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Topological Interactions in Broken Gauge Theories Topological Interactions in Broken Gauge Theories ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus Prof. dr. P.W.M. de Meijer arXiv:hep-th/9511195v1 27 Nov 1995 ten overstaan van een door het college van dekanen ingestelde commissie in het openbaar te verdedigen in de Aula der Universiteit op dinsdag 19 september 1995 te 13.30 uur door Mark Dirk Frederik de Wild Propitius geboren te Amsterdam Promotor: Prof. dr. ir. F.A. Bais Promotiecommisie: Prof. dr. P.J. van Baal Prof. dr. R.H. Dijkgraaf Prof. dr. G. ’t Hooft Prof. dr. A.M.M. Pruisken Prof. dr. J. Smit Prof. dr. H.L. Verlinde Faculteit der Wiskunde, Informatica, Natuur- en Sterrenkunde Instituut voor Theoretische Fysica This thesis is based on the following papers: F.A. Bais, P. van Driel and M. de Wild Propitius, Phys. Lett. B280 (1992) 63. • F.A. Bais, P. van Driel and M. de Wild Propitius, Nucl. Phys. B393 (1993) 547. • F.A. Bais, A. Morozov and M. de Wild Propitius, Phys. Rev. Lett. 71 (1993) • 2383. F.A. Bais and M. de Wild Propitius, in The III International Conference on • Mathematical Physics, String Theory and Quantum Gravity, Proceedings of the Conference, Alushta, 1993, Theor. Math. Phys. 98 (1994) 509. M. de Wild Propitius and F.A. Bais, Discrete gauge theories, to appear in the • proceedings of the CRM-CAP Summer School ‘Particles and Fields 94’, Bannf, Alberta, Canada, August 16-24, 1994. Voor mijn moeder Who needs physics when we’ve got chemistry? -Peggy Sue Got Married, Francis Coppola Contents Preface 1 1 Discrete gauge theories 9 1.1 Introduction................................... 9 1.2 Braidgroups................................... 10 1.3 ZN gaugetheory ................................ 14 1.3.1 Coulombscreening . .. .. 15 1.3.2 Survival of the Aharonov-Bohm effect . .. 18 1.3.3 Braid and fusion properties of the spectrum . .... 24 1.4 Nonabeliandiscretegaugetheories . ..... 29 1.4.1 Classification of stable magnetic vortices . ...... 30 1.4.2 Fluxmetamorphosis . .. .. 34 1.4.3 Includingmatter ............................ 39 1.5 Quantumdoubles................................ 42 1.5.1 D(H) .................................. 43 1.5.2 Truncatedbraidgroups. 48 1.5.3 Fusion, spin, braid statistics and all that . ............. 50 1.6 D¯ 2 gaugetheory................................. 55 1.6.1 Aliceinphysics ............................. 57 1.6.2 Scattering doublet charges off Alice fluxes . .... 61 1.6.3 Nonabelianbraidstatistics . .. 64 1.A Aharonov-Bohmscattering. .. 68 1.B B(3, 4) and P (3, 4) ............................... 71 2 Abelian Chern-Simons theories 75 2.1 Introduction................................... 75 2.2 Group cohomology and symmetry breaking . .... 78 2.3 Cohomologyoffiniteabeliangroups. .... 79 2.3.1 Hn(H, U(1))............................... 79 2.3.2 Chern-Simons actions for finite abelian groups . ...... 82 2.4 Chern-Simons actions for U(1)k gaugetheories. 84 2.5 Quasi-quantumdoubles. 86 i ii CONTENTS 2.6 U(1)Chern-Simonstheory . 92 2.6.1 Dirac monopoles and topological mass quantization . ....... 94 2.6.2 Dynamics of the Chern-Simons Higgs medium . .. 96 2.6.3 ZN Chern-Simonstheory . .102 2.7 U(1) U(1)Chern-Simonstheory. .107 2.7.1× Unbroken phase with Dirac monopoles . 108 2.7.2 Higgsphase ...............................110 2.7.3 Z (1) Z (2) Chern-SimonstheoryoftypeII. 112 N × N 2.8 Z2 Z2 Z2 Chern-Simonstheory . .114 2.8.1× Spectrum× ................................115 2.8.2 Nonabelian topologicalinteractions . .118 2.8.3 Electric/magnetic duality . 121 2.9 Dijkgraaf-Witteninvariants . .124 2.A Cohomologicalderivations. 126 3 Nonabelian discrete Chern-Simons theories 131 3.1 Dω(H) for nonabelian H ............................131 3.2 Discrete Chern-Simons Alice electrodynamics . ........134 3.3 D¯ N Chern-Simonstheory. .140 3.A Conjugatedcohomology . 146 Bibliography 151 Samenvatting 159 Dankwoord 161 Preface Symmetry has become one of the main guiding principles in physics during the twentieth century. Over the last ten decades, we have progressed from external to internal, from global to local, from finite to infinite, from ordinary to supersymmetry and recently arrived at the notion of quantum groups. In general, a physical system consists of a finite or infinite number of degrees of free- dom which may or may not interact. The dynamics is prescribed by a set of evolution equations which follow from varying the action with respect to the different degrees of freedom. A symmetry then corresponds to a group of transformations on the space time coordinates and/or the degrees of freedom that leave the action and therefore also the evo- lution equations invariant. External symmetries have to do with invariances (e.g. Lorentz invariance) under transformations on the space time coordinates. Symmetries not related with transformations of space time coordinates are called internal symmetries. We also discriminate between global symmetries and local symmetries. A global or rigid symme- try transformation is the same throughout space time and usually leads to a conserved quantity. Turning a global symmetry into a local symmetry, i.e. allowing the symmetry transformations to vary continuously from one point in space time to another, requires the introduction of additional gauge degrees of freedom mediating a force. This so-called gauge principle has eventually led to the extremely successful standard model of the strong and electro-weak interactions between the elementary particles based on the local gauge group SU(3) SU(2) U(1). × × A symmetry of the action is not automatically a symmetry of the groundstate of a physical system. If the action is invariant under some symmetry group G and the ground- state only under a subgroup H of G, the symmetry group G is said to be spontaneously broken down. The symmetry is not completely lost though, for the broken generators of G transform one groundstate into another. The physics of a broken global symmetry is quite different from a broken local gauge symmetry. The signature of a broken continuous global symmetry group G in a physical system is the occurrence of massless scalar degrees of freedom, the so-called Goldstone bosons. Specifically, each broken generator of G gives rise to a massless Goldstone boson field. Well-known realizations of Goldstone bosons are the long range spin waves in a ferromagnet, in which the rotational symmetry is broken below the Curie temperature through the appearance of spontaneous magnetization. A beautiful example in particle physics is the low energy physics of the strong interactions, where the spontaneous break- 1 2 PREFACE down of (approximate) chiral symmetry leads to (approximately) massless pseudoscalar particles such as the pions. In the case of a broken local gauge symmetry, on the other hand, the would be massless Goldstone bosons conspire with the massless gauge fields to form a massive vector field. This celebrated phenomenon is known as the Higgs mechanism. The canonical example in condensed matter physics is the ordinary superconductor. In the phase transition from the normal to the superconducting phase, the U(1) gauge symmetry is spontaneously broken by a condensate of Cooper pairs. This leads to a mass MA for the photon field in the superconducting medium as witnessed by the Meissner effect: magnetic fields are expelled from a superconducting region and have a characteristic penetration depth which in proper units is just the inverse of the photon mass MA. The Higgs mechanism also plays a key role in the unified theory of weak and electromagnetic interactions, that is, the Glashow-Weinberg-Salam model where the product gauge group SU(2) U(1) is broken to the U(1) subgroup of electromagnetism. Here, the massive vector particles× correspond to the W and Z bosons mediating the short range weak interactions. More speculative applications of the Higgs mechanism are those where the standard model of the strong, weak and electromagnetic interactions is embedded in a grand unified model with a large simple gauge group. The most ambitious attempts invoke supersymmetry as well. In addition to the characteristics in the spectrum of fundamental excitations described above, there are in general other fingerprints of a broken symmetry in a physical system. These are usually called topological excitations or just defects, see for example the ref- erences [44, 93, 108, 110] for reviews. Defects are collective degrees of freedom carrying ‘charges’ or quantum numbers which are conserved for topological reasons, not related to a manifest symmetry of the action. It is exactly the appearance of these topological charges which renders the corresponding collective excitations stable. Topological excitations may manifest themselves as particle-like, string-like or planar-like objects (solitons), or have to be interpreted as quantum mechanical tunneling processes (instantons). Depending on the model in which they occur, these excitations carry evocative names like kinks, domain walls, vortices, cosmic strings, Alice strings, monopoles, skyrmions, texture, sphalerons and so on. Defects are crucial for a full understanding of the physics of systems with a broken symmetry and lead to a host of rather unexpected and exotic phenomena, which are in general of a nonperturbative nature. The prototypical example of a topological defect is the Abrikosov-Nielsen-Olesen flux tube in the type II superconductor with broken U(1) gauge symmetry [1, 99]. The topolog- ical quantum number characterizing these defects is the magnetic flux, which indeed can only take discrete values. A beautiful example in particle physics is the ’t Hooft-Polyakov monopole [67, 105] occurring in any grand unified model in which a simple gauge group G is broken to a subgroup H which contains the electromagnetic U(1) factor. Here, it is the quantized magnetic charge which is conserved for topological reasons. In fact, the presence of magnetic monopoles in these models reconciles the two well-known arguments for the quantization of electric charge, namely Dirac’s argument based on the existence of a magnetic monopole [49] and the obvious fact that the U(1) generator should be compact PREFACE 3 as it belongs to a larger compact gauge group.
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