Permuting actions, moment maps and the generalized Seiberg-Witten equations

by Martin Callies
Doctoral thesis
Date of Examination:2016-02-09
Date of issue:2016-04-21
Advisor:Prof. Dr. Viktor Pidstrygach
Referee:Prof. Dr. Viktor Pidstrygach
Referee:Prof. Dr. Thomas Schick
Persistent Address: http://dx.doi.org/10.53846/goediss-5618

 

 

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Abstract

English

In this thesis, we study properties and the geometry related to the generalization of the Seiberg-Witten equations introduced by Taubes and Pidstrygach. A crucial ingrediant to these equations is a hyperkähler manifold M with a permuting Sp(1)-action. We study the differential forms induced on M and construct cocycles of degree 2 and 4 in the Cartan model for equivariant cohomology and the corresponding (generalizations of) moment maps in hyperkähler and multi-symplectic geometry. We generalize this and provide a natural and explicit construction of such a homotopy moment map for each cocycle in the Cartan model (of arbitrary degree). Coming back to the generalized Seiberg-Witten equations, we study properties of the generalized Dirac operator and provide new Lichnerowicz-Weitzenböck formulas in dimension 3. Finally, we give a list of examples of the generalized Seiberg-Witten equations, which have been studied in the literature.
Keywords: generalized Seiberg-Witten equations; moment maps; n-plectic geometry; Dirac operator; permuting action; hyperkähler; Weitzenböck formula
 
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