The general concept of quantization is discussed, which provides the starting point for the further developments in this book. Starting with the concepts familiar from quantum mechanics, a number of quantum spaces defined via explicit operators on Hilbert space are discussed in detail, including compact and non-compact examples.

With the form of the target theory built up over the previous two chapters, we move to a geometric description of gravitational motion. By recasting the relative dynamics of a pair of falling objects as the deviation of nearby geodesic trajectories in a spacetime with a metric, Einstein’s equation is motivated. To describe geodesic deviation quantitatively, the Riemann tensor is introduced, and its role in characterizing spacetime structure is developed. With the full field equation of general relativity in place, the linearized limit is carefully developed and compared with the gravito-electro-magnetic theory from the first chapter.

This final chapter provides a discussion of the BFSS matrix model, which defines the matrix quantum mechanics, providing another basis for Matrix theory. The relation between M-theory and the IKKT model is briefly discussed.

In this core chapter, the one-loop effective action for Matrix theory on 3 + 1 dimensional branes is elaborated, and the Einstein–Hilbert term is obtained in the presence of fuzzy extra dimensions. Some justification for the stability of the background is given.

This chapter discusses the central models of interest, dubbed Yang–Mills matrix models. We explain how quantum spaces are obtained as nontrivial backgrounds or vacua of these models. Their quantization is discussed, both from a perturbative as well as a nonperturbative point of view.

This chapter discusses the IKKT matrix model and its quantization. This model is the basis for Matrix theory, which is distinguished by maximal supersymmetry, leading to benign properties at the quantum level. The one-loop effective action is elaborated explicitly.

This chapter discusses the class of covariant quantum spaces, which admit a large symmetry group. This includes the four-dimensional fuzzy sphere, the four-dimensional fuzzy hyperboloid, and a near-realistic 3 + 1 dimensional cosmological FLRW quantum spacetime.

Noncommutative field theory is the analog of classical field theory on quantum spaces. We discuss both classical as well as quantum aspects of such field theories. In particular, a transparent understanding of the crucial phenomenon of UV/IR mixing is obtained using the novel tool of string modes. These exhibit the stringy nature of noncommutative field theory, and will play an important role in the following. In particular, noncommutative gauge theory is defined via Yang–Mills matrix models.

The basic properties of Lie groups and related concepts are collected. This includes a discussion of coadjoint orbits and their symplectic structure, which helps to understand the semi-classical origin of imortant quantum spaces discussed in later chapters.

The basic mathematical concepts and tools for differentiable manifolds are provided as needed in the following, with emphasis on symplectic manifolds.