This chapter is also introductory/motivational in nature. After some warm–up in two space–time dimensions (and in one), we go to our preferred susy laboratory: D = 3. Three is a very nice number of dimensions. It is the first element in the magic sequence of real division algebras

ℝ ↔ ℂ ↔ ℍ ↔ O

whose entries correspond, respectively, to susy in 3, 4, 6 and 10 dimensions. The relation between 3D supergravity and D ≥ 4 sugra is like diet Coke versus the original drink: no sugar, no caffeine, but all the flavor. Our 3D dietsugra has no propagating graviton, no propagating gravitino, no propagating gauge vector, but still has all the field–theoretic, algebraic, and geometric structures of higher–dimensional supergravity. In Sections 2.8 and 2.9 we add plenty of sugar and uplift our results to D ≥ 4 rigid susy. Finally, in Section 2.10 we add the caffeine, and study the geometrical structures emerging from extended 4D sugra.

General references for supergravity include refs. [82], [144], [146], [266], and [289].

SUSY in diverse dimensions

Supersymmetry is the (only) symmetry that relates bosons to fermions. In fact, it is the only S–matrix quantum symmetry that may connect states of different spins [96, 182]. A classical symmetry between bosons of different spins is usually not preserved at the quantum level, unless there is a supersymmetry to protect it. Hence, in most instances, (enough) supersymmetry is required for the classical geometric structures of field theory to make sense at the quantum level.

We assume the reader has some familiarity with the susy algebra (the Haag–Łopuszanski–Sohnius theorem [182]) and its general implications, as well as a knowledge of the representations of the algebra [139, 281]. We will not insist on these topics, but just mention them when needed.

The detailed form of (Poincaré) supersymmetry depends on the zoology of spinors existing in the various dimensions (and space–time signatures): Weyl, Majorana, Majorana–Weyl, symplectic–Majorana, etc. However, the general structure is quite universal, so we shall state the properties of the algebra in the different dimensions without elaborating on their derivation.