We take a supergravity theory to be one that has some supersymmetry and contains gravity, but no higher spin fields. In ten and eleven dimensions, there exist only four supergravity theories which are distinguished by their different underlying supersymmetry algebras. The types of spinor one can have in different dimensions was studied in chapter 5 and it is key to understanding the different possible supersymmetry algebras. In turn, the number of components of the supercharge determines the number of degrees of freedom and the spins of the particles in the supermultiplet. By analysing the irreducible representations of supersymmetry algebras in general dimensions it was shown [10.10] than there is only one supersymmetry multiplet in eleven dimensions containing no spin higher than gravity, no such multiplets above eleven dimensions and only four such supermultiplets in ten dimensions. A Majorana spinor in ten and eleven dimensions has 32 = 210/2 components. In eleven dimensions there is only one supergravity theory [13.1] whose supersymmetry algebra contains a supercharge which is a Majorana spinor. In ten dimensions we can have Majorana–Weyl spinors, which have 16 components. In this dimension there are two maximal supergravity theories: the IIA supergravity theory [10.11–10.13] which has a supersymmetry algebra with one Majorana spinor and so two Majorana–Weyl spinors of opposite chirality, and the IIB supergravity theory [10.14–10.16], which contains a supersymmetry algebra with two Majorana-Weyl spinors of the same chirality.