Dissertation der Fakult¨atf¨urPhysik der Ludwig-Maximilians-Universit¨atM¨unchen Nonperturbative Type IIB Model Building in the F-Theory Framework vorgelegt von Benjamin Helmut Friedrich Jurke aus Rheda-Wiedenbr¨uck M¨unchen 2010 Benjamin Helmut Friedrich Jurke Nonperturbative Type IIB Model Building in the F-Theory Framework 1. Gutachter: PD Dr. Ralph Blumenhagen Max-Planck-Institut f¨urPhysik, M¨unchen, Germany 2. Gutachter: Prof. Dr. Ilka Brunner Ludwig-Maximilians Universit¨atM¨unchen, Germany Datum der m¨undlichen Pr¨ufung: 28. Februar 2011 to whom it may concern... (and in particular my mother) Zusammenfassung Diese Dissertation behandelt das Gebiet der nicht-st¨orungstheoretischen Stringtheo- rie, die allgemein als vielversprechendster Ansatz zu einer konsistenten Beschreibung der Quantengravitation angesehen wird. Die f¨unfbekannten zehn-dimensionalen perturba- tiven Stringtheorien sind durch zahlreiche Dualit¨atenmiteinander verkn¨upft,sodass eine zugrundeliegende nicht-perturbative elf-dimensionale Theorie, genannt M-Theorie, postuliert wird. Uber¨ deren fundamentale Objekte ist aufgrund diverser technischer Schwierigkeiten allerdings nur wenig bekannt. Zur Typ-IIB-Stringtheorie existiert auch noch ein alternativer nicht-perturbativer Zugang, die F-Theorie. Diese geometrisiert die SL(2; Z)-Selbstdualit¨atder IIB-Theorie in Form einer elliptischen Faserung ¨uber der Raumzeit. Dar¨uber hinaus sind auch h¨oherdimensionaleObjekte wie etwa 7-Branen als Singularit¨atenin der geometrischen Beschreibung enthalten. Dieser formal elegante Ansatz erfordert allerdings einen großen technischen Aufwand in der Konstruktion akzeptabler Kompaktifizierungsgeometrien, da sehr viele Aspekte zwangsl¨aufiggleichzeitig behandelt werden m¨ussen. Daf¨urist aber eine im Vergleich zur perturbativen Stringtheorie einfachere Erzeugung essen- tieller Bausteine f¨urvereinheitlichte Theorien (GUTs) m¨oglich, beispielsweise bestimm- te Yukawa-Kopplungen oder Spinor-Darstellung. Ziel der Untersuchungen ist es daher eine vereinheitlichte Theorie innerhalb der F-Theorie zu formulieren, welche gewisse ph¨anomenologische Grundbedingungen erf¨ullt. Im Rahmen dieser Arbeit werden zun¨achst E3-Bran-Instantonen der Typ-IIB-String- theorie {{also vier-dimensionale Objekte, die sich ausschließlich um die unsichtbaren Dimensionen der Raumzeit wickeln {{mit M5-Branen in der F-Theorie in Beziehung gesetzt. Diese Objekte sind von großer Bedeutung f¨urdie Erzeugung ben¨otigterYukawa- Kopplungen oder etwa die Stabilisierung diverser freier Parameter einer Theorie. Be- stimmte Eigenschaften der M5-Branen erlauben es dann eine neue Bedingung zu for- mulieren, wann E3-Branen zum Superpotential beitragen k¨onnen. Im Anschluss zu dieser Analyse werden verschiedene Kompaktifizierungsgeometrien konstruiert und ihre prinzipielle Tauglichkeit zur Beschreibung grundlegend realistischer vereinheitlichter Theorien gepr¨uft. Ein entscheidender Punkt ist dabei den Eichfluss auf den enthaltenen 7-Branen korrekt zu beschreiben. Uber¨ die Methode der spek- tralen Uberdeckungen¨ {{die zun¨achst noch weiterer Verfeinerungen bedarf {{l¨asstsich vii viii dadurch dann chirale Materie erzeugen und zugleich die vereinheitlichte Eichgruppe zum Standardmodell hin reduzieren. Letztlich gelingt es in dieser Arbeit ein konkretes, ver- einheitlichtes Modell mit der Eichgruppe SU(5) im Rahmen der F-Theorie zu konstru- ieren, welches eine akzeptable Ph¨anomenologieaufzeigt und zudem die beobachteten drei chiralen Materie-Generationen reproduziert. Summary This dissertation is concerned with the topic of non-perturbative string theory, which is generally considered to be the most promising approach to a consistent description of quantum gravity. The five known 10-dimensional perturbative string theories are all interconnected by numerous dualities, such that an underlying non-perturbative 11- dimensional theory, called M-theory, is postulated. Due to several technical obstacles, little is known about the fundamental objects in this theory. There exists an alternative non-perturbative description to type IIB string theory, namely F-theory. Here the SL(2; Z) self-duality of IIB theory is geometrized in the form of an elliptic fibration over the space-time. Moreover, higher-dimensional objects like 7-branes are included via singularities into the geometric picture. This formally elegant description, however, requires significant technical effort for the construction of suitable compactification geometries, as many different aspects necessarily have to be dealt with at the same time. On the other hand, the generation of essential GUT building blocks like certain Yukawa couplings or spinor representations is easier com- pared to perturbative string theory. The goal of this study is therefore to formulate a unified theory within the framework of F-theory, that satisfies basic phenomenological constraints. Within this thesis, at first E3-brane instantons in type IIB string theory {{4-dimen- sional objects that are entirely wrapped around the invisible dimensions of space-time{{ are matched with M5-branes in F-theory. Such objects are of great importance in the generation of critical Yukawa couplings or the stabilization of the free parameters of a theory. Certain properties of M5-branes then allow to derive a new criterion for E3-branes to contribute to the superpotential. In the aftermath of this analysis, several compactification geometries are constructed and checked for basic properties that are relevant for semi-realistic unified model build- ing. An important aspect is the proper handling of the gauge flux on the 7-branes. Via the spectral cover description{{which at first requires further refinements{{chiral mat- ter can be generated and the unified gauge group can be broken to the Standard Model. Ultimately, in this thesis an explicit unified model based on the gauge group SU(5) is constructed within the F-theory framework, such that an acceptable phenomenology and the observed three chiral matter generations are obtained. ix Acknowledgments My particular gratitude goes to my advisor Ralph Blumenhagen, who constantly{{in his very own iconic way {{encouraged me and let me partake on his numerous insights into string theory and the required mathematics to delve deeper into the subject. I could not have asked for a more stimulating atmosphere in our work group during the countless discussions and research debates. I am also indebted to Dieter L¨ustfor his constant support and the effort he puts into providing all members of the string theory group with extraordinary opportunities for travel and research. I am very grateful to my collaborators Andr´esCollinucci, Thomas W. Grimm and Timo Weigand, from whom I learned at an unsurpassed pace especially during the first half of my PhD time. It was truly an unique opportunity to work with such a group of gifted people. Likewise, I would also like to thank my later collaborators and (former) office mates Thorsten Rahn and Helmut Roschy for the many productive and humorous hours back in the \cave", which was recently joined by Andreas Deser. Our time in the office was and still is a constant source of many memorable moments. Thanks to Oliver Schlotterer for many discussions (and disputes) as well as the oc- casional shared bottle of wine, a statement which in the same spirit can be extended to my old friend and former office mate Florian K¨uhnel. For comments on the manuscript of my thesis I would in particular like to thank Florian K¨uhnel,Thorsten Rahn, Helmut Roschy and Oliver Schlotterer. Furthermore, I would like to thank{{in alphabetical order{{Martin Ammon, Volker Braun, Ilka Brunner, James Gray, Michael Haack, James Halverson, Daniel H¨artl, Johannes Held, Johanna Knapp, Philipp Kostka, Sebastian Moster, Erik Plauschinn, Felix Rennecke, Maximilian Schmidt-Sommerfeld and Stephan Stieberger for various discussions and the sharing of thoughts and ideas. Moreover, I am grateful to my undergraduate advisor Reinhart K¨ogerlerfor pointing me in the right direction. An important cornerstone for me have always been my longstanding friends from back home, namely Daniel Altemeier, Cathrin and Sebastian Becker, Michael Brock- amp, Markus and Martin Menze, Christian Mester and Johannes R¨ottig.I am looking forward to each opportunity to meet the old circle again. I would also like to mention Philipp Nordhus. Words seem inadequate for the level of gratitude I have for my mother, who despite xi xii difficult circumstances made my venture into academia possible in the first place and supported me through all those years {{thanks, mum! Likewise, I am deeply thankful to my family, in particular my grandmother Gertrud as well as my godfathers Helmut and Friedrich for countless instances of help and support, which also includes my aunt Hannelore. Thanks for all your understanding and encouragement. This work is part of the String Theory program at the Max Planck Institute of Physics (MPP) in collab- oration with the Arnold Sommerfeld Center (ASC) of the Ludwig-Maximilians-Universit¨atM¨unchen (LMU) and supported by the International Max Planck Research School (IMPRS) on Elementary Particle Physics (EPP) of the Max Planck Society in Munich, Germany. Publications During the preparation of this thesis the following five research projects {{in col- laboration with the indicated authors {{and their respective results were published in peer-reviewed journals, listed in chronological order of publication: Global F-theory GUT model building and related matters: Ralph Blumenhagen, Thomas W. Grimm, Benjamin