Squashed extra dimensions in gauge theory and matrix models

One of the great outstanding problems in theoretical physics is to find a satisfactory quantum theory of all fundamental interactions including gravity. It is widely expected that the observable low-energy physics emerges from an underlying simpler, unified theory. Progress in recent years has led to the remarkable insight that even geometry can emerge, starting from simple models with little or no geometry. This occurs in certain Yang-Mills gauge theory models, and in matrix models which in turn are related to string theory. The underlying mechanism is a geometric variant of the well-known Higgs mechanism. This insight is based on the recognition that suitable matrix configurations, which arise as solutions, can be interpreted in terms of a generalized “quantum” geometry. This quantum geometry might play the role of physical space-time, or arise as tiny “fuzzy” extra dimensions which extend our 3+1-dimensional space-time. The project focuses on novel realizations of such extra dimensions arising within a very prominent example of a 3+1-dimensional gauge theory given by the maximally supersymmetric Yang-Mills theory, and certain variants thereof including the IKKT matrix model.

The main focus of this project are new solutions in super-Yang-Mills theory and matrix models, which describe such fuzzy extra dimensions. These solutions are characterized by a rich self-intersecting geometry, which leads to the essential structures required for low-energy particle physics. This leads in partiuclar to fermions which have the desired chiral properties that are essential in particle physics. In this project, the properties of the (generalized) Higgs sector and the corresponding (modified) vacua which arise from these solutions will be studied. In a further step, the resulting low-energy physics will be determined, and the potential as a possible description of elementary particle will be assessed. To this end the resulting low-energy modes on the non-trivial vacua will be determined, and the symmetry breaking pattern will be elaborated. Quantum corrections will be computed to leading order. Another goal is to find more general solutions with analogous properties, and to adapt the results in the context of matrix models and space-times with non-vanishing curvature.