Variational Problems on Supermanifolds
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Institut f¨ur Mathematik Arbeitsgruppe Geometrie Variational Problems on Supermanifolds Dissertation zur Erlangung des akademischen Grades “doctor rerum naturalium” (Dr. rer. nat.) in der Wissenschaftsdisziplin Mathematik/Differentialgeometrie eingereicht an der Mathematisch-Naturwissenschaftlichen Fakult¨at der Universit¨at Potsdam von Florian Hanisch Potsdam, den 21.09.2011 This work is licensed under a Creative Commons License: Attribution - Noncommercial - Share Alike 3.0 Germany To view a copy of this license visit http://creativecommons.org/licenses/by-nc-sa/3.0/de/ Published online at the Institutional Repository of the University of Potsdam: URL http://opus.kobv.de/ubp/volltexte/2012/5975/ URN urn:nbn:de:kobv:517-opus-59757 http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59757 Abstract In this thesis, we discuss the formulation of variational problems on supermanifolds. Super- manifolds incorporate bosonic as well as fermionic degrees of freedom. Fermionic fields take values in the odd part of an appropriate Grassmann algebra and are thus showing an anti- commutative behaviour. However, a systematic treatment of these Grassmann parameters requires a description of spaces as functors, e.g. from the category of Grassmann algberas into the category of sets (or topological spaces, manifolds,. ). After an introduction to the general ideas of this approach, we use it to give a description of the resulting supermanifolds of fields/maps. We show that each map is uniquely characterized by a family of differential operators of appropriate order. Moreover, we demonstrate that each of this maps is uniquely characterized by its component fields, i.e. by the coefficients in a Taylor expansion w.r.t. the odd coordinates. In general, the component fields are only locally defined. We present a way how to circumvent this limitation. In fact, by enlarging the supermanifold in question, we show that it is possible to work with globally defined components. We eventually use this formalism to study variational problems. More precisely, we study a super version of the geodesic and a generalization of harmonic maps to supermanifolds. Equations of motion are derived from an energy functional and we show how to decompose them into components. Finally, in special cases, we can prove the existence of critical points by reducing the problem to equations from ordinary geometric analysis. After solving these component equations, it is possible to show that their solutions give rise to critical points in the functor spaces of fields. In dieser Dissertation wird die Formulierung von Variationsproblemen auf Supermannig- faltigkeiten diskutiert. Supermannigfaltigkeiten enthalten sowohl bosonische als auch fermion- ische Freiheitsgrade. Fermionische Felder nehmen Werte im ungeraden Teil einer Grassman- nalgebra an, sie antikommutieren deshalb untereinander. Eine systematische Behandlung dieser Grassmann-Parameter erfordert jedoch die Beschreibung von R¨aumen durch Funk- toren, z.B. von der Kategorie der Grassmannalgebren in diejenige der Mengen (der topologis- chen R¨aume, Mannigfaltigkeiten, ...). Nach einer Einf¨uhrung in das allgemeine Konzept dieses Zugangs verwenden wir es um eine Beschreibung der resultierenden Supermannigfaltigkeit der Felder bzw. Abbildungen anzugeben. Wir zeigen, dass jede Abbildung eindeutig durch eine Familie von Differentialoperatoren geeigneter Ordnung charakterisiert wird. Dar¨uber hinaus beweisen wir, dass jede solche Abbildung eineindeutig durch ihre Komponentenfelder, d.h. durch die Koeffizienten einer Taylorentwickelung bzgl. ungerader Koordinaten bestimmt ist. Im Allgemeinen sind Komponentenfelder nur lokal definiert. Wir stellen einen Weg vor, der diese Einschr¨ankung umgeht: Durch das Vergr¨oßern der betreffenden Supermannigfaltigkeit ist es immer m¨oglich, mit globalen Koordinaten zu arbeiten. Schließlich wenden wir diesen Formalismus an, um Variationsprobleme zu untersuchen, genauer betrachten wir eine super- Version der Geod¨ate und eine Verallgemeinerung von harmonischen Abbildungen auf Super- mannigfaltigkeiten. Bewegungsgleichungen werden von Energiefunktionalen abgeleitet und wir zeigen, wie sie sich in Komponenten zerlegen lassen. Schließlich kann in Spezialf¨allen die Existenz von kritischen Punkten gezeigt werden, indem das Problem auf Gleichungen der gew¨ohnlichen geometrischen Analysis reduziert wird. Es kann dann gezeigt werden, dass die L¨osungen dieser Gleichungen sich zu kritischen Punkten im betreffenden Funktor-Raum der Felder zusammensetzen. Acknowledgments I would like to thank my advisor Christian B¨ar for suggesting this interesting topic to me, for his continued support and for his patience even though the work took far more time than we originally thought. Special thanks also to the members of the Potsdam geometry group, especially to Frank Pf¨affle, Christoph Stephan, Christian Becker, Volker Branding and my former roommate Dennis Koh for various enlightening discussions and feedback. It was (and still is) a great time. I am grateful to Gregor Weingart for very helpful email conversations concerning the theory of jet bundles and differential operators, as well as to Christoph Sachse for explaining parts of the Molotkov-Sachse approach to supergeometry to me at points where I got stuck. Moreover, I want to thank Vicente Cort´es, Sylvie Paycha and Thomas Schick, who were referees or member of the board of examination, for making very useful suggestions for im- provement of this text. Finally, I would like to thank the “IMPRS for Geometric Analysis, Gravitation and String Theory” at the Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut) and the Sonderforschungsbereich Raum-Zeit-Materie (SFB 647) for financial support during the time I worked on this thesis. Statement of Originality This thesis contains no material which has been accepted for the award of any other de- gree or diploma at any other university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my the- sis, when deposited in the University Library, being made available for loan and photocopying. Potsdam, September 21, 2011 Florian Hanisch 3 4 Contents 1 Introduction i 2 Elements of Superalgebra 1 2.1 Supermodulesandalgebras . ... 1 2.2 Freemodulesandlinearalgebra. ..... 5 2.3 Extension and restriction of scalars . ........ 8 3 Supermanifolds 10 3.1 Elementary structure of supermanifolds . ........ 10 3.2 Morphismsbetweensupermanifolds. ...... 14 3.3 Supervectorbundles.............................. 16 3.4 Tangent sheaf and tangent spaces . ..... 19 3.5 Metrics, frames and integration . ...... 22 3.6 Connections..................................... 27 4 Categorical aspects of supergeometry 31 4.1 The categorical approach to algebra and geometry . ......... 32 4.2 The Molotkov-Sachse approach to supergeometry . ......... 37 4.3 Thespaceofmorphisms ............................. 42 5 The fine structure of the space of morphisms 45 5.1 Classical formalism of jets and differential operators . ............ 45 5.2 Algebraic theory of super differential operators and -jets............ 46 5.3 Thestructureofhigherpoints. ..... 55 5.4 Application to the Batchelor picture . ....... 63 5.5 Componentformalism .............................. 66 5.6 Components on general supermanifolds . ....... 72 6 Variational Problems on Supermanifolds 75 6.1 Functionals and critical points . ....... 75 6.2 Harmonic maps and elliptic theory . ..... 79 6.3 Supergeodesics .................................. 82 6.4 Superharmonicmaps............................... 90 A Appendix: Elements of category theory 99 5 6 i 1 Introduction The research in this thesis addresses the problem of the formulation of geometric variational problems on supermanifolds and the development of technics to find and describe its critical points. The problems in question generalize the notions of geodesics and harmonic maps to supergeometry. The development of supergeometry took its origin from concepts of theoretical physics: In quantum physics, there are two different kinds of particles, bosons and fermions. Whereas bosons behave according to the Bose-Einstein statistic and are allowed to occupy the same quantum state, fermions are not able to do so, they behave according to the Fermi-Dirac statistic. As a consequence, bosonic fields are described by operators Φ1, Φ2 which commute among each other up to order , whereas fermionic fields Ψ1, Ψ2 lead to anticommutator relations: [Φ1, Φ2]= O() {Ψ1, Ψ2} = O() In a classical limit → 0, this leads to a theory of commuting and anticommuting classi- cal fields. Supergeometry is the attempt to incorporate commuting and anticommuting (i.e. bosonic and fermionic) objects in a unified framework in order to obtain a geometric approach to physics which treats both types of fields on the same footing. There are several reasons to look for such a framework. In quantum physics, the path integral formalism for fermions requires a classical field theory which includes anticommuting quantities. In general, it is then desirable to be able to work on an arbitrary curved space. Moreover, the concept of superspaces allows for the construction of theories which are manifestly supersymmetric in a nice geometric fashion (although not all supersymmetric theories arise in this way). Finally, the subject is also interesting from a mathematical point of view. It leads to interesting gen- eralizations