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Sy^ Massive Kaluza-Klein Theories and Their Spontaneously Broken DE06FA435 DEUTSCHES ELEKTRONEN-SYNCHROTRON ( £|sY^ in der HELMHOLTZ-GEMEINSCHAFT DESY-THESIS-2006-013 July 2006 Massive Kaluza-Klein Theories and their Spontaneously Broken Symmetries by O. Hohm ISSN 1435-8085 NOTKESTRASSE 85 - 22607 HAMBURG DESY behält sich alle Rechte für den Fall der Schutzrechtserteilung und für die wirtschaftliche Verwertung der in diesem Bericht enthaltenen Informationen vor. DESY reserves all rights for commercial use of information included in this report, especially jn case of filing application for or grant of patents. To be sure that your reports and preprints are promptly included in the HEP literature database send them to (if possible by air mail): DESY DESY Zentralbibliothek Bibliothek Notkestraße 85 Platanenallee 6 22607 Hamburg 15738Zeuthen Germany Germany Massive Kaluza-Klein Theories and their Spontaneously Broken Symmetries Dissertation zur Erlangung des Doktorgrades des Departments Physik der Universit¨at Hamburg Vorgelegt von Olaf Hohm aus Hamburg Hamburg 2006 ii Gutachter der Dissertation: Jun. Prof. Dr. Henning Samtleben Prof. Dr. Jan Louis Gutachter der Disputation: Jun. Prof. Dr. Henning Samtleben Prof. Dr. Klaus Fredenhagen Datum der Disputation: 04. Juli 2006 Vorsitzender des Prufungsaussc¨ husses: Dr. Hans Dierk Ruter¨ Vorsitzender des Promotionsausschusses: Prof. Dr. Gun¨ ter Huber Dekan der Fakult¨at Mathematik, Informatik und Naturwissenschaften: Prof. Dr. Arno Fruh¨ wald iii Abstract In this thesis we investigate the effective actions for massive Kaluza-Klein states, focusing on the massive modes of spin-3/2 and spin-2 fields. To this end we determine the spontaneously broken gauge symmetries associated to these `higher-spin' states and construct the unbroken phase of the Kaluza-Klein theory. We show that for the particular background AdS S3 S3 a consistent coupling of the first massive spin- 3 × × 3/2 multiplet requires an enhancement of local supersymmetry, which in turn will be partially broken in the Kaluza-Klein vacuum. The corresponding action is constructed as a gauged maximal supergravity in D = 3. Subsequently, the symmetries underlying an infinite tower of massive spin-2 states are analyzed in case of a Kaluza-Klein compactification of four-dimensional gravity to D = 3. It is shown that the resulting gravity{spin-2 theory is given by a Chern-Simons action of an affine algebra and also allows a geometrical interpretation in terms of `algebra-valued' differential geometry. The global symmetry group is determined, which contains an affine extension of the Ehlers group. We show that the broken phase can in turn be constructed via gauging a certain subgroup of the global symmetry group. Finally, deformations of 3 3 the Kaluza-Klein theory on AdS3 S S and the corresponding symmetry breakings are analyzed as possible applications× ×for the AdS/CFT correspondence. Zusammenfassung In dieser Arbeit untersuchen wir die effektiven Wirkungen fur¨ massive Kaluza-Klein Zust¨ande, mit besonderer Beachtung von massiven Spin-3/2 und Spin-2-Moden. Zu diesem Zweck bestimmen wir die diesen ‘h¨oheren Spin' Feldern assoziierten spontan gebrochenen Symmetrien und konstruieren die ungebrochene Phase der Kaluza-Klein Theorie. Wir zeigen fur¨ den speziellen Hintergrund AdS S3 S3, dass die konsis- 3 × × tente Kopplung des ersten massiven Spin-3/2 Multipletts eine Erweiterung der lokalen Supersymmetrie erfordert, die wiederum im Kaluza-Klein Vakuum partiell gebrochen ist. Die korrespondierende Wirkung wird als eine geeichte maximale Supergravita- tionstheorie in D = 3 konstruiert. Anschliessend werden die Symmetrien analysiert, die im Falle einer Kaluza-Klein Kompaktifizierung von vierdimensionaler Gravitation nach D = 3 einem unendlichen Turm massiver Spin-2-Zust¨ande zugrundeliegen. Es wird gezeigt, dass die resultierende Gravitations{Spin-2 Theorie gegeben ist durch die Chern-Simons Wirkung einer affinen Algebra und dass sie weiterhin eine geometrische Interpretation vermittels `Algebra-wertiger' Differentialgeometrie erlaubt. Die glo- bale Symmetriegruppe, die eine affine Erweiterung der Ehlers-Gruppe enth¨alt, wird bestimmt. Wir zeigen, dass durch Eichung einer bestimmten Untergruppe der glob- alen Symmetriegruppe die gebrochene Phase konstruiert werden kann. Schlussendlich werden Deformationen der Kaluza-Klein Theorie auf AdS S3 S3 und die kor- 3 × × respondierenden Symmetriebrechungen als m¨ogliche Anwendungen der AdS/CFT - Korrespondenz analysiert. iv Acknowledgments I am greatly indebted to Henning Samtleben for his excellent supervision. I would also like to thank Jan Louis for supporting this PhD thesis in many ways. For scientific discussions and/or social interactions I would like to thank Iman Benmachiche, Christian Becker, Marcus Berg, Jens Fjelstad, Thomas Grimm, Hans Jockers, Paolo Merlatti, Falk Neugebohrn, Thorsten Prustel,¨ Sakura Sch¨afer-Nameki, Bastiaan Spanjaard, Silvia Vaula, Martin Weidner and Mattias Wohlfarth. Contents 1 Introduction 1 1.1 String theory as a candidate for quantum gravity . 1 1.2 Holography and Kaluza-Klein theories . 3 1.3 Outline of the thesis . 7 2 Higher-spin fields and supergravity 9 2.1 Consistency problems of higher-spin theories . 9 2.2 Gauged supergravity . 13 2.2.1 Generalities . 13 2.2.2 Gauged = 8 and = 16 supergravity in D = 3 . 18 N N 3 Massive spin-3/2-multiplets in supergravity 21 3.1 IIB supergravity on AdS S3 S3 S1 . 21 3 × × × 3.1.1 The 10-dimensional supergravity solution . 21 3.1.2 The Kaluza-Klein spectrum . 23 3.2 Effective supergravities for spin-1/2 and spin-1 multiplets . 30 3.3 The spin-3/2 multiplet . 32 3.3.1 Gauge group and spectrum . 34 3.3.2 The embedding tensor . 36 3.3.3 Ground state and isometries . 38 3.3.4 The scalar potential for the gauge group singlets . 41 4 Massive spin-2 fields and their infinite-dimensional symmetries 43 4.1 Is there a spin-2 Higgs effect? . 43 4.2 Kac-Moody symmetries in Kaluza-Klein theories . 45 4.3 Unbroken phase of the Kaluza-Klein theory . 48 4.3.1 Infinite-dimensional spin-2 theory . 48 4.3.2 Geometrical interpretation of the spin-2 symmetry . 52 4.3.3 Non-linear σ-model and its global symmetries . 56 vi CONTENTS 4.3.4 Dualities and gaugings . 59 4.4 Broken phase of the Kaluza-Klein theory . 61 4.4.1 Local Virasoro invariance for topological fields . 61 4.4.2 Virasoro-covariantisation for scalars . 66 4.5 Spin-2 symmetry for general matter fields . 69 4.6 Consistent truncations and extended supersymmetry . 70 5 Applications for the AdS/CFT correspondence 73 5.1 The AdS/CFT dictionary . 73 5.2 Marginal = (4; 0) deformations . 77 N 5.2.1 Non-linear realization of SO(3)+ SO(3)− . 78 × 5.2.2 Resulting = (4; 0) spectrum . 80 N 5.2.3 Lifting the deformation to D = 10 . 82 5.3 Marginal = (3; 3) deformations . 83 N 5.3.1 Deformations in the Yang-Mills multiplet . 83 5.3.2 Resulting = (3; 3) spectrum . 84 N 6 Outlook and Discussion 89 7 Appendices 93 A The different faces of E8(8) . 93 A.1 E8(8) in the SO(16) basis . 93 A.2 E8(8) in the SO(8; 8) basis . 94 A.3 E in the SO(8) SO(8) basis . 94 8(8) × A.4 E in the SO(4) SO(4) basis . 96 8(8) × B Kac-Moody and Virasoro algebras . 97 C Kaluza-Klein action on R3 S1 with Yang-Mills type gauging . 100 × D Spin-2 theory for arbitrary internal manifold . 102 Chapter 1 Introduction 1.1 String theory as a candidate for quantum gravity By now it is common wisdom that one of the most important and also most challenging problems of theoretical physics is the reconciliation of gravity, i.e. of general relativity, with quantum theory. Any `naive' approach to the problem of quantizing gravity along the lines of conventional field theories has failed, in that the resulting theories are non-renormalizable, so that they can be viewed at best only as an effective description. It is generally appreciated that the conceptually different nature of general rela- tivity lies at the heart of the problem of formulating a consistent theory of quantum gravity. As a so-called background-independent theory it is invariant under the full diffeomorphism group of the underlying manifold. In accordance with that there is no preferred reference frame and no fixed background space-time. Instead, space-time is completely determined dynamically. Since any conventional formulation of quantum field theory relies in contrast on a background space-time (as Minkowski space) with its preferred notion of distance, causality and time, it is not surprising that the quan- tization of gravity is a complicated problem. Therefore, quantum gravity struggles with a number of conceptually severe problems, like the famous so-called `problem of time' (for a pedagogical introduction see, e.g., [1].) Among the approaches to quantize gravity are those which try to maintain the conventional ideas of quantum field theory, like the notion of particles, S-matrix etc., while the others aim to take seriously the lessons which general relativity has taught us by maintaining general covariance also in the quantum theory. For instance, `loop quantum gravity' belongs to the latter [2, 3, 4]. While these theories are truly background-independent, they suffer from the humbling drawback that it is not clear how to get a semi-classical limit or even how to identify the observables [5]. String theory on the other hand is a perfectly well-defined perturbative quantum theory of gravity [6, 7]. Even though it extends the standard concepts of quantum field theory in the sense that it replaces the notion of a point-particle by a one-dimensional string, it still yields a conventional formulation in that it basically predicts certain S-matrix 2 Introduction elements. It provides a framework to compute finite results for, say, graviton-graviton- scattering on a Minkowski background to arbitrary accuracy. Accordingly, string theory in its original formulation is not background-independent, since it quantizes the propagation of strings on a fixed 10-dimensional background space-time (which is usually taken to be Minkowski space).
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