Supersymmetry is defined as a Bose–Fermi symmetry. Spinors are defined in general dimensions. The Wess–Zumino model is defined first in two dimensions on-shell, where the invariance of the action is proven using Majorana spinor identities. The susy algebra is defined, and using Fierz identities, one proves the closure of the algebra and resulting off-shell susy. Then, the four-dimensional free off-shell Wess–Zumino model is defined as a simple generalization.

To define irreducible representations, spinors with dotted and undotted indices are defined. Irreducible representations of susy are defined, first in the massless case, then in the massive case, with and without central charges. The R-symmetry of the algebras is defined. The Lagrangians for the d = 4 multiplets, N = 1 chiral, N = 1 vector, their coupling and the N = 2 models, and finally the unique N = 4 model, are described.

This short chapter touches on the limitations of the SM. The SM does not include gravity, and it does not explain the major components of the mass–energy budget of the universe, dark matter and dark energy, the latter being probably the cosmological constant. CP violation in the quark sector is too small to explain the matter–antimatter asymmetry of the Universe, but, if confirmed, the non-SM CP violation in the neutrino sector might be large enough. The ‘strong CP violation’ problem might be solved with the existence of a very light particle, the axion; experiments are reaching the requested sensitivity. Supersymmetric particles present in some extensions of the SM have been searched for, but not found so far.

The SM contains too many free parameters: the masses of the fermions and of the bosons, and the mixing angles. The masses of the fermions, from neutrinos to the top quark, span 13 orders of magnitude. Why such big difference? Why is mixing small in the quark sector, and large in the neutrino sector? Why do the proton and the electron have exactly equal (and opposite) charges? Why are there just three families? Are there any spatial dimensions beyond the three we know? And so on.

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.

We use the the supersymmetric harmonic oscillator to introduce supersymmetry, and to motivate how world-sheet supersymmetry leads to space-time fermions. After a short review of Poincaré Lie superalgebras, we introduce the RNS-model, both as an example of a supersymmetric field theory and as a stepping stone to type-II superstrings. We motivate the GSO-projection through modular invariance and show how the massless spectra as well as the D-branes of type-II string theories arise. Type-I and heterotic strings are briefly introduced, and we conclude with an overview of the dualities that relate the five modular invariant supersymmetric string theories to one another, and to eleven-dimensional M-theory.