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D-Branes in Topological String Theory

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D-Branes in Topological String Theory DISSERTATION D–Branes in Topological String Theory ausgef¨uhrt zum Zwecke der Erlangung des akademischen Grades eines Doktor der Naturwissenschaften unter der Leitung von Ao. Univ. Prof. Dipl.-Ing. Dr. techn. Maximilian Kreuzer E136 Institut f¨ur Theoretische Physik und Prof. Wolfgang Lerche CERN eingereicht an der Technischen Universit¨at Wien Fakult¨at f¨ur Physik von Dipl.-Ing. Johanna Knapp Matrikelnummer: 9925349 Pfarrhofweg 3 A–3485 Haitzendorf [email protected] Wien, am 14. August 2007 to my family and my partner Kurzfassung Das Hauptthema dieser Doktorarbeit sind D–branes in topologischer Stringtheorie. Topol- ogische Stringtheorie beschreibt einen Untersektor der vollen Stringtheorie, der sich auf die Nullmoden der physikalischen Felder beschr¨ankt. Man erh¨alt eine solche Theorie durch den so genannten topologischen Twist, welcher, aus dem Blickwinkel der Weltfl¨achenwirkung, einer mit der Theorie vertr¨aglichen Redefinition der Spins der Felder entspricht. Der topologische Twist kann auf zwei Arten durchgef¨uhrt werden, woraus zwei unterschiedliche und a priori unabh¨angige Theorien resultieren, die als A– und B–Modell bezeichnet werden. F¨ur offene Strings kann man Randbedingungen definieren, welche mit dem topologischen Twist konsistent sind. Diese werden als A– bzw. B–branes bezeichnet. Vor Kurzem wurde eine neue Beschreibung von topologischen D–branes im B–Modell, welches durch ein Landau–Ginzburg Modell realisiert werden kann, gefunden. Solche Landau–Ginzburg Theorien sind durch ein Superpotential charakterisiert und B–branes werden durch Matrix- faktorisierungen dieses Superpotentials beschrieben. Diese Doktorarbeit besch¨aftigt sich mit den Eigenschaften und Anwendungen dieses Formalismus. Eine der Problemstellungen be- fasst sich mit der Berechnung des effektiven Superpotentials . Dieses Objekt ist von Weff ph¨anomenologischem Interesse, da man es als vierdimensionales Superpotential einer N = 1 supersymmetrischen Calabi–Yau Kompaktifizierung interpretieren kann. Es wird gezeigt, wie man diese Gr¨oße f¨ur minimale Modelle berechnen kann. Diese Modelle sind n¨utzliche Spielzeugmodelle f¨ur Stringkompaktifizierungen. Als eines der Hauptresultate dieser Dok- torarbeit zeigen wir, dass es m¨oglich ist zu berechnen, indem man die allgemeinsten Weff nicht–lineare Deformationen von Matrixfaktorisierungen berechnet. Ein bemerkenswertes Re- sultat in der Stringtheorie besagt, dass A-Modell und B–Modell durch eine Dualit¨at miteinan- der verbunden sind, die sogenannte Mirrorsymmetrie. Diese besagt, dass das A–Modell auf einer bestimmten Calabi–Yau Mannigfalitgkeit ¨aquivalent ist zum B–Modell auf der Mirror– Calabi–Yau. Mirrorsymmetrie kann erweitert werden um auch offene Strings zu beschreiben. Diese Du- alit¨at ist bis jetzt noch nicht ausreichend verstanden, insbesondere f¨ur kompakte Calabi–Yau Mannigfalitgkeiten. In dieser Doktorarbeit werden wir uns ausserdem mit Mirrorsymmetrie f¨ur offene Strings f¨ur die einfachste kompakte Calabi–Yau befassen, f¨ur den Torus. Unter der Verwendung von Matrixfaktorisierungen werden Dreipunktfunktionen im B–Modell berechnet und es wird gezeigt, dass diese mit den entsprechenden Gr¨oßen im A–Modell ¨ubereinstimmen. i Abstract The main focus of this thesis are D–branes in topological string theory. Topological string theory describes a subsector of the full string theory, that only captures the zero–modes of the physical fields. It can be obtained by an operation called topological twist, which is, at the level of the worldsheet action, a consistent redefinition of the spins of the fields. The topological twist can be done in two ways. This leads to two different, a priori independent, models, called the A–model and the B–model. If we consider open string theory, there are boundary conditions which are consistent with the topological twist. These boundary conditions are called A–type and B–type D–branes, respectively. Recently, a new description of D–branes in the topological B–model has been found. The B–model can be realized in terms of a Landau–Ginzburg theory, which is characterized by a superpotential. B–type D–branes are described by matrix factorizations of this superpotential. This thesis is devoted to exploring the properties and applications of this formalism. One of the central challenges of this thesis is the calculation of the effective superpotential . This quantity is of phenomenological interest since it has an interpretation as a four– Weff dimensional space–time superpotential in N = 1 supersymmetric string compactifications. We show how to calculate it for the minimal models, a subclass of Landau–Ginzburg theories, which serve as useful toy models for string compactifications. One of the main results of this thesis is that can be obtained by calculating the most general, non–linear deformation Weff of a matrix factorization. A remarkable result in string theory states that the A–model and the B–model are related by a duality known as mirror symmetry. The statement is that the A–model on a Calabi–Yau manifold is equivalent to the B–model on the mirror Calabi–Yau. Mirror symmetry can be extended to open strings. Open string mirror symmetry is by now poorly understood, in particular for compact Calabi–Yau spaces. In this thesis we will verify open string mirror symmetry for the simplest compact Calabi–Yau, the torus. Using matrix factorization techniques we calculate three–point correlators in the B–model and show that they match with the correlators we compute in the A–model. ii Acknowledgements I want to thank my supervisor at CERN, Wolfgang Lerche, for accepting me as his student and for suggesting research problems which are well suited for making one’s first steps in scientific work. I appreciate that I was left the time and the freedom to come to terms with the subject in my own way. Furthermore I want to thank Wolfgang for giving me insight into the beautiful field of topological string theory and for various short discussions which immediately cured my confusion when I was stuck in some calculation. It is a pleasure to thank my supervisor in Vienna, Maximilian Kreuzer, for constant support all through my theoretical physics career, and in particular for suggesting to me to do my PhD at CERN which I never would have dared without his encouragement. I also want to thank him for pleasant and instructive discussions during my stay in Vienna in February 2007. I want to express my gratitude to Hans Jockers for expertly answering many of my questions and for discussions which helped me see the things I was working on in a wider context. Fur- thermore I want to thank Emanuel Scheidegger for helpful discussions on topological string theory and modular forms. Thanks also to Marcos Mari˜no for organizing such a great student seminar. As promised, Marlene Weiss gets her own paragraph in this acknowledgements section. It is hard to find such a great office mate. I really enjoyed the many discussions we had on a broad variety of topics, sometimes even physics. I want to thank her for invaluable company and support during the last two years. Furthermore I want to thank James Bedford, Cedric Delaunay, I˜naki Garcia–Etxebarria, Ste- fan Hohenegger, Tristan Maillard, Are Rachlow, Fouad Saad, John Ward and all the other students at CERN for many pleasant encounters. I am deeply grateful to my boyfriend for more support than I could possibly ask for. I want to thank my friends and family for their continuing support, in particular Gige for taking care of all the administrative stuff in Austria. Thanks also to Mrs. M¨ossmer for help with the paper- work related to registering my thesis. Finally, I want to thank all the people who ever cooked for me, and Guide Michelin and Gault Millau for leading the way to most enjoyable decadence. I also want to acknowledge the recently deceased Professor Wolfgang Kummer, who taught me a great share of my theoretical physics general eductation and who has positively influenced my decision to focus on theory. iii Contents 1 Introduction 1 1.1 Motivation ..................................... 1 1.2 SummaryandOutline ............................... 5 2 Topological Strings and D–branes 6 2.1 Aspects of N =2Theories............................. 7 2.2 D–branes ...................................... 18 3 Boundary Landau–Ginzburg Models and Matrix Factorizations 26 3.1 Introduction.................................... 26 3.2 TheBoundaryLandauGinzburgModel . 27 3.3 Matrix Factorizations from a Physics Point of View . .......... 30 3.4 Matrix Factorizations from a Mathematics Point of View . .......... 36 3.5 Construction of Matrix Factorizations . ........ 39 3.6 Further Applications of Matrix Factorizations . ........... 42 4 The Effective Superpotential 43 4.1 Interpretation of the Effective Superpotential . ........... 43 4.2 Construction of the Effective Superpotential via Deformation Theory . 48 4.3 The Effective Superpotential from Consistency Constraints .......... 60 5 D–Branes in Topological Minimal Models 66 5.1 MinimalModels .................................. 67 5.2 The A3 MinimalModel–AToyExample . 68 5.3 The E6 MinimalModel .............................. 74 5.4 Relation to Kazama–Suzuki Coset Models . ...... 82 5.5 MoreResults .................................... 84 6 The Torus and Homological Mirror Symmetry 88 6.1 Introduction.................................... 88 6.2 Matrix Factorizations . 90 6.3 Cohomology..................................... 91 6.4 CorrelatorsintheB–model . 94 6.5 The“exceptional”D2Brane. 99 6.6 The A–Model and Homological Mirror
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