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On Supertwistor Geometry and Integrability in Super Gauge Theory Martin Wolf On Supertwistor Geometry And Integrability In Super Gauge Theory Institut f¨ur Theoretische Physik Fakult¨at f¨ur Mathematik und Physik Leibniz Universit¨at Hannover Copyright c 2006 by Martin Wolf. All rights reserved. ITP–UH–18/06 Institut f¨ur Theoretische Physik Gottfried Wilhelm Leibniz Universit¨at Hannover Appelstraße 2, 30167 Hannover, Germany [email protected] To My Family On Supertwistor Geometry And Integrability In Super Gauge Theory Von der Fakultat¨ fur¨ Mathematik und Physik der Gottfried Wilhelm Leibniz Universitat¨ Hannover Zur Erlangung des Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von Dipl.-Phys. Martin Wolf geboren am 19. Februar 1979 in Zwickau (Sachsen) Hannover 2006 Tag der Promotion: 21. Juli 2006 Betreuer: Prof. Dr. Olaf Lechtenfeld und Dr. Alexander D. Popov Referent: Prof. Dr. Olaf Lechtenfeld Korreferent: Prof. Dr. Holger Frahm eÒÝÊ dÙÖiÊ aÖaÊ ÛÓÖia; ÔedÓ ÛeÐÐÓÊ a ÛiÒÓº iÛ ÊaÖÚi haÊ eËaªØ; CeÐebÖi«bÓÖ Ó eÖejiÓÊ ØeÌaªØ i ÌiA hiʺ Acknowledgments This work could not have been carried out without the support of many people. It is therefore a pleasure to acknowledge their help. First of all, I would like to thank Olaf Lechtenfeld for allowing me to be a member of his research group and for giving me the opportunity to do my Ph.D. studies here in Hannover. I am grateful to him for his help with advice and expertise, and for numerous helpful discussions. I am also deeply indebted to Alexander Popov who was always a brilliant advisor. I would like to express my gratitude towards him for patiently teaching me a lot of mathematics and physics, for our countless conversations on various topics which crossed the way, and for always helping with advice on any sort of problem. He has always encouraged my interest in mathematical physics since the time I came to Hannover. Furthermore, I would like to thank Alexander Popov and Christian S¨amann for very enjoyable collaborations. I am also grateful to Holger Frahm for sparing his invaluable time reviewing the manuscript of my thesis, and to Elmar Schrohe for kindly agreeing to be the head of the examination board. In addition, I am thankful to the following people for their help and for numerous discussions: Chong-Sun Chu, Louise Dolan, Norbert Dragon, Klaus Hulek, Alexander Kling, Niall J. MacKay, Stefan Petersen, Ronen Plesser, Elmar Schrohe, Christian S¨amann, Richard Szabo, Sebastian Uhlmann, Kirsten Vogeler and Robert Wimmer. Furthermore, I would like to thank a number of wunderful people for having provided such an enjoyable time here in Hannover. Therefore, many thanks to all of them: Hendrik Adorf, Alexandra De Castro, Henning Fehrmann, Carsten Grabow, Matthias Ihl, Alexander Kling, Michael Klawunn, Michael K¨ohn, Marco Krohn, Carsten Luckmann, Andreas Osterloh, Klaus Osterloh, Guillaume Palacios, Leonardo Quevedo, Christian S¨amann, Sebastian Uhlmann and Carsten von Zobeltitz. In particular, I have to thank Stefan Petersen, Kirsten Vogeler and Robert Wimmer for a wonderful friendship. They made it an unforgetable time here in Hannover. In addition, our countless discussions, their comments, suggestions and ideas were always a great source of inspiration to me. Furthermore, I am also very grateful to Ingmar Glauche and Cindy Rockstroh for always helping and supporting me. Thank you! I want to express my deepest gratitude towards those this thesis has been dedicated to, i.e., to my parents Bettina and Hans-Peter Wolf and to my brother Henrik Wolf. I would not have been in the position to carry out this work at all without their support and encouragement. Finally, I would like to thank the Deutsche Forschungsgemeinschaft (DFG) for continued support of my graduate work through the Graduiertenkolleg No. 282 entitled Quantenfeldtheo- retische Methoden in der Teilchenphysik, Gravitation, Statistischen Physik und Quantenoptik. Zusammenfassung In der vorliegenden Arbeit werden im Rahmen des Twistorzugangs verschiedene Aspekte der Integrabilit¨at supersymmetrischer Eichtheorien betrachtet. Nach einer kurzen Darstellung der Grundlagen der Twistortheorie im ersten Kapitel, untersuchen wir zun¨achst selbstduale supersymmetrische Yang-Mills-Theorie (SYM). Insbesondere wird deren Twistorformulierung erl¨autert und ausgehend von dieser, werden weitere selbstduale Modelle vorgestellt. Jene folgen durch geeignete Reduktion aus selbstdualer SYM-Theorie und k¨onnen zum Teil im Rahmen topologischer Feldtheorien verstanden werden. F¨ur deren Twistorformulierung schlagen wir be- stimmte gewichtete projektive Superr¨aume als Twistorr¨aume vor. Interessanterweise sind diese Calabi-Yau-Supermannigfaltigkeiten, so daß es ebenfalls m¨oglich ist, geeignete Wirkungsprinzi- pien zu formulieren. Im dritten Kapitel dieser Arbeit befassen wir uns mit der Twistorformu- lierung eines supersymmetrischen Bogomolny-Modells in drei Raumzeitdimensionen. Die nicht- supersymmetrische Variante dieses Modells beschreibt statische Yang-Mills-Higgs-Monopole im Prasad-Sommerfield-Limes. Insbesondere wird eine supersymmetrische Erweiterung des Minitwistorraumes betrachtet und, folgend aus einem der Grundprinzipien der Twistortheorie, betten wir diesen in eine konkrete Doppelfaserung ein. Diese Methode hat den Vorteil der M¨oglichkeit der Formulierung eines Chern-Simons ¨ahnlichen Wirkungsfunktionals, welches fast holomorphe Vektorb¨undel ¨uber dem entsprechenden Korrespondenzraum beschreibt. Letzterer kann mit einer Cauchy-Riemann-Struktur ausgestattet werden. Weiterhin formulieren wir holomorphe BF-Theorie auf dem Minisupertwistorraum und beweisen, daß die Modulr¨aume dieser drei Theorien bijektiv zueinander sind. Im Anschluß werden bestimmte Deformationen der komplexen Struktur des Minisupertwistorraumes betrachtet und ein daraus resultierendes supersymmetrisches Bogomolny-Modell mit massiven Feldern konstruiert. Im vierten Kapitel wird dann die Twistorformulierung von SYM-Theorien vorgestellt. Im letzten Kapitel vertiefen wir die Untersuchung selbstdualer SYM-Theorien bez¨uglich ihrer (klassischen) Integrabilit¨at. Insbesondere wird die Twistorkonstruktion unendlich dimensionaler Algebren nichtlokaler Sym- metrien behandelt. Diese sind zum einen affine Erweiterungen globaler und zum anderen affine Erweiterungen superkonformer Symmetrien. Im weiteren betrachten wir die Konstruktion von self-dualer SYM-Hierarchien. Jene beschreiben unendlich viele Fl¨usse auf den entsprechenden L¨osungsr¨aumen, wobei die niedrigsten Raumzeittranslationen darstellen. Dies bedeutet, daß eine gegebene L¨osung zu den Feldgleichungen in eine unendlich dimensionale Familie neuer L¨osungen eingebettet werden kann. Weiterhin werden unendlich viele erhaltene nichtlokale Str¨ome konstruiert. Schlagworte: Supertwistorgeometrie, Supersymmetrische Eichtheorien, Integrabilit¨at Abstract In this thesis, we report on different aspects of integrability in supersymmetric gauge theories. Our main tool of investigation is supertwistor geometry. In the first chapter, we briefly review the basics of twistor geometry. Afterwards, we discuss self-dual super Yang-Mills (SYM) theory and some of its relatives. In particular, a detailed twistor description of self-dual SYM theory is presented. Furthermore, we introduce certain self-dual models which are, in fact, obtainable from self-dual SYM theory by a suitable reduction. Some of them can be interpreted within the context of topological field theories. To provide a twistor description of these models, we pro- pose weighted projective superspaces as twistor spaces. These spaces turn out to be Calabi-Yau supermanifolds. Therefore, it is possible to write down appropriate action principles, as well. In chapter three, we then deal with the twistor formulation of a certain supersymmetric Bogomolny model in three space-time dimensions. The nonsupersymmetric version of this model describes static Yang-Mills-Higgs monopoles in the Prasad-Sommerfield limit. In particular, we consider a supersymmetric extension of mini-twistor space. This space is in turn a part of a certain double fibration. It is then possible to formulate a Chern-Simons type theory on the correspondence space of this fibration. As we explain, this theory describes partially holomorphic vector bundles. It should be noticed that the correspondence space can be equipped with a Cauchy-Riemann structure. Moreover, we formulate holomorphic BF theory on mini-supertwistor space. We then prove that the moduli spaces of all three theories are bijective. In addition, complex struc- ture deformations on mini-supertwistor space are investigated eventually resulting in a twistor correspondence involving a supersymmetric Bogomolny model with massive fields. In chapter four, we review the twistor formulation of non-self-dual SYM theories. The remaining chapter is devoted to a more detailed investigation of (classical) integrability of self-dual SYM theories. In particular, we explain the twistor construction of infinite-dimensional algebras of hidden sym- metries. Our discussion is exemplified by deriving affine extensions of internal and space-time symmetries. Furthermore, we construct self-dual SYM hierarchies and their truncated versions. These hierarchies describe an infinite number of flows on the respective solution space. The low- est level flows are space-time translations. The existence of such hierarchies allows us to embed a given solution to the equations of motion of self-dual SYM theory into an infinite-parameter family of new solutions. The dependence of the self-dual SYM fields on the additional
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