Üç boyutta süperkonformal sigma modeller

In this thesis, we study superconformal sigma models in three dimensions. These models are formulated as field theories of smooth maps from a three-dimensional world-volume into a target space, subject to supersymmetry and conformal invariance. The geometric constraints imposed by these symmetries on the world-volume and on the target space are analyzed in detail. It is established that conformal invariance forces the target space to be a Riemannian cone, regardless of whether supersymmetry is present. The minimal supersymmetric extension of the model does not impose additional restrictions on the target space geometry. Instead, supersymmetry invariance constrains the geometry of the three-dimensional world-volume, since it requires the supersymmetry parameter to be a conformal Killing spinor. Three-dimensional manifolds admitting local conformal Killing spinors have been classified in both Euclidean and Lorentzian signatures. Regardless of the metric signature, the conformal Killing spinor equation admits four independent local solutions on a three-dimensional manifold if the manifold is locally conformally flat. In the Euclidean case, no solutions exist otherwise. In contrast, in Lorentzian signature, if the manifold is not locally conformally flat but is locally conformally equivalent to a pp-wave metric, then exactly one conformal Killing spinor may exist.