MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Gauge Theories of Dirac Type

Jürgen Tolksdorf and Torsten Thumstädter


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.

MSC Codes:
51P05, 53C07, 70S05, 70S15, 83C22
02.40.Hw, 02.40.Ma, 03.50.Kk, 03.65.Sq, 04.20.Cv
clifford modules, dirac type operators, bundle reduction, spontaneous symmetry breaking, fermionic mass operator

Related publications

2006 Repository Open Access
Jürgen Tolksdorf and Torsten Thumstädter

Gauge theories of dirac type

In: Journal of mathematical physics, 47 (2006) 8, p. 082305