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.

One of the great outstanding problems in theoretical physics is to find a quantum theory of all fundamental interactions including gravity. It is generally expected that this requires a novel quantum structure of space-time and geometry. This project studies such quantum geometries which arise within well-known models, namely Yang-Mills gauge theories or related matrix models. In the first part of the project, the focus is on a class of self-intersecting quantum geometries, which can arise as extra dimensions, and lead to an effective low-energy physics close to the standard model of elementary particles. It was shown that the class of geometries under consideration not only lead to the correct pattern of chiral fermions, but also that the necessary Higgs-type condensate does indeed occur. A rich class of new exact solutions was found, which consist of 6-dimensional self-intersecting spaces with non-vanishing Higgs condensate, which glues the sheets at their intersection and gives masses to the fermions. Its possible role in the context of particle physics was studied. In the second part of the project, a new class of 4-dimensional covariant quantum geometries was discovered and studied, which can play the role of our cosmological space- time. They are compatible with the group of isometries, but nevertheless can be viewed as a sort of lattice, with an intrinsic length scale which marks the shortest possible distances. This is exactly what is expected in a full quantum theory of gravity. Indeed, it was shown within this project that these covariant quantum spaces are solutions of certain Yang-Mills matrix models, and lead to a gauge theory which includes all the essential ingredients of gravity, notably a spin 2 graviton. The necessary mathematical tools were developed for the 4-dimensional sphere and hyperboloid, which is the basis for a near-realistic cosmological space-time solution. The basic properties of these gauge theories were established, which leads to a very promising candidate for a full-fledged quantum theory of gravity. Moreover these space admit similar types of squashed extra dimensions as studied in the first part of the project.