Matrix Models, Quantum Spaces, and Gravity

The aim of the proposed project is to develop further a novel framework for the theory of fundamental interactions and elementary particles, which allows to go beyond the classical concepts of space-time and geometry. This framework is based on a certain type of Yang-Mills Matrix Models. On the one hand, these matrix models are known to describe gauge theory on non-commutative or quantized spaces. On the other hand, they also arise in the context of string theory, and were conjectured to contain gravity. The underlying mechanism for gravity operating in these models was recently clarified by the applicant, resolving several problems and obstructions for an application of these models in the theory of fundamental interactions. This leads to a novel approach to gauge theory and gravity on non-commutative spaces, based on a simple type of matrix models describing both space- time and matter. The metric responsible for gravity arises only effectively in the semi-classical low-energy limit. The resulting "emergent" gravity theory can be expected to be better suited for quantization than general relativity. This project studies the theoretical foundations and the physical consequences of these Yang-Mills Matrix Models, focusing on emergent gravity. Gravity is related in a specific way with the quantum structure of space-time, relating the metric with a dynamical Poisson tensor. The initial focus will be on the development of formal and theoretical aspects of this emergent gravity. This includes a more detailed understanding of the resulting geometry, the one-loop quantization, and the low-energy effective action. In particular, the gravitational equations of motion and solutions both with and without matter will be studied. Special emphasis will be on possible deviations from general relativity. In a second stage, the focus will be on physical aspects and consequences. This includes also aspects of high-energy physics and the perspectives for unified models of fundamental interactions.

This project studies a novel type of matrix model, which is a promising candidate for a comprehensive quantum theory of fundamental interactions including gravity. The most important aspect of the model is that it does not assume any classical concept of space and geometry, but is based on the most basic mathematical objects, i.e. matrices. Physical space-time and its geometry arise as solutions of the model, along with physical matter and fields. Remarkably, the resulting space-time has an intrinsic quantum nature, which means that it is not a continuum but fuzzy at very short distances. In this project, the properties of the resulting or emergent space-time were studied. One emphasis was on the effective gravity theory, which is expected to arise from the model. New solutions were found, and significant progress was made towards the clarification of the effective gravity which arises in the model. This involved developing suitable mathematical tools and techniques. A mechanism for deviations from general relativity was uncovered, which may be significant at cosmological scales. Furthermore a possibility to recover the basic ingredients of particle physics within this model was found. The leading part of the quantum corrections to the model were computed, which is essential towards establishing the model as consistent and physically viable.