Conformal Properties of Generalized Dirac Operator

by Varun Thakre
Doctoral thesis
Date of Examination:2013-06-05
Date of issue:2013-07-24
Advisor:Prof. Dr. Viktor Pidstrygach
Referee:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Max Wardetzky
Persistent Address: http://dx.doi.org/10.53846/goediss-3957

 

 

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Abstract

English

In this thesis we study the non-linear Dirac operator in dimension four and the associated generalization of the Seiberg-Witten equations in dimension four. The central object of this generalization is a hyperK ahler manifold M, admitting certain symmetries. The non-linear Dirac operator acts on generalized spinors, which are equivariant maps taking values in M. Restricting to a special case of Swann bundles allows us to study the behaviour of the non-linear Dirac operator under the conformal change of metrics on the base manifold. Harmonic spinors are generalizations of aholomorphic maps between hyperK ahler manifolds. The Weitzenb ock formula for the non-linear Dirac operator can be interpreted as an energy identity for generalized spinors, analogous to the one satisfi ed by maps between hyperK ahler manifolds. In the light of this comparison, we analyze the behaviour of the energies under smooth deformations of the base manifold.This is the fi rst step in deriving a blow-up condition for harmonic spinors with bounded energies, as in the case of aholomorphic maps. In the fi nal part, we prove that restricted to the case when the target hyperK ahler manifold is a hyperK ahler reduction of a flat-space, a harmonic spinor is L-infinity bounded. We conclude with some remarks towards understanding the singular set of harmonic spinors.
Keywords: generalised dirac operator, non-linear sigma model, generalised seiberg-witten, conformal behaviour, gauge theory
 
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