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BPS State Counting Using Wall-Crossing, Holomorphic Anomalies and Modularity

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BPS State Counting Using Wall-Crossing, Holomorphic Anomalies and Modularity BPS state counting using wall-crossing, holomorphic anomalies and modularity Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn von Thomas Wotschke aus Iserlohn Bonn 2013 Dieser Forschungsbericht wurde als Dissertation von der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Bonn angenommen und ist auf dem Hochschulschriftenserver der ULB Bonn http://hss.ulb.uni-bonn.de/diss_online elektronisch publiziert. 1. Gutachter: Prof. Dr. Albrecht Klemm 2. Gutachter: Prof. Dr. Hans-Peter Nilles Tag der Promotion: 11.07.2013 Erscheinungsjahr: 2013 Abstract In this thesis we examine the counting of BPS states using wall-crossing, holomorphic anomalies and modularity. We count BPS states that arise in two setups: multiple M5-branes wrapping P × T 2, where P denotes a divisor inside a Calabi-Yau threefold and topological string theory on elliptic Calabi-Yau threefolds. The first setup has a dual description as type IIA string theory via a D4-D2-D0 brane system. Furthermore it leads to two descriptions depending on the size of P and T 2 relative to each other. For the case of a small divisor P this setup is described by the (0; 4) Maldacena-Strominger-Witten conformal field theory of a black hole in M-theory and for the case of small T 2 the setup can by described by N = 4 topological Yang-Mills theory on P. The BPS states are counted by the modified elliptic genus, which can be decomposed into a vector-valued modular form that provides the generating function for the BPS invariants and a Siegel-Narain theta function. In the first part we discuss the holomorphic anomaly of + the modified elliptic genus for the case of two M5-branes and divisors with b2 (P) = 1. Due to the wall-crossing effect the change in the generating function is captured by an indefinite theta function, which is a mock modular form. We use the Kontsevich-Soibelman wall-crossing formula to determine the jumps in the modified elliptic genus. Using the regularisation procedure for mock modular forms of Zwegers, modularity can be restored at the cost of holomorphicity. We show that the non-holomorphic completion is due to bound states of single M5-branes. At the attractor point in the moduli space we prove the holomorphic anomaly equation, which is compatible with the holomorphic anomaly equations observed in the context of N = 4 Yang-Mills theory on P2 and E-strings on a del Pezzo surface. We 2 calculate the generating functions of BPS invariants for the divisors P ; F0; F1 and the del Pezzo surface 1 dP8 and dP9 2 K3 . In the second part we study the quantum geometry of elliptic Calabi-Yau threefolds and examine topological string theory on these spaces. We find a holomorphic anomaly equation for the topological amplitudes with respect to the base that is recursive in the genus and in the base class. The topological amplitudes with respect to the base can be expressed in terms of quasi-modular forms, which resembles the holomorphic anomaly. In particular this generalises a holomorphic anomaly discovered for the 1 2 K3. For genus zero and base F1 we prove the holomorphic anomaly by using mirror symmetry and we motivate our holomorphic anomaly equation by establishing the connection to the holomorphic anomaly equations of Bershadsky-Cecotti-Ooguri-Vafa. Using T-duality allows us to relate this anomaly to the anomaly for the case of D4-D2-D0 BPS state counting. We calculate the generating function of BPS 1 invariants of 2 K3 for higher rank branes in topological string theory and by using algebraic-geometric techniques that were developed in the context of stability of sheaves. iii Danksagungen Walk on walk on with hope in your heart And you’ll never walk alone Zunächst möchte ich meinem Doktorvater Prof. Dr. Albrecht Klemm danken, dass er mir die Möglich- keit gegeben hat in seiner Arbeitsgruppe und dem spannenden Feld der Stringtheorie zu arbeiten. Die Möglichkeit mathematische Erkenntnisse für physikalische Ergebnisse und umgekehrt zu verwenden, haben mich seit meinem Studium fasziniert und insbesondere mein Doktorvater hat meine Faszination in zahlreichen inspirierende Diskussionen und durch seine permanente Unterstützung über die Jahre meiner Doktorarbeit weiter vertieft. Neben den stets guten Möglichkeiten in Bonn, ist auch seine Unter- stützung bei der Teilnahme an verschiedenen Schulen und Konferenzen zu nennen. Prof. Dr. Hans-Peter Nilles danke ich für die Übernahme der Zweitkorrektur. Ohne die zahlreichen Diskussionen mit meinen Kooperationspartnern wäre diese Arbeit nicht mög- lich gewesen. Ich danke Ihnen, dass sie ihre Einsichten mit mir geteilt haben und immer wieder neue Anregungen und Herausforderungen geliefert haben. Somit haben sie wesentlich zu der produktiven At- mosphäre während meiner Doktorarbeit beigetragen. Ich danke Dr. Murad Alim, Dr. Babak Haghighat, Dr. Michael Hecht, Dr. Jan Manschot und Dr. Marco Rauch. Dr. Murad Alim und Dr. Michael Hecht sei besonders für zahlreiche hilfreiche Diskussionen und Aufenthalte in München gedankt. Dr. Marco Rauch danke ich für eine herausragende gemeinsame Zeit während des Studiums, im gemeinsamen Büro während der Promotion und darüber hinaus. Mein Dank gilt ebenfalls den aktuellen und ehemaligen Mitgliedern der Arbeitsgruppe von Prof. Klemm, die stets für Fragen und Anregungen bereit waren. Während meiner Zeit im Büro 104 und im BCTP konnte ich stets auf ein produktives und konstruktives Arbeitsumfeld bauen. Ich danke Dr. Thomas Grimm, Dr. Tae-Won Ha, Dr. Hans Jockers, Dr. Denis Klevers, Dr. Daniel Lopes, Dr. Ma- soud Soroush, Dr. Piotr Sulkowski, Navaneeth Gaddam, Jie Gu, Maximilian Poretschkin, Jonas Reuter und Marc Schiereck. Insbesondere danke ich Maximilian Poretschkin für die gemeinsame Zeit im Büro 2.010 im BCTP, viele anregende Diskussionen und sein stets kollegiales, verlässliches und freundschaft- liches Verhalten. Weiterhin gebührt ihm Dank für das Korrekturlesen meiner Arbeit und viele hilfreiche, konstruktive Vorschläge. Ebenfalls Dr. Maxim Mai sei für seine hilfreichen Anregungen gedankt. Ich danke der Deutsche Telekom Stiftung für die Förderung während meiner Doktorarbeit. Die Stipendiatentreffen, Seminare und weitere Veranstaltungen haben meine Doktorarbeit zu einer einzig- artigen Zeit werden lassen. Ich danke Frau Christiane Frense-Heck für die Betreuung und Prof. Dr. Johann-Dietrich Wörner für seine engagierte Tätigkeit als Mentor. Ebenso danke ich der Bonn-Cologne Graduate School of Physics and Astronomy für die Förderung meiner Doktorarbeit. Meine Zeit während der Promotion wurde auch von vielen Freunden außerhalb der Forschung ge- prägt. Ich danke Dr. Maxim Mai und Konstantin Ottnad für viele Kaffeepausen und die gemeinsame Zeit seit dem Studium. Ich danke Patrick Bavink für viele gute Ratschläge und die gemeinsam erlebten Momente an der Platte und außerhalb. Weiterhin danke ich Christian Burgdorf für zahlreiche Plakate und gemeinsam geleitete Kurse. Meinen Eltern Magdalene und Dieter Wotschke sowie meiner Freundin Christina Cappenberg danke ich vom ganzen Herzen für ihre Unterstützung und Liebe – danke für alles. iv Contents 1 Introduction and Motivation 1 2 Counting BPS states in string theory 13 2.1 BPS states and the supersymmetry algebra . 14 2.2 BPS states and gauge theories . 16 2.3 BPS states and black holes . 19 2.4 Stability of BPS states . 22 2.4.1 Walls of marginal stability and BPS indices . 23 2.4.2 The Kontsevich-Soibelmann wall-crossing formula . 24 2.4.3 Stability conditions . 26 2.4.4 Verification of the Kontsevich-Soibelman wall-crossing formula from physics . 29 2.5 An example: SU(2) Seiberg-Witten theory . 31 2.5.1 The setup and the solution . 31 2.5.2 Quiver description of SU(2) Seiberg-Witten theory . 34 2.6 Modular forms . 35 2.6.1 Elliptic and Siegel modular forms . 35 2.6.2 Mock modularity . 47 2.7 Topological string theory . 57 2.7.1 The chiral ring structure of topological string theory . 57 2.7.2 Deformations . 59 2.7.3 Non-linear sigma model realisation . 59 2.7.4 Topological field theories . 60 2.7.5 Mirror symmetry . 62 2.7.6 Holomorphic anomaly equations . 66 2.7.7 Stable pairs and enumerative invariants . 69 2.8 Counting D4-D2-D0 BPS states . 70 2.8.1 The Maldacena-Strominger-Witten conformal field theory . 71 2.8.2 Counting BPS states a la Gaiotto-Strominger-Yin and using split attractor flows 77 2.8.3 N = 4 Super Yang-Mills, E-strings and bound-states . 79 2.8.4 Counting BPS states via wall-crossing and sheaves . 82 3 Geometries and their construction 85 3.1 Calabi-Yau manifolds as hypersurfaces and complete intersections in weighted project- ive space . 85 3.2 Complex surfaces . 90 v Contents 3.3 Classical geometry of elliptically fibred Calabi-Yau spaces . 90 3.3.1 The classical geometrical data of elliptic fibrations . 92 3.3.2 Realisations in toric ambient spaces . 94 4 Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes 99 4.1 Generating functions from wall-crossing . 99 4.2 Wall-crossing and mock modularity . 100 4.2.1 D4-D2-D0 wall-crossing . 101 4.2.2 Relation of Kontsevich-Soibelman to Göttsche’s wall-crossing formula . 102 4.2.3 Holomorphic anomaly at rank two . 104 4.3 Applications and extensions . 106 4.3.1 Blow-up formulae and vanishing chambers . 106 + 4.3.2 Applications to surfaces with b2 = 1....................... 107 4.3.3 Extensions to higher rank and speculations . 110 5 Quantum geometry of elliptic fibrations 113 5.1 Quantum cohomology, modularity and the anomaly equations . 113 5.2 The B-model approach to elliptically fibred Calabi-Yau spaces . 116 5.3 Modular subgroup of monodromy group . 118 5.4 Derivation of the holomorphic anomaly equation . 122 5.4.1 The elliptic fibration over F1 ........................... 122 5.4.2 Derivation from BCOV .
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