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Structure of Coset Models Structure of Coset Models Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨aten der Georg-August-Universit¨at zu G¨ottingen arXiv:math-ph/0308031v1 26 Aug 2003 vorgelegt von Soren¨ Koster¨ aus Aachen G¨ottingen 2003 D 7 Referent: Prof. Dr. Karl-Henning Rehren1 Korreferent: Prof. Dr. Detlev Buchholz2 Tag der mundlichen¨ Prufung:¨ 3. Juni 2003 1Institut fur¨ Theoretische Physik der Universit¨at G¨ottingen 2Institut fur¨ Theoretische Physik der Universit¨at G¨ottingen .Συνηθε´ιας δ´η, o⁀֓ιµαι, δ´εoιτ֓αν ֓´ , ει֓ µ´ελλoι τα` ανω֓´ ֓´oψεσθαι Πoλιτειας Z [P, (516a4)]. Zusammenfassung: Wir untersuchen Einbettungen lokaler, chiraler, konformer Quantentheorien , die mit einer gegebenen Untertheorie ver- tauschen; die UntertheorienC ⊂ B werden als Coset-Modelle bezeichnet.A ⊂ B Die C ⊂ B meisten Ergebnisse dieser Arbeit sind modellunabh¨angig, jedoch wird diese Un- tersuchung motiviert durch Einbettungen von Stromalgebren und deren Coset- Modelle. Wir zeigen, dass es zu jeder gegebenen Untertheorie eine eindeutige, innere Darstellung U A gibt, die die konforme SymmetrieA auf ⊂ B der Untertheorie verwirklicht. Die lokalen beobachtbaren Gr¨oßen von , die mit U A vertauschen, bilden das maximale Coset-Modell . B Cmax Unter der Annahme, dass U A durch Integrale eines innewohnenden Quan- tenfeldes erzeugt wird, zeigen wir: Die Einbettung derA Untertheorie und ihrer Coset-Modelle steht in unmittelbarer Analogie zur Inklusion chiraler Theo- rien in einer 1+1-dimensionalen lokalen, konformen Quantentheorie. Die lokalen beobachtbaren Gr¨oßen von zu einem bestimmten Gebiet sind gerade diejeni- Cmax gen Gr¨oßen von , die mit den lokalen beobachtbaren Gr¨oßen der Untertheorie B zu demselben Gebiet vertauschen. A Wir geben einige Anwendungen unserer Ergebnisse und diskutieren m¨ogliche Verallgemeinerungen unserer Vorgehensweise. Abstract: We study inclusions of local, chiral, conformal quantum theories which commute with a given subtheory . These subtheories areC ⊂ called B Coset models. Most of our results areA ⊂ model-independent, B althoughC ⊂ B our analysis is motivated by the inclusions of current algebras and their Coset models. We prove that to every given there is a unique, inner representation U A which implements conformal symmetryA ⊂ B on the subnet. The local observables of which commute with U A form the maximal Coset model . B Cmax Assuming U A to be generated by integrals of a quantum field affiliated with the subnet , we show: The inclusion of the subnet and of its Coset models is directly analogousA ⊂ B to the inclusion of chiral observables in a local, conformal theory in 1+1 dimensions. The local observables of the maximal Coset model associated with a given region are found to be characterised by their commuting with the local observables of associated with the very same region. We give applications andA discuss possible generalisations of our methods. Contents I Introduction 1 II Coset pairs of chiral subtheories 7 1 Assumptions, conventions, and notions . 7 1.1 Chiralconformalsymmetry . 7 1.2 Chiral nets, chiral subnets and their Coset models .... 9 1.3 Discussingtheassumptions . 12 2 Conformaltransformationsasobservables . 22 2.1 Borchers-Sugawara construction . 24 2.2 Direct applications to chiral subnets . 30 2.3 Discussion on the Borchers-Sugawara construction . 34 III Subnets of chiral current algebras 36 1 Quarkmodelsofchiralcurrentalgebras. 37 1.1 Chiral currents as Wick squares .............. 37 1.2 Sugawara construction for chiral current algebras . 41 2 Chiralconformalnetsofcurrentalgebras . 44 2.1 Compactpictureofchiralcurrentalgebras . 44 2.2 Integratingchiralcurrentalgebras. 46 2.3 Fromloopalgebrastochiralcurrentalgebras. 50 3 Current subalgebras as instructive examples . 53 4 Conformalcovariancesubalgebras . 56 4.1 Characterisation of conformal inclusions . 57 4.2 Covariant and invariant colours . 60 IV Local Nature of Coset Models 64 1 Net-endomorphismProperty . 65 2 Chiralholography........................... 70 2.1 The holographic quasi-theory in 1+1 dimensions . 71 2.2 Sharp geometrical action and time-like commutativity . 75 3 Solvingtheisotonyproblem ..................... 78 i ii V Coset pairs of finite Index 82 1 On local inclusions of Coset pairs ................. 82 2 Coset pairsasnetsofsubfactors . 87 2.1 The localised representation of a Coset pair........ 88 2.2 On spatial Coset pairs ................... 94 3 Coset construction of normal CTPS ................ 99 3.1 The Coset pairs of Goddard, Kent and Olive . 101 3.2 Extension (A10, E6) of ir 21 as a Coset model . 103 V 22 3.3 A normal CTPS in E(8) ................. 107 L 1 VI The conformally covariant derivatives of the U(1) current 111 1 The chiral nets generated by the Φ(n) ................ 112 2 No stress-energy tensor in Φ(n)-models ............... 116 VII Discussion 118 1 Summary ............................... 118 2 Outlook ................................ 122 A Appendix 125 1 Lemmas on PSL(2, R) and PSL(2, R)∼ ............... 125 1 1 ∼ 2 Lemmas on Diff+(S ) and Diff+(S ) ................ 128 3 AlternativeargumentforlemmaIV.2 . 130 4 Detailed proof of the BMT theorem ................ 131 1 5 Diff+(S )-Symmetry and PSL(2, R)-Covariance. 132 Frequently used symbols 152 Danksagungen 153 Chapter I Introduction The first quarter of the twentieth century saw two great revolutions is physics: the uncovering of quantum physics and the discovery of relativity. The quest for unifications of both in systems with infinitely many degrees of freedom, the development of relativistic quantum field theories, has led to remarkable suc- cesses, especially in elementary particle physics. The description of the physical laws valid in this context and our present understanding of the fundamental in- teractions, mainly summarised in the standard model for the strong, weak and electro-magnetic forces, has not yet reached the stage of a mathematically con- sistent theory. On the other hand, the underlying physical principles are clear: The primary objects are expectation values of observable quantities. Not all observables are compatible, ie some sets of observables can not be measured si- multaneously with arbitrary accuracy, like eg position and momentum due to Heisenberg’s uncertainty relation. The laws of physics are independent of the choice of frame of reference, ie they may be formulated covariantly with respect to the relativistic spacetime symmetry group. The principle of locality states that it is meaningful to talk of observables which can be measured in bounded space- time regions and that observables are always compatible, if they are spacelike separated. There are two concise mathematical frameworks which capture possible ways to formulate relativistic quantum field theories. In the first, one assigns to points xµ in spacetime operator-valued distributions φ(xµ), the quantum fields, which may be viewed as quantised versions of the field strengths known from classical field theories. The principles of relativistic quantum field theory are reflected in certain requirements on the quantised fields, mostly known as Wightman’s axioms [WG65]. Successes of this framework are summarised eg in [Jos65, SW64, Tod65, BLT75]. Another approach was proposed by Haag and Kastler [HK64]. Here, one selects regions in spacetime and assigns to them (topological -) algebras ( ) of bounded operators.O The local algebras ( ) are regarded as∗ the algebrasA O of A O observations possible within the region and their elements are called the local O 1 2 Introduction I observables. The principles of relativistic quantum field theory can be expressed in this setting in a natural manner, leading to the framework of local quantum physics. This way of describing relativistic quantum field theory can be regarded as an extension of the picture of quantum fields and it has the advantage that one does not need to deal with particular sets of “coordinates” of the theory (the quantum fields). Especially structural and conceptual problems can be discussed within the setting of local quantum physics successfully (cf [Haa92, Buc00]). The physics of a local quantum theory is mainly encoded in the isotony inclusions of local algebras: if a region isA contained in another region the O1 O2 property of isotony means that the local algebra ( 1) is contained in the algebra ( ). Put differently: isotony says that theA amountO of possible observations A O2 increases with the localisation region they can be made in. This appears to be an almost trivial statement, but the isotony inclusions give structure of physical relevance to the local algebras which, ignoring this substructure, are (essentially) all isomorphic on quite general grounds [BDF87]. By isotony, a local quantum theory is a net of local algebras. Within the framework of Haag and Kastler it is natural to consider sub- theories, ie inclusions of one local quantum theory, , in another one, , given by the inclusion of their local algebras: ( ) ( ).A The origin of theseB inclusions is encoded in two properties: they areA covariantO ⊂ B O with respect to the action of the spacetime symmetry group and they are consistent with the net-structure of the subtheory, ie for we have: ( ) ( ). O1 ⊂ O2 A O1 ⊂A O2 Obviously, there are non-trivial subtheories like, for example, observables A included as gauge invariants in a local quantum theory with a compact gauge symmetry. This example gives one motivation for studyingF subtheories: By the Doplicher-Roberts reconstruction [DR90] all the relevant information is in fact encoded in already.
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