TY - THES A1 - Hanisch, Florian T1 - Variational problems on supermanifolds T1 - Variationsprobleme auf Supermannigfaltigkeiten N2 - In this thesis, we discuss the formulation of variational problems on supermanifolds. Supermanifolds 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 anticommutative 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. N2 - In dieser Dissertation wird die Formulierung von Variationsproblemen auf Supermannigfaltigkeiten diskutiert. Supermannigfaltigkeiten enthalten sowohl bosonische als auch fermionische Freiheitsgrade. Fermionische Felder nehmen Werte im ungeraden Teil einer Grassmannalgebra an, sie antikommutieren deshalb untereinander. Eine systematische Behandlung dieser Grassmann-Parameter erfordert jedoch die Beschreibung von Räumen durch Funktoren, z.B. von der Kategorie der Grassmannalgebren in diejenige der Mengen (der topologischen Räume, Mannigfaltigkeiten, ...). Nach einer Einführung 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. Darüber hinaus beweisen wir, dass jede solche Abbildung eineindeutig durch ihre Komponentenfelder, d.h. durch die Koeffizienten einer Taylorentwickelung bzgl. von ungeraden Koordinaten bestimmt ist. Im Allgemeinen sind Komponentenfelder nur lokal definiert. Wir stellen einen Weg vor, der diese Einschränkung umgeht: Durch das Vergrößern der betreffenden Supermannigfaltigkeit ist es immer möglich, mit globalen Koordinaten zu arbeiten. Schließlich wenden wir diesen Formalismus an, um Variationsprobleme zu untersuchen, genauer betrachten wir eine super-Version der Geodäte und eine Verallgemeinerung von harmonischen Abbildungen auf Supermannigfaltigkeiten. Bewegungsgleichungen werden von Energiefunktionalen abgeleitet und wir zeigen, wie sie sich in Komponenten zerlegen lassen. Schließlich kann in Spezialfällen die Existenz von kritischen Punkten gezeigt werden, indem das Problem auf Gleichungen der gewöhnlichen geometrischen Analysis reduziert wird. Es kann dann gezeigt werden, dass die Lösungen dieser Gleichungen sich zu kritischen Punkten im betreffenden Funktor-Raum der Felder zusammensetzt. KW - Supergeometrie KW - Variationsrechnung KW - Differentialoperatoren KW - Funktorgeometrie KW - supergeometry KW - variational calculus KW - differential operators KW - functor geometry Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-59757 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - A remark on the index of symmetric operators N2 - We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol. T3 - Preprint - (1998) 04 KW - manifolds with singularities KW - differential operators KW - index KW - 'eta' invariant Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25169 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - The index of higher order operators on singular surfaces N2 - The index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points contains the Atiyah-Singer integral as well as two additional terms. One of the two is the 'eta' invariant defined by the conormal symbol, and the other term is explicitly expressed via the principal and subprincipal symbols of the operator at conical points. In the preceding paper we clarified the meaning of the additional terms for first-order differential operators. The aim of this paper is an explicit description of the contribution of a conical point for higher-order differential operators. We show that changing the origin in the complex plane reduces the entire contribution of the conical point to the shifted 'eta' invariant. In turn this latter is expressed in terms of the monodromy matrix for an ordinary differential equation defined by the conormal symbol. T3 - Preprint - (1998) 03 KW - manifolds with singularities KW - differential operators KW - index KW - 'eta' invariant KW - monodromy matrix Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25127 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - On the index formula for singular surfaces N2 - In the preceding paper we proved an explicit index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points. Apart from the Atiyah-Singer integral, it contains two additional terms, one of the two being the 'eta' invariant defined by the conormal symbol. In this paper we clarify the meaning of the additional terms for differential operators. T3 - Preprint - (1997) 31 KW - manifolds with singularities KW - differential operators KW - index KW - 'eta' invariant KW - monodromy matrix Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25116 ER -