Gauge Theories of Dirac Type
Jürgen Tolksdorf and Torsten Thumstädter
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Submission date: 23. Mar. 2005 (revised version: December 2005)
published in: Journal of mathematical physics, 47 (2006) 8, art-no. 082305
DOI number (of the published article): 10.1063/1.2218673
MSC-Numbers: 51P05, 53C07, 70S05, 70S15, 83C22
PACS-Numbers: 02.40.Hw, 02.40.Ma, 03.50.Kk, 03.65.Sq, 04.20.Cv
Keywords and phrases: clifford modules, dirac type operators, bundle reduction, spontaneous symmetry breaking, fermionic mass operator
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A specific class of gauge theories is geometrically described in terms of fermions. In particular, it is shown how the geometrical frame presented naturally includes spontaneous symmetry breaking of Yang-Mills gauge theories without making use of a Higgs potential. In more physical terms, it is shown that the Yukawa coupling of fermions, together with gravity, necessarily yields a symmetry reduction provided the fermionic mass is considered as a globally well-defined concept. The structure of this symmetry breaking is shown to be compatible with the symmetry breaking that is induced by the Higgs potential of the minimal Standard Model. As a consequence, it is shown that the fermionic mass has a simple geometrical interpretation in terms of curvature and that the (semi-classical) ``fermionic vacuum'' determines the intrinsic geometry of space-time. We also discuss the issue of ``fermion doubling'' in some detail and introduce a specific projection onto the ``physical sub-space'' that is motivated from the Standard Model.