........................ 87 3.4 Two equations in one ....................... 90 3.5 Dy has no normalisable zero modes . 91 3.6 A formula for the index of D ................... 92 3.7 Another formula for I(M 2) .................... 92 3.8 Some properties of kernels .................... 93 3.9 Regularity properties of the propagator . 94 3.10 The ¯rst term doesn't contribute . 97 3.11 Another expression for the degree of the map Á . 99 6 [email protected] draft version of 8/10/1998 3.12 A coordinate-free expression for r .