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Geometric and Non-Geometric Backgrounds from String Dualities

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Geometric and Non-Geometric Backgrounds from String Dualities Geometric and non-geometric backgrounds from string dualities Valent´ıVall Camell M¨unchen2018 Geometric and non-geometric backgrounds from string dualities Valent´ıVall Camell Dissertation an der Fakult¨atf¨urPhysik der Ludwig{Maximilians{Universit¨at M¨unchen vorgelegt von Valent´ıVall Camell aus Lloren¸cdel Pened`es M¨unchen, den 2. August 2018 Erstgutachter: Prof. Dr. Dieter L¨ust Zweitgutachter: Prof. Dr. Ilka Brunner Tag der m¨undlichen Pr¨ufung:17. September 2018 Zusammenfassung In dieser Arbeit werden bestimmte Klassen von nicht-geometrischen String-Hintergr¨unden, sowie weitere Aspekte im Wechselspiel zwischen Kompaktifizierungsmodellen und String- Dualit¨aten,analysiert. Konkreter ausgedr¨uckt, werden die durchgef¨uhrtenAnalysen durch String-Zielraum-Dualit¨aten(T- S- und U- Dualit¨aten)motiviert, die Beziehungen zwischen verschiedenen R¨aumen sind, die, wenn sie als String-Hintergr¨undeverwendet werden, zu den selben Quantentheorien f¨uhren. Insbesondere beginnen wir mit der Konstruktion einer Klasse von Kodimension-zwei L¨osungender String-Hintergrundgleichungen, die aus nicht-trivialen Zwei-Torus-Faserungen ¨uber einer zweidimensionalen Basis B bestehen. Nachdem ein Degenerationspunkt auf B eingekreist wurde, werden die Fasern mit einem beliebigen Element der T-Dualit¨ats- gruppe geklebt. Diese Gruppe schließt Transformationen ein, die weder Diffeomorphismen noch Eichtransformationen sind, die zu nicht-geometrischen Konfigurationen f¨uhren.Die entsprechenden Defekte, die wir T-fects nennen, k¨onnenmit der T-Dualit¨atsmonodromie um sie herum identifiziert werden. Wir bestimmen alle m¨oglichen derartigen Geometrien, indem wir meromorphe Funktionen finden, die sie charakterisieren.. Das bedeutet, dass die Konfigurationen einige Supersymmetrie bewahren. Mit Hilfe einer Hilfsfl¨ache, die ¨uber B gefasert ist, wird ein geometrisches Bild entwickelt, in dem die Monodromie als ein Produkt von Dehn-Twists auf dieser Oberfl¨ache beschrieben werden kann. Die Zwei-Torus-Faserung Struktur bricht auf den T-fects zusammen, und um die voll- st¨andigeString-L¨osungzu erhalten, muss man eine mikroskopische Beschreibung der De- generationen einkleben, die typischerweise einige Isometrien der Faser bricht. Wir analysieren die Physik in diesem Gebiet f¨urparabolische T-fects durch Untersuchung der Dynamik von Strings mit nicht-verschwindenden Windungszahlen um den Defekt. Daraus ergibt sich, dass die Physik in der N¨ahedes Kerns im Allgemeinen Windungsmoden beinhaltet, die die Supergravitationsann¨aherungnicht erfasst. Aus diesem Grund werden auch T-fects und deren Physik in der N¨aheihres Kerns innerhalb Double Field Theory analysiert und es wird diskutiert, in welchem Ausmaß die gefundenen Windungsmoden in so einem Formalismus kodiert werden k¨onnen.Eine der untersuchten Geometrien entspricht der kompaktifiziert NS5-Brane, f¨urdie auch T-Dualit¨atentlang anderer Isometrierichtungen analysiert wird und die Ergebnisse werden mit der kompaktifiziert NS5-Brane verglichen. Eine nat¨urliche Generalisierung f¨urT-fects L¨osungenbesteht darin, analoge Faserungen in M- und Type II-Theorien zu betrachten und U-Dualit¨at-Monodromien zu erlauben. In diesem Fall schr¨anktjedoch Supersymmetrie die Elemente der Dualit¨atsgruppe ein, die als vi Zusammenfassung Monodromien verwendet werden k¨onnen.Insbesondere ergibt sich, dass nur Konfiguratio- nen, die zu T-fects dual sind, konstruiert werden k¨onnen.Wir untersuchen diese Konfig- urationen anhand des neulich entwickelten Formalismus von generalisierten G-Strukturen in Exceptional Field Theory, dass die U-dualit¨atGeneralisierung von Double Field Theory ist. Ein solcher Formalismus ist ein m¨achtiges Werkzeug f¨urdie Untersuchung von super- symmetrischen Flusskompaktifizierungen von Type II- und M- Theorien, da es Metrische- und Flussfreiheitsgrade vereint. Die entwickelten Techniken werden auch dazu verwen- det, um Kompaktifizierungen von Type IIB zu AdS6 Vakua zu untersuchen, wobei bekan- nte Ergebnisse in der Literatur durch eine einfachere Formulierung mit nat¨urlichen ge- ometrischen Objekten reproduziert werden. Eine solche Formulierung erlaubt es auch, notwendige Bedingungen f¨urdie allgemeinsten konsistenten Trunkierungen mit Vektor- multiplets um diese Vakua zu finden. Diese konsistenten Trunkierungen sind wichtige Werkzeuge f¨urdie holographische Untersuchung dieser Vakua. Abstract In this thesis certain classes of non-geometric string backgrounds, as well as other aspects in the interplay between compactification models and string dualities, are analysed. Con- cretely, the performed analyses are motivated by string target space dualities (T- S- and U- dualities), which are relations between different spaces that, when used as backgrounds on which strings consistently propagate, give rise to the same quantum theory. In particular, we begin by constructing a class of codimension-two solutions to the string background equations consisting of non-trivial two-torus fibrations over a two dimensional base B. After encircling a degeneration point on B, the fiber is glued with an arbitrary element of the T-duality group, including transformations that are neither diffeomorphisms nor gauge transformations, which lead to non-geometric configurations. The correspond- ing defects, which we call T-fects, can be identified with the T-duality monodromy around them. We determine all possible such geometries by finding meromorphic functions of the base characterising them, which implies that the configurations preserve some supersym- metry. Using an auxiliary surface fibered over B, we develop a geometric picture where the monodromy can be described as a product of Dehn twists on this surface. The two-torus fibration structure breaks down at the T-fects and to obtain the com- plete string solution, one needs to glue in a microscopic description of the degenerations, which typically breaks some of the isometries of the fiber. We analyse the physics in these regions for parabolic T-fects by studying dynamics of strings with non-zero winding numbers around the defect. We deduce that, in general, physics near the core involves winding modes that the supergravity approximation fails to capture. For this reason, we also analyse T-fects and its near-core physics within Double Field Theory, an extension of supergravity designed to capture momentum and winding modes in the same footing, and discuss to which extent the encountered physics can be encoded within this formalism. One of the geometries we study corresponds to the compactified NS5 brane, for which we also analyse T-duality along other isometry directions and compare the results with the compact NS5 brane. A natural generalisation for T-fect solutions is to consider analogous fibrations in M- and type II theories and allow for U-duality monodromies. In this case, however, super- symmetry restricts the elements of the duality group that can be used as monodromies. In particular, we find that only configurations dual to T-fects can be constructed. We study these configurations using the recently developed formalism of generalised G-structures in Exceptional Field Theory, the U-duality generalisation of Double Field Theory. viii Abstract This formalism is a powerful tool for the study of supersymmetric flux compactifications of type II and M- theories, since it unifies metric and fluxes degrees of freedom. We use the developed techniques to study compactifications of type IIB to AdS6 vacua, rederiving known results in the literature using a simpler formulation in terms of natural geometric objects. This formulation also allows one to establish necessarily conditions for the most general consistent truncations with vector multiplets around these vacua. These consistent truncations are important tools for the holographic study of these vacua. Acknowledgements I would like to start thanking my supervisor Dieter L¨ustwho gave me the opportunity to initiate me in the field of String Theory and opened me the doors of his research group. During my PhD, I very much benefited of his vast knowledge of the field. I also wanted to thank Ilka Brunner for accepting co-supervising this thesis, as well as Ralph Blumenhagen who was always very supportive and helpful. Moreover, I am deeply indebted to Stefano Massai, Erik Plauschinn and Emanuel Malek, who supervised my work during different stages of my PhD. I want to specially thank them for sharing their knowledge and their research with me, for their patience answering all my questions and for their always very valuable advices. I am also grateful to Felix Rudolph, for very valuable discussions, and to Ismail Achmed-Zade, for our talks about physics and life, and to the rest of the members of the string theory group. Furthermore, all these work would have been hardly possible without the daily support and understanding from J´ulia,who was there to share the happiness of the successes but also my sorrows and concerns. I also want to express my deepest gratitude to my parents, Valent´ıand Helena, who have always supported and encouraged me, to my sister Maria, to Ignasi and to little Bruna, whose smile is our future. Finally my warm thanks to Aina and Laia, who every Tuesday evening found a way to keep me away from my problems, and to all my family and friends. x Aknowledgements Contents Zusammenfassung v Acknowledgements ix 1 Introduction 1 1.1 Why string theory? . .1 1.2 Basics of String Theory . .4 1.3 String dualities . .9 1.4 Non-geometric compactification models . 12 1.5 Generalised Geometry and extended space formalisms . 16 1.6 Summary and overview . 19 2 Toroidal fibrations, T-folds and T-fects 21 2.1 T-duality monodromies . 22 2.1.1 Monodromy of mapping tori . 23 2.2 Three-manifolds and T-folds . 25 2.2.1 Geometric τ monodromies . 26 2.2.2 T-folds and ρ monodromies . 31 2.2.3 General case . 33 2.3 Torus fibrations over a surface and T-fects . 34 2.3.1 Geometric τ-branes . 34 2.3.2 NS5 and exotic ρ-branes . 41 2.3.3 Colliding degenerations . 45 2.3.4 Remarks on global constructions . 47 2.4 T-fects in extended space theories . 48 2.4.1 Actions of the T-duality group O(2; 2; Z)............... 48 2.4.2 Doubled torus fibrations . 51 2.4.3 T-fects in Double Field Theory . 53 3 Physics of winding modes close to the T-fects 55 3.1 Exact metrics and T-duality .
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