DISS. ETH NO. 17977 TOPOLOGICAL STRINGS, MATRIX MODELS, AND NONPERTURBATIVE EFFECTS A dissertation submitted to ETH ZURICH for the degree of Doctor of Sciences presented by MARLENE JULIA WEISS Dipl. Phys., ETH Zurich Date of birth 18.08.1980 citizen of Germany accepted on the recommendation of J¨urg Fr¨ohlich Marcos Mari˜no 2008 Acknowledgements First of all, I would like to thank my supervisor Marcos Mari˜no for his continuous support and guidance throughout all stages of this work, for teaching me countless things on string theory, for many animated discussions, and also for organizing great graduate students’ seminars at CERN. I am similarly indebted to my second advisor J¨urg Fr¨ohlich for all his support, many insightful comments, and for tirelessly insisting on clear, rigorous statements. One could hardly wish for two more committed and inspiring thesis advisors. I also wish to thank Thomas Grimm, Ricardo Schiappa and especially Albrecht Klemm for pleasant collaborations in which I had the honour to participate. Furthermore, I take the op- portunity to thank my teachers in Zurich, Paris and Lausanne, in particular Dietmar Salamon, Misha Shaposhnikov, Michael Spira, Gian-Michele Graf, Jean-Pierre Rivoal and Eduard Zehn- der. They have very much shaped my view on physics and mathematics. All the members of the ITP at ETH Zurich, as well as my students, have contributed to making my weekly visits very delightful, and I thank them for creating such a friendly atmosphere. It has been a great pleasure to share office with James Bedford, Johanna Knapp and Are Raklev. Johanna gets extra thanks for countless mostly non-physics discussions, hilarious sar- castic comments on many occasions, and for competent Linux support. Thanks for the latter also to the worldwide Ubuntu community. I also thank the adorable staff at CERN, especially Nanie Perrin and Suzy Vascotto. I am grateful to many people I met at CERN, in Zurich and at conferences, among oth- ers Murad Alim, Luis Alvarez-Gaum´e,´ Fran¸cois Arl´eo, Matthias Blau, Vincent Bouchard, Ilka Brunner, Frederik Denef, Ron Donagi, Alexander Flossdorf, Fran¸cois Gelis, Gero von Gersdorff, Alessandra Gnecchi, Chris Herzog, Gabi Honecker, Max Kreuzer, Wolfgang Lerche, Tristan Maillard, Sara Pasquetti, Emanuel Scheidegger, Cornelius Schmidt-Colinet, Joan Sim´on, Stefan Stieberger, Andy Strominger, David Tong and Aleksi Vuorinen, for very enjoyable encounters, and to some of the above also for interesting discussions on physics. I am especially indebted to James Bedford for help with the manuscript and to Emanuel Scheidegger for sharing his knowl- edge on Gromov-Witten theory and related topics. Mauro Papinutto’s and Kevin Schnelli’s skiing, climbing and italian lessons and friendship are gratefully acknowledged, and I thank Babis Anastasiou for providing greek food, fun company, and putting up with my grammar obsession. I thank Yannis, Henri, Th´er`ese Burnier and family for all their support and advice, and for introducing me to windsurfing. Without my friends Stephan Debrunner, Elisabeth Furtw¨angler, Nora Graser, Andrea Har- bach, Barbara Loisch, David Noth, Stefan Schwarz and Daniel Weiller, I would not be the person I am today, and I thank them for everything. Last but by no means least, I am grateful to my parents and sisters, who were always there. Kurzfassung Diese Dissertation befasst sich mit topologischer Stringtheorie und ihrer Beziehung zu Matrix- modellen und Instantoneffekten. Topologische Stringtheorie ist ein Sektor der Stringtheorie. Sie wird konstruiert, indem in einem = (2, 2) supersymmetrischen Sigmamodell auf dem N Worldsheet durch den sogenannten topologischen Twist die Lorentzstr¨ome neu definiert wer- den, bevor es an zweidimensionale Gravitation gekoppelt wird. Das Ergebnis ist eine String- theorie mit metrikunabh¨angigen Observablen. Da der topologische Twist auf zwei konsistente Weisen definiert werden kann, gibt es zwei topologische Stringtheorien, das A- und das B- Modell, die mittels Mirrorsymmetrie zueinander ¨aquivalent sind. Ihre Amplituden entsprechen effektiven Ladungen in der Typ IIA beziehungsweise IIB Theorie, kompaktifiziert auf der Calabi- Yau-Mannigfaltigkeit, in die das Sigmamodell das Worldsheet abbildet. Vom mathematischen Standpunkt aus beschreiben sie die Gromov-Witten-Invarianten des betreffenden Calabi-Yaus. Das B-Modell zeigt die sogenannte holomorphe Anomalie, die sich durch rekursive Differen- tialgleichungen ausdr¨ucken l¨asst, die die B-Modell-Amplituden erf¨ullen. Die durch diese Gle- ichungen beschriebene Nicht-Holomorphizit¨at der Amplituden steht in enger Beziehung zu ihrer Modularit¨at. Ein Resultat dieser Arbeit ist eine Methode, die holomorphen Anomaliegleichun- gen effizient zu integrieren, wobei die modulare Struktur der Amplituden genutzt wird. Ein weiterer Aspekt der topologischen Amplituden ist ihre Bedeutung in Verbindung mit der Dualit¨at zwischen heterotischer und Typ II-Stringtheorie. Wir berechnen die topologischen Amplituden auf der heterotischen Seite f¨ur eine grosse Klasse von vorgeschlagenen Paaren von heterotischen und Typ II-Theorien. In einem Teil der F¨alle sind die Gromov-Witten-Invarianten der betreffenden Calabi-Yau-Mannigfaltigkeit bekannt und stimmen mit unseren Ergebnissen ¨uberein, was einen weiteren pr¨azisen Test der Dualit¨at darstellt. Gegenstand des zweiten Teils dieser Arbeit ist die Verbindung zwischen topologischer String- Theorie, Matrixmodellen und Instantoneffekten. Es wird seit einigen Jahren vermutet, dass das topologische B-Modell in vielen F¨allen ¨aquivalent zu einem Matrixmodell ist. Andererseits besteht eine klassische Beziehung zwischen dem asymptotischen Verhalten der St¨orungsreihe und der Zustandssumme der Instanton¨uberg¨ange. Wir zeigen, dass diese Beziehung auch f¨ur Matrix- modelle gilt. Mittels der Dualit¨at zwischen Matrixmodellen und topologischen Strings k¨onnen wir aus der Instanton-Zustandssumme im Matrixmodell auch Vorhersagen f¨ur das asymptotische Verhalten topologischer Stringtheorien machen, unter Umgehung der zugeh¨origen Instantonam- plitude innerhalb der Stringtheorie. Wir testen unsere Vorhersagen f¨ur eine Reihe von Matrix- modellen und topologischen Stringtheorien und finden in allen F¨allen hervorragende Uberein-¨ stimmung. Die Matrix-Formulierung des B-Modells liefert so eine m¨ogliche nicht-perturbative Erweiterung, die mit perturbativen Methoden ¨uberpr¨ufbar ist. Abschliessend zeigen wir, dass L¨osungen von Matrixmodellen, die die Eigenwerte auf Gebiete um mehrere Extrema des Potentials verteilen, sogenannte Multi-Cut-Modelle, sich mithilfe eines Multi-Instanton-Formalismus aus den Single-Cut-Modellen heraus beschreiben lassen. Abstract This thesis is about topological string theory and its relation with matrix models and instanton effects. Topological string theory is a sector of string theory. It is constructed by redefining the Lorentz currents on the worldsheet of an = (2, 2) supersymmetric Sigma model by means of N a so-called topological twist, before coupling it to two-dimensional gravitation. The result is a string theory with metric-independent observables. As the topological twist can be defined in two consistent ways, there are two topological string theories, the A model and the B model, equivalent to one another by mirror symmetry. Their amplitudes correspond to effective charges in the type IIA and IIB theories, respectively, compactified on the Calabi-Yau manifold which is the target of the Sigma model. In mathematical terms, the topological string describes the Gromov-Witten invariants of that Calabi-Yau manifold. The B-model exhibits a so-called holomorphic anomaly. This can be expressed in recursively defined differential equations which are satisfied by the amplitudes. The deviation of the amplitudes from being holomorphic, as described by the differential equations, is closely related to their modularity. A key result of this thesis is a method for integrating the anomaly equations efficiently by exploiting the modular structure of the amplitudes. Another important aspect of the topological amplitudes is their role in the duality between heterotic and type II string theory. On the heterotic side, we calculate the topological amplitudes for a large class of conjectured pairs of heterotic and type II theories. In some of these examples, the Gromov-Witten invariants of the appropriate Calabi-Yau manifolds are known and agree with our calculations. This provides further data in favor of heterotic-type II duality. Furthermore, we explore the connections between topological string theory, matrix models and instanton effects. In recent years it has been conjectured that the topological B model is equivalent to a matrix model. On the other hand, there is a classical relationship between the asymptotic behavior of the perturbation series and the instanton corrections to the partition sum. We use this relationship to make predictions concerning the asymptotic behavior of the perturbation series based on the partition sum. By means of the duality between matrix models and topological strings, we are then also able to make predictions for the asymptotic behavior of topological string theories, avoiding the corresponding instanton amplitude within string theory. These predictions are tested for some matrix models and topological string theories. In all cases tested we find precise agreement. In this way, the matrix formulation of the B model leads to a possible nonperturbative extension which can be tested with perturbative methods. Finally, we show that solutions of matrix