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Limit Theories and Continuous Orbifolds

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Limit Theories and Continuous Orbifolds Limit theories and continuous orbifolds Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Physik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakult¨at I der Humboldt-Universit¨at zu Berlin von Cosimo Restuccia geb. am 02.02.1984 in Florenz (Italien) Pr¨asident der Humboldt-Universit¨at zu Berlin: Prof. Dr. Jan-Hendrik Olbertz Dekan der Mathematisch-Naturwissenschaftlichen Fakult¨at I: Prof. Stefan Hecht, Ph.D. arXiv:1310.6857v1 [hep-th] 25 Oct 2013 Gutachter/innen: 1. Prof. Dr. Hermann Nicolai 2. Prof. Dr. Volker Schomerus 3. Prof. Dr. Matthias Staudacher Tag der mundlichen¨ Prufung:¨ 5. Juli 2013 Typeset in LATEX Abstract Relativistic quantum field theories are in general defined by a collection of effective actions, describing the dynamics of quantum fields at different energy scales. The consequent natural idea of a space of theories is still nowadays a rather imprecise notion, since a detailed knowledge or a classification of quantum field theories is out of reach. In two space-time dimensions the situation is in a better shape: in this context conformal field theories are under control in many instances, and we know sequences of rational theories emerging as end-points of renormalisation group flows. The present thesis explores the behaviour of sequences of rational two-dimensional conformal field theories when the central charge approaches its supremum. In particular, after a review of various notions useful to study limit theories, like the concept of averaged fields and the construction of continuous orbifolds, we analyse in detail the limit of sequences of N = 2 supersymmetric conformal field theories that are connected by renormalisation group flows. Remarkably, as we show explicitly for N = 2 minimal models in the c ! 3 regime, the limit is not unique, since with extended Virasoro symmetry one can choose different scalings for the labels of the spectrum in the limit. We construct explicitly two conformal field theories emerging as the large level limit of minimal models, and we identify both of them: one is the N = 2 superconformal field theory of two uncompactified free real bosons and two free real fermions, the other is its continuous orbifold by U(1). We compare spectrum, torus partition function, correlators and boundary conditions. The neatest interpretation of this result is given by studying the realisation of N = 2 minimal models as gauged Wess Zumino Witten models: taking the two different limits amounts to zooming into two different regions of the target-space geometry. We furthermore conjecture that one of the limit theories emerging from the limit of su(n+1) Kazama-Suzuki models (n = 1 being the minimal models case) is a U(n) continuous orbifold of a free theory. We motivate this idea by comparing the boundary spectra. At the end we speculate about the possible extensions and generalisations of the idea presen- ted in this thesis. Keywords: Two-dimensional conformal field theories, string theory, supersymmetric minimal models, con- tinuous orbifolds Zusammenfassung Relativistische Quantenfeldtheorien sind im Allgemeinen durch eine Vielzahl effektiver Wir- kungen definiert. Diese beschreiben die Dynamik der Quantenfelder bei gewissen Energieskalen. Die daraus folgende Idee von einem Raum von Theorien ist heutzutage immer noch eine eher unpr¨azise Bezeichnung, da eine genaue Kenntnis oder Klassifizierung von Quantenfeldtheorien sich außer Reichweite befindet. In zwei Dimensionen ist die Situation vorteilhafter: hier sind konforme Feldtheorien in vielen F¨allen besser verstanden, und wir kennen Folgen von rationalen Theorien, die durch Renormie- rungsgruppenflussen¨ verbunden sind. Diese Dissertation untersucht das Verhalten von Folgen rationaler zweidimensionaler konfor- mer Feldtheorien wenn sich die zentrale Ladung ihrem Supremum n¨ahert. Nach einer Diskussion von verschiedenen, sich fur¨ das Studium von Grenzwerttheorien nutzlich¨ erweisenden Aspekten wie das Konzept gemittelter Felder und der Konstruktion von kontinu- ierlichen Orbifolds, analysieren wir im Detail den Grenzwert einer Folge von N = 2 super- symmetrischen konformen Feldtheorien. Wir zeigen explizit fur¨ N = 2 minimale Modelle im Limes c ! 3, dass bemerkenswerterweise der Grenzwert nicht eindeutig ist, da mit erweiterter Virasoro-Symmetrie verschiedene Skalierungen fur¨ die Quantenzahlen des Spektrums im Grenz- wert gew¨ahlt werden k¨onnen. Wir konstruieren explizit zwei konforme Feldtheorien, die aus dem Grenzwert fur¨ große Level der minimalen Modelle hervorgehen, und identifizieren beide: Eine ist die N = 2 superkonforme Feldtheorie von zwei nichtkompaktifizierten freien reellen Bosonen und zwei freien reellen Fermionen, die andere ist dessen kontinuierlicher U(1)-Orbifold. Wir verglei- chen Spektrum, Torus-Zustandsummen, Korrelatoren und Randbedingungen. Die eing¨angigste Interpretation dieses Ergebnisses wird die Realisierung von N = 2 Minimalen Modellen als ge- eichte Wess-Zumino-Witten-Modelle gegeben: das Auftauchen zweier verschiedener Grenzwerte kann als das Hineinzoomen in zwei verschiedene Regionen der Geometrie im Zielraum interpre- tiert werden. Desweiteren vermuten wir, dass eine der Grenzwerttheorien, die aus dem Grenzwert von su(n + 1) Kazama-Suzuki Modellen (n = 1 ist der Fall fur¨ Minimale Modelle) ein kontinuierli- cher U(n)-Orbifold einer freien Theorie ist. Wir motivieren diese Idee durch den Vergleich der Randspektren. Am Ende spekulieren wir uber¨ die M¨oglichkeit des Auftretens von kontinuierlichen Orbifolds in unterschiedlichen F¨allen. Schlagw¨orter: Zweidimensionale Konforme Feldtheorien, Stringtheorie, Supersymmetrische Minimale Modelle, Kontinuierliche Orbifolds Contents Introduction 1 1 Limit theories 11 1.1 Generalities . 12 1.2 Large radius limit of one free boson on a circle . 16 1.3 Large level limit of unitary Virasoro minimal models . 21 1.A Free boson on a circle of radius R ........................... 24 2 Discrete and continuous orbifolds in CFT 27 2.1 Discrete orbifolds in CFT and in BCFT . 28 2.2 Continuous orbifolds in CFT and in BCFT . 34 2.A Notations for orbifolds . 40 3 N = 2 minimal models 41 3.1 N = 2 superconformal algebra . 42 3.2 Spectrum of N = 2 minimal models . 44 3.3 N = 2 minimal models as supersymmetric parafermions . 47 3.4 Bulk correlators . 53 3.5 Superconformal boundary conditions . 54 3.6 WZW description and geometry . 56 3.A Kazama-Suzuki Grassmannian cosets . 62 4 Limit of N = 2 minimal models: geometry 71 4.1 Two different limit theories . 71 4.2 Geometric interpretation of the limits: bulk . 72 4.3 Geometric interpretation: boundary . 74 5 The free theory limit 77 5.1 Partition function . 77 5.2 Fields and two-point function . 79 5.3 Three-point function . 81 5.4 A-type boundary conditions . 84 5.5 B-type boundary conditions . 85 vii viii CONTENTS 6 New c = 3 theory 89 6.1 The spectrum . 89 6.2 Partition function . 92 6.3 Fields and correlators . 93 6.4 Two-point function . 94 6.5 Three-point functions . 95 6.6 Boundary conditions and one-point functions . 99 7 Continuous orbifold interpretation 103 7.1 The orbifold . 103 7.2 Partition function . 104 7.3 Boundary conditions . 108 8 Limits of Kazama-Suzuki models 113 8.1 Kazama-Suzuki models: limit and continuous orbifold . 113 8.2 Boundary conditions match . 114 8.3 Limit of boundary conditions on the SU(3)=U(2) model . 115 8.4 Continuous orbifold C2=U(2) ............................. 117 Summary and outlook 121 A Characterology 125 A.1 Theta functions, and their modular properties . 125 A.2 Characters . 126 A.3 Supersymmetric partition functions and GSO projections . 132 A.4 A non-trivial modular transformation . 134 B Asymptotics of Wigner 3j-symbols 137 B.1 Notations and preliminaries . 137 B.2 Wigner's estimate . 138 B.3 Deeply in the allowed region . 140 B.4 On the transition region . 143 C Elements of boundary CFT 147 Bibliography 158 Aknowledgements 159 Introduction The discovery of a Higgs-like boson at CERN last year has produced a renewed excitement in the world of subatomic physics. Once again the Standard Model of particle physics has shown its merits in forecasting the existence of new fundamental particles. The conceptual and mathematical scheme on which the Standard Model is based is the framework of relativistic quantum field theories (QFTs). Since the early days of their (roughly) 70 years long history, QFTs have been plagued by infinities, ubiquitously appearing as outcome of computations of observables. The modern ap- proach to deal conceptually with this problem, pioneered by Wilson [1], consists in looking at QFTs as families of effective actions. At a fixed energy (or length) scale, the field modes of a QFT that are more energetic than the scale are unlikely to be excited, but crucially modify the form of the action, and therefore also the dynamics of the modes with energies comparable with the scale. This point of view suggests the picture of a space of actions, and the speculative idea that a given theory (a description of a physical system) manifests itself as a trajectory in this space, by going from higher to lower energy scales. Unfortunately, this concept is nowadays still rather heuristic, since we do not have a proper classification of QFTs, or a general definition of a metric and of a topology for a space of quantum theories [2]. In this introduction we describe the point of view of our work in this ideal framework. This thesis is devoted to explore the limits of sequences of N = 2 supersymmetric two dimensional conformal field theories. We start by briefly
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