Compactified branes, gauge theory and gravity in matrix models

Finding a consistent quantum theory which incorporates all fundamental interactions including gravity is an outstanding problem in theoretical physics. The aim of this project is to develop further one of the most promising approaches towards this goal, based on matrix models of Yang-Mills type. These models describe not only gauge theory on non-commutative or "quantized" spaces but also gravity, and allow go beyond the classical concept of a space-time continuum. In recent years, there has been substantial progress in the understanding of gravity within these models. The metric underlying gravity emerges only effectively in the semi-classical low-energy limit, on a space-time which becomes "fuzzy" at very short scales. One specific model - known as IKKT model - is well- suited for quantization. It was originally introduced as non-perturbative description of string theory. Its quantization can be carried out using well-known techniques from Yang-Mills gauge theory, which presents a great simplification compared with general relativity. In the first part of the project, the quantization of the matrix model will be carried out systematically to leading (one-loop) order. This will allow a detailed analysis of the physical implications of emergent gravity, and a comparison with general relativity. Furthermore, the powerful symmetries of the matrix model will be exploited to develop new techniques and gain new insights into maximally supersymmetric Yang-Mills gauge theory, which also arises in the matrix model. In the second part of the project, the relation with the standard model of strong and electroweak forces will be studied via intersecting branes. This will be done based on novel types of solutions with compactified extra dimensions. The stability of these solutions will be studied. The aim is to understand the resulting low-energy effective gauge theory at experimentally relevant scales.

The aim of this project was to develop further a promising approach towards a consistent quantum theory of all fundamental interactions, based on matrix models of Yang-Mills type. These models incorporate gauge theory as well as (some form of) gravity on non-commutative or ``quantized'' space-time. This means that the classical concept of a space-time continuum is replaced by a quantum notion of geometry. Space behaves as usual for large scales, but becomes fuzzy and non-classical at very short scales. Geometric concepts such as the metric are not fundamental but emerge effectively in the semi-classical limit. The specific model under consideration - known as IKKT or IIB model - is particularly well-suited for this endeavour; it can also be viewed as a non-perturbative approach to string theory. One focus of this project was a novel type of solutions where such fuzzy spaces arise as small extra dimensions. The main goal was to understand the resulting physics, in particular the relation with particle physics, but also the implications to gravity. For the particle physics aspects, once can approximate the matrix model by a (standard) supersymmetric Yang-Mills gauge theory of a particular type. We succeeded in finding at least two different solutions for such extra-dimensional branes in these models, which lead to a low-energy particle physics similar to that of the standard model. One is based on intersecting fuzzy spheres, and one is based on a novel type of squashed 4-dimensional self-intersecting quantum space. Fermionic matter then arises at the intersections, which allows to apply novel geometric insights in standard gauge theory. Progress was also made on the understanding of the geometric perturbations of quantum spaces and their possible relation with gravity, suggesting promising directions to future research.