The evolution equations for Dirac-harmonic Maps

This thesis investigates the gradient flow of Dirac-harmonic maps. Dirac-harmonic maps are critical points of an energy functional that is motivated from supersymmetric field theories. The critical points of this energy functional couple the equation for harmonic maps with spinor fields. At present, many analytical properties of Dirac-harmonic maps are known, but a general existence result is still missing. In this thesis the existence question is studied using the evolution equations for a regularized version of Dirac-harmonic maps. Since the energy functional for Dirac-harmonic maps is unbounded from below the method of the gradient flow cannot be applied directly. Thus, we first of all consider a regularization prescription for Dirac-harmonic maps and then study the gradient flow. Chapter 1 gives some background material on harmonic maps/harmonic spinors and summarizes the current known results about Dirac-harmonic maps. Chapter 2 introduces the notion of Dirac-harmonic maps in detail and presents a regularization prescription for Dirac-harmonic maps. In Chapter 3 the evolution equations for regularized Dirac-harmonic maps are introduced. In addition, the evolution of certain energies is discussed. Moreover, the existence of a short-time solution to the evolution equations is established. Chapter 4 analyzes the evolution equations in the case that the domain manifold is a closed curve. Here, the existence of a smooth long-time solution is proven. Moreover, for the regularization being large enough, it is shown that the evolution equations converge to a regularized Dirac-harmonic map. Finally, it is discussed in which sense the regularization can be removed. In Chapter 5 the evolution equations are studied when the domain manifold is a closed Riemmannian spin surface. For the regularization being large enough, the existence of a global weak solution, which is smooth away from finitely many singularities is proven. It is shown that the evolution equations converge weakly to a regularized Dirac-harmonic map. In addition, it is discussed if the regularization can be removed in this case.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Potsdam

Year

2012

Author dc:creator

• Branding, Volker

Contributors dc:contributor

• Bär, Christian

• Dirac-harmonische Abbildungen
• Gradientenfluss
• Wärmefluss
• Spin Geometrie
• nichtlineare partielle Differentialgleichung
• Dirac-harmonic maps
• Gradient flow
• Heat Flow
• Spin Geometry
• nonlinear partial differential equations

Statement dc:rights

• Creative Commons - Namensnennung, Nicht kommerziell, Weitergabe zu gleichen Bedingungen 3.0 Deutschland

Repository record source_url

https://publishup.uni-potsdam.de/frontdoor/index/index/docId/6232

OAI identifier oai:identifier

oai:kobv.de-opus4-uni-potsdam:6232