The heat flow for Dirac-harmonic maps

The main result of Chapter 1 is short time existence of the heat flow for Dirac-harmonic maps on closed manifolds. Dirac-harmonic maps are the critical points of a functional motivated by the supersymmetric non-linear sigma model from quantum field theory. Finding non-trivial examples for Dirac-harmonic maps turned out to be a rather challenging task and not many examples were known. With the aim to get a general existence program for Dirac-harmonic maps, the heat flow for Dirac-harmonic maps was introduced by Chen, Jost, Sun, and Zhu. The flow consists of a second order harmonic map type system coupled with a first order Dirac type system. For source manifolds with boundary Chen, Jost, Sun, and Zhu obtained short time existence. This heat flow approach to obtain Dirac-harmonic maps was fully legitimized when the existence of a global weak solution was established by Jost, Liu, and Zhu, from which they deduced existence results for Dirac-harmonic maps (for source manifolds with boundary). Our strategy to show short time existence on closed manifolds roughly is as follows: first, we solve the first order Dirac type system, then we take its solution, plug it into the second order harmonic map type system and solve the latter with a contraction argument. A main ingredient for the contraction argument are estimates for Dirac operators along maps which we will develop. The subject of Chapter 2 is the existence and genericness of minimal kernels of Dirac operators along maps. In particular, the existence results we achieve yield many suitable initial values for the short time existence result of Chapter 1. In Chapter 3 and 4 we deal with a certain Banach bundle that has as base space the Banach manifold of k-times continuously differentiable maps between a closed manifold and a connected manifold without boundary. These results are the basis of our original ansatz to solve the first order Dirac type system. The content of Chapter 5 is the computation of the curvature of the Bourguignon-Gauduchon connection in the semi-Riemannian case. The Bourguignon-Gauduchon connection is an important tool that allows to compare spinors for different metrics. We use it for example in Chapter 2