Unitary supermultiplets of OSp( 8 4, R) and the spectrum of the S⁷ compactification of 11-dimensional supergravity

The spectrum of the S⁷ compactification of 11-dimensional supergravity is constructed very simply by using the oscillator method. The spectrum consists of a special class of positive energy unitary irreducible representations (UIR's) of the N = 8 anti-de Sitter supergroup OSp( 8 4, R). This class of UIR's correspond to the short supermultiplets whose spin range is 2. The ultra-short supermultiplet, the singleton, whose spin range is 1 2 has no Poincaré limit and does not occur in the spectrum. The method described here can be used to obtain all positive energy unitary supermultiplets of OSp( 8 4, R) and not just those that arise in the S⁷ compactification. In addition, the oscillator method can be used to obtain the positive energy UIR's of the non-compact supergroups OSp( 2n 2m, R), OSp( 2n^* 2m) and SU(n, m p).