Spinors, harmonic maps and Dirac operators

Who: Kristoffer Dyrgaard Sørensen
When: Wednesday, December 19, 2012 at 17:30
Where: U49

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Kristoffer Dyrgaard Sørensen defends his Master Thesis on Spinors, harmonic maps and Dirac operators. His Master thesis advisor was Martin Svensson.

Abstract:

The main topic for this thesis is harmonic maps. After an introduction of preliminary mathematical tools, some examples of harmonic maps shall be given as motivation for the remaining examinations, e.g., Calabi’s results on harmonic maps S2 → Sn. Hereafter, harmonic maps are related to twistor lifts and the twistor approach is used for the construction of harmonic maps and properties of harmonic maps shall follow. The rather general twistor approach is applied towards harmonic maps into Lie groups in two settings: for loop groups in general and in the Grassmannian model in the case of U(n) (or Lie subgroups hereof) constituting the codomain. These two approaches are related and both reduce the complexity of the harmonic equation. The twistor approach verifies that a large subset of harmonic maps can be generated purely from finite collections of unitons. In particular, a result of Uhlenbeck [39] shows that this subset constitutes the entire set of harmonic maps from the Riemann sphere. This result is strengthened by a direct approach that generates these maps by arbitrary holomorphic data.

The concept of harmonic maps is generalised in the context of Dirac-harmonic maps. These elements are generated from smooth maps between manifolds and twisted spinor fields and are shown to be critical values for a generalised energy expression. The Dirac-harmonic maps inherit properties from harmonic maps and in general their complexity is increased. Even though, existence results of harmonic maps are still present. Existence of Dirac-harmonic maps is given under certain assumptions on the base map and the spaces of Dirac-harmonic maps are given lower bounds on their dimension. Examples in accordance which this result are presented.

The thesis is available for download in PDF format.