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Bicoloured Torus Loop Groups Bicoloured torus loop groups Roles of promotor and copromotor: The primary and secondary PhD supervisors, as these terms are commonly understood in the international research community, are the people listed on page ii as ‘copromotor’ and ‘promotor’, respectively. The reason for this reversal is that Dutch law requires a PhD thesis to be approved by a full professor. Assessment committee: Prof.dr. M.N. Crainic, Universiteit Utrecht Prof.dr. C.L. Douglas, University of Oxford Prof.dr. K.-H. Neeb, Friedrich-Alexander-Universität Erlangen-Nürnberg Prof.dr. K.-H. Rehren, Georg-August-Universität Göttingen Prof.dr. C. Schweigert, Universität Hamburg Printed by: Ipskamp Printing, Enschede ISBN: 978-90-393-6743-8 © Shan H. Shah, 2017 Bicoloured torus loop groups Bigekleurde toruslusgroepen (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnificus, prof.dr. G.J. van der Zwaan, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op woensdag 15 maart 2017 des ochtends te 10.30 uur door Shan Hussain Shah geboren op 6 mei 1989 te Lelystad Promotor: Prof.dr. E.P.van den Ban Copromotor: Dr. A.G. Henriques Funding: The research that resulted in this thesis was supported by a Graduate Funding Grant for the project ‘The classification of abelian Chern–Simons theories’, obtained from the ‘Geometry and Quantum Theory’ (GQT) mathematics cluster of the Netherlands Organisation for Scientific Research (NWO). Contents 1 Introduction 1 1.1 Von Neumann algebras 2 1.2 Conformal nets 5 1.3 Defects between conformal nets 14 1.4 Bicoloured loop groups 21 1.5 Main results 24 1.6 Organisation of the text 28 1.7 Notations and conventions 29 2 Unicoloured torus loop groups 31 2.1 Unicoloured torus loop groups and their structure 32 2.2 Central extensions associated to lattices 34 1 2.3 Actions of Diff+(S ) on central extensions 39 2.4 Actions of lifts of lattice automorphisms on central extensions 45 2.5 Irreducible, positive energy representations 49 3 Bicoloured torus loop groups 71 3.1 Bicoloured torus loop groups and their structure 72 3.2 Central extensions associated to spans of lattices 86 1 3.3 Actions of covers of Diff+(S ) on central extensions 98 3.4 Irreducible, positive energy representations 101 4 Outlook 113 4.1 Defects between lattice conformal nets 113 4.2 Generalising to different tori at the two defect points 114 4.3 Generalising from tori to non-abelian Lie groups 116 A Background material 117 A.1 Lattices 117 A.2 Central extensions of groups 124 iii iv CONTENTS A.3 Group representations 129 A.4 Heisenberg groups 142 Bibliography 153 Samenvatting 161 Unigekleurde toruslusgroepen 161 Bigekleurde toruslusgroepen 163 Acknowledgements 165 Curriculum vitae 167 Chapter 1 Introduction The material collected and developed in this thesis is motivated by the following question: Given that certain centrally extended loop groups and their positive energy representations give rise to conformal nets, does there exist a generalisation of the theory of loop groups which allows one to analogously construct defects between conformal nets? We focus exclusively on a special class of loop groups, namely the torus loop groups which give rise to lattice conformal nets, and manage to make progress towards answering the above question. Our results can be summarised by the claim Torus loop groups generalise to so called bicoloured torus loop groups. The latter enjoy many properties one expects them to have for constructing defects between lattice conformal nets. Our first aim in this Introduction will be to explain the various terms used in the above question. We begin with a short review of von Neumann algebras in Section 1.1 because these feature prominently in the definition of conformal nets that follows next in Section 1.2. In that section we list some (mathematical) motivations for the theory of conformal nets and basic examples of these objects. We notably give an overview of the construction of certain conformal nets from central extensions of loop groups. We then turn to introducing the notion of a defect between two conformal nets in Section 1.3 of which we present elementary examples. The scarcity of these examples will immediately spark the question of finding other methods of constructing defects. To answer this question we formulate in Section 1.4 a 1 2 INTRODUCTION loose, hypothetical notion of a bicoloured loop group as a method of producing conformal net defects. Our proposal starts properly in Section 1.5, where we make precise the special case of a definition of a bicoloured torus loop group and summarise the results we obtain in this thesis on this new notion. We refer to Section 1.7 and Appendix A for various notations, conventions and definitions used in this Introduction. Remark 1.0.1 (Advice to the reader). The question posed above and the material on von Neumann algebras, conformal nets and defects treated in Sections 1.1 to 1.3 has been included only to explain the context of our studies, and it will not make a relevant reappearance until Section 4.1. The bulk of our investigations in Chapters 2 and 3 does not strictly require knowledge of these matters. The reader may therefore safely skip Sections 1.1 to 1.3 without missing technical background needed for the rest of the thesis. 1.1 Von Neumann algebras Von Neumann algebras were introduced by F. Murray and J. von Neumann in a series of papers in the 1930s and 40s with applications to representation theory and physics in mind. Their relevance to this Introduction is their appearance in the definitions of conformal nets and defects we give in Sections 1.2 and 1.3, respectively, and they will make a brief reappearance in Chapter 4. We assume a passing familiarity with the weak, the strong operator and ultraweak topology on the algebra of bounded operators on a Hilbert space from the reader, as treated in for example [Con90, Chapters IV and IX]. Let be a Hilbert space and denote by ( ) its algebra of bounded H B H operators. For a subset S ( ), its commutant S0 is defined as the set of bounded operators on which⊆ B H commute with all operators of S. We list some algebraic properties ofH taking commutants. It reverses inclusions of subsets, and we have S S00 and S000 = S0. This means that the operation of taking the ⊆ commutant does not continue indefinitely. It is always true that S0 is a unital subalgebra of ( ) and if S is self-adjoint, meaning that S∗ = S, then S0 is a unital -subalgebra.B H ∗ A topological property of forming the commutant is that S0, and hence also S00 is weakly closed in ( ). The following important result strengthens this fact dramatically whenBS isH a unital -algebra. ∗ Theorem 1.1.1 (Von Neumann’s Bicommutant Theorem). The double commu- tant A00 of a unital -subalgebra A of ( ) is equal to both the closure of A in the weak, and in the∗ strong operator topology.B H 1.1 Von Neumann algebras 3 That is, the algebraic operation of taking the double commutant of a unital -subalgebra of ( ) can be expressed in two, equivalent topological terms. It∗ implies that theB followingH definition is unambiguous. Definition 1.1.2 ((Concrete) von Neumann algebras). A (concrete) von Neu- mann algebra is a unital -subalgebra A of the algebra ( ) of bounded operators on a Hilbert space∗ which equivalently B H H • is closed in the weak operator topology on ( ), B H • is closed in the strong operator topology on ( ), or B H • satisfies A00 = A. One can furthermore use the Bicommutant Theorem to prove Corollary 1.1.3. The smallest von Neumann algebra vN(S) containing a subset S ( ) equals both (S S∗)00 and the closure of the unital -subalgebra generated⊆ B H by S in either the weak[ or strong operator topology. ∗ So elements of vN(S) can be thought of as limits (in the weak or strong operator topology) of polynomials with (non-commuting) variables in S S∗. We give some examples of von Neumann algebras. The most obvious[ one is of course ( ) itself. Slightly more interesting are B H Example 1.1.4 (Algebras of essentially bounded functions). Let (X , Σ, µ) be 2 2 a σ-finite measure space and L (X ) := L (X , Σ, µ) the Hilbert space of all measurable functions f : X C which are square integrable, modulo functions ! that are zero almost everywhere. Define next L1(X ) := L1(X , Σ, µ) to be the set of measurable functions f : X C which are essentially bounded, 2 divided out by the same equivalence relation! as for L (X ). With the pointwise multiplication, the -operation f ∗ := f and the essential supremum norm this ∗ is a C∗-algebra. There is a unital, isometric -homomorphism ∗ 2 L1(X ) , L (X ) , f mf , !B 7! where mf is left multiplication by f . It can then be proved that L1(X )0 = L1(X ) (see [Con90, Theorem 6.6]), which implies that L1(X ) is a von Neu- 2 mann subalgebra of (L (X )). B The above are examples of abelian von Neumann algebras. It can in fact be shown that all abelian von Neumann algebras on a separable Hilbert space are of this form (see [Con90, Theorem 7.8]). This is the reason why the general 4 INTRODUCTION theory of von Neumann algebras is sometimes referred to as ‘non-commutative measure theory’. The von Neumann algebras that are relevant to us in this Introduction are far from abelian, though.
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