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Double Field Theory on Group Manifolds

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Double Field Theory on Group Manifolds Double Field Theory on Group Manifolds Falk Haßler München 2015 Double Field Theory on Group Manifolds Falk Haßler Dissertation an der Fakultät für Physik der Ludwig–Maximilians–Universität München vorgelegt von Falk Haßler aus Eberswalde-Finow München, den 01.06.2015 Erstgutachter: Prof. Dr. Dieter Lüst Zweitgutachter: Priv.-Doz. Dr. Ralph Blumenhagen Tag der mündlichen Prüfung: 15.07.2015 Zusammenfassung Die vorliegende Arbeit befasst sich mit Double Field Theory (DFT), einer effektiven Feld- theorie, die die Dynamik von geschlossenen Strings mit niedriger Energie auf einem Torus beschreibt. Alle Observablen, die aus dieser Dynamik entstehen, stimmen in bestimmten Familien von Hintergrundraumzeiten überein. Solche verschiedenen Hintergründe sind durch T-Dualität miteinander verbunden. DFT macht T-Dualität auf einem Torus explizit sichtbar, indem sie zusätzlich zu den D Koordinaten der Raumzeit weitere D Windungskoordinaten einführt. Der Strong Constraint, eine wichtige Konsistenzbedingung der Theorie, schränkt allerdings die Abhängigkeit aller physikalischen Felder auf die Hälfte der Koordinaten des entstehenden gedoppelten Raumes ein. Eine wichtige Anwendung findet DFT bei verallgemeinerten Scherk-Schwarz Kompakti- fizierungen. Sie führen zu halbmaximalen, elektrisch geeichten Supergravitationen, welche durch den Embedding Tensor Formalismus klassifiziert werden. Er beschreibt die Einbettung ihrer Eichsymmetrie in die Gruppe O(n; n). Da der Strong Constraint nicht mit allen Lö- sungen des Embedding Tensors kompatibel ist, wird er in der DFT Flussformulierung durch den schwächeren Closure Constraint ersetzt. Dadurch werden Kompaktifizierungen auf Hin- tergründen möglich, die nicht T-dual zu geometrisch wohldefinierten Hintergründen sind. Sie können nur mit Hilfe von nicht-geometrischen Flüssen beschrieben werden und sind auf Grund ihrer speziellen Eigenschaften von großem phänomenologischem Interesse. Allerdings verschleiert die Verletzung des Strong Constraints ihren Uplift zur vollen String Theorie. Des Weiteren gibt es technische Unklarheiten bei der Verallgemeinerung von herkömmlichen Scherk-Schwarz Kompaktifizierungen auf den gedoppelten Raum der DFT. So ist zum Bei- spiel nicht bekannt, wie der Twist der Kompaktifizierung im Allgemeinen zu konstruieren ist. Nachdem die grundlegenden Konzepte von DFT und verallgemeinerten Scherk-Schwarz Kompaktifizierungen wiederholt wurden, wird DFTWZW, eine Verallgemeinerung der bishe- rigen Formulierung, vorgestellt. Sie beschreibt die Niederenergiedynamik von geschlossenen, bosonischen Strings auf einer kompakten Gruppenmanigfaltigkeit und erlaubt die genannten Probleme zu lösen. Ihre klassische Wirkung und die dazugehörenden Eichtransformationen ergeben sich aus Closed String Field Theory bis zur kubischen Ordnung in den masselosen Feldern. Die so gewonnen Ergebnisse werden durch eine verallgemeinerte Metrik ausgedrückt und auf alle Ordnungen erweitert. Es zeigt sich eine explizite Trennung zwischen Hintergrund und Fluktuationen. Damit die Eichalgebra schließt, müssen die Fluktuationen einem modi- fiziertem Strong Constraint genügen, während für den Hintergrund der Closure Constraint ausreicht. Zusätzlich zu den aus der bisherigen Formulierung bekannten verallgemeinerten Diffeomorphismen ist DFTWZW auch unter gewöhnlichen Diffeomorphismen des gedoppelten Raumes invariant. Werden diese durch den extended Strong Constraint, eine optionale Zu- satzbedingung, gebrochen, ergibt sich die traditionelle DFT. Für die neue Theorie wird eine Flussformulierung hergeleitet und der Zusammenhang zu verallgemeinerten Scherk-Schwarz Kompaktifizierungen beleuchtet. Ein möglicher tree-level Uplift von einem wahrhaft nicht- geometrischen Hintergrund (nicht T-dual zu geometrischen) wird präsentiert. Weiterhin kön- nen die Unklarheiten bei der Konstruktion von geeigneten Twists beseitigt werden. Auf diese Weise ensteht ein allgemeineres Bild von DFT und der ihr zugrunde liegenden Strukturen. Abstract This thesis deals with Double Field Theory (DFT), an effective field theory capturing the low energy dynamics of closed strings on a torus. All observables arising from those dynam- ics match on certain families of background space times. These different backgrounds are connected by T-duality. DFT renders T-duality on a torus manifest by adding D windig co- ordinates in addition to the D space time coordinates. An essential consistency constraint of the theory, the strong constraint, only allows for fields which depend on half of the coordinates of the arising doubled space. An important application of DFT are generalized Scherk-Schwarz compactifications. They give rise to half-maximal, electrically gauged supergravities which are classified by the embed- ding tensor formalism, specifying the embedding of their gauge group into O(n; n). Because it is not compatible with all solutions of the embedding tensor, the strong constraint is replaced by the closure constraint of DFT’s flux formulation. This allows for compactifications on backgrounds which are not T-dual to well-defined geometric ones. Their description requires non-geometric fluxes. Due to their special properties, they are also of particular phenomeno- logical interest. However, the violation of the strong constraint obscures their uplift to full string theory. Moreover, there is an ambiguity in generalizing traditional Scherk-Schwarz compactifications to the doubled space of DFT: There is a lack of a general procedure to construct the twist of the compactification. After reviewing DFT and generalized Scherk-Schwarz compactifications, DFTWZW, a gen- eralization of the current formalism is presented. It captures the low energy dynamics of a closed bosonic string propagating on a compact group manifold and it allows to solve the problems mentioned above. Its classical action and the corresponding gauge transformations arise from Closed String Field Theory up to cubic order in the massless fields. These results are rewritten in terms of a generalized metric and extended to all orders in the fields. There is an explicit distinction between background and fluctuations. For the gauge algebra to close, the latter have to fulfill a modified strong constraint, while for the former the closure constraint is sufficient. Besides the generalized diffeomorphism invariance known from the traditional formulation, DFTWZW is invariant under standard diffeomorphisms of the doubled space. They are broken by imposing the totally optional extended strong constraint. In doing so, the traditional formulation is restored. A flux formulation for the new theory is derived and its connection to generalized Scherk-Schwarz compactifications is discussed. Further, a possible tree-level uplift of a genuinely non-geometric background (not T-dual to any geomet- ric configuration) is presented. Finally, the ambiguity in constructing the compactification’s twist is eliminated. Altogether, a more general picture of DFT and the structures it is based on emerges. Acknowledgments First, I would like to express my special gratitude to my supervisor Dieter Lüst for his irre- placeable support, encouragement and trust. He offered me, a total string theory newcomer, the great opportunity to make my first contact with this wonderfully diverse field of research. Working with him, I constantly benefited from his vivid ideas, physical intuition, striking knowledge of the literature and last but not least his widespread connections to the scientific community. Further, I am deeply indebted to Ralph Blumenhagen who answered my numerous ques- tions with impressive patience. His ability to spot an ambiguity immediately – even in the most complicated setup – was essential for our fruitful collaborations. I also had the pleasure to collaborate with Pascal du Bosque whom I would like to thank especially. His unbreakable endurance while fighting with me through many very technical calculations was indispensable. I also want to thank (in alphabetical order) Lara Anderson, André Betz, Ilka Brunner, Livia Ferro, Michael Fuchs, James Gray, Michael Haack, Daniela Herschmann, Olaf Hohm, Daniel Junghans, Florian Kurz, Stefano Massai, Christoph Mayrhofer, Stefan Groot Nibbe- link, Peter Patalong, Erik Plauschinn, Daniel Plencner, Felix Rennecke, Cornelius Schmidt- Colinet, Maximilian Schmidt-Sommerfeld and Rui Sun for many great discussions, patience while answering my questions and removing my concerns. All members of the String Theory Group in Munich and of the Max-Planck-Institute for Physics made it fun to work here. The three years of my doctoral studies went by much too fast. Finally, I want to mention that all the work I have done here would have been hardly possible without the support and understanding my wife and my parents gave me. Living with a physicist who sometimes forgets the most basic things in everyday life while thinking about some abstract question is a challenge. Thank you for mastering it everyday. Contents Abstract vii Acknowledgments ix 1 Introduction 1 1.1 Unification . .1 1.2 String theory . .3 1.2.1 Worldsheet . .3 1.2.2 Target space . .4 1.2.3 T-duality . .5 1.3 Superstring theory . .7 1.4 Low energy effective theory . .8 1.4.1 T-duality . .9 1.4.2 Compactification . 10 1.4.3 Non-geometric backgrounds . 12 1.4.4 Applications . 14 1.5 Outline and summary . 14 2 Double Field Theory 17 2.1 Action and its symmetries . 17 2.1.1 Action . 17 2.1.2 Generalized diffeomorphisms . 18 2.1.3 Double Lorentz symmetry . 19 2.2 Equations of motion . 21 2.3 Flux formulation . 22 2.3.1 Closure constraint . 23 2.3.2 Action and equations of
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