The Standard Model as a geometric variational problem

The standard model of elementary particle physics is one of the most successful models in modern theoretical physics. It describes both the electromagnetic as well as the nuclear forces in a unified manner and is able to make predictions with high accuracy for experiments at particle colliders. The success of the standard model is also highlighted by a number of Nobel Prizes which have been awarded in its favor. Despite its tremendous success it does not seem possible to unify the standard model with gravity in a consistent way which is mostly due to the lack of its mathematical understanding. The current form of the standard model is based on the outcome of experiments at particle colliders such that its underlying mathematical structure is not worked out in full detail. One of the central aims of this project is to close this gap. In physics one usually studies the standard model on a flat spacetime. On the one hand this assumption leads to many simplifications in the mathematical analysis, on the other hand this is justified as our spacetime is almost flat. However, the full mathematical structure of the standard model only becomes visible when it is formulated in an abstract geometric manner which will be the first goal of this project. After expressing the standard model in the desired framework we will derive the equations that govern its (classical) dynamics. In the physics literature one is mostly interested in the quantized version of these equations which however tacitly assumes that there exist solutions of the classical equations of motion. For equations of this particular form (non-linear wave equations) new and powerful mathematical tools have been established in recent years which will be employed to investigate the equations of the standard model with full mathematical rigor. A further important building block of the standard model, which will be studied within this project, is the so-called Higgs-mechanism. This mechanism is important as it ensures that the elementary particles which are described by the standard model gain a mass. Due to the particular mathematical structure of the standard model it is not possible to directly assign a mass to each elementary particle. More precisely, it is necessary to create the masses in a second step via the Higgs- mechanism, but there are many open questions concerning its mathematical structure. In addition to the creation of masses for the elementary particles contained in the standard model, the Higgs- mechanism also bears a number of unphysical particles. On a flat spacetime it is always possible to remove these unphysical particles from the theory, but it is not known if this will also be possible when the theory is formulated in an abstract geometric manner and we will also investigate this aspect of the standard model within this project.