Abstract. We calculate the spectrum of the S5 compactification of the chiral N = 2, D = 10 supergravity theory. The modes on S5 fall into unitary irreducible representations of the D = 5, N = 8 anti-de Sitter supergroup U(2,2/4). These unitary supermultiplets involve fields of spin s 2 with quantised 'mass' eigenvalues. The massless multiplet contains fifteen vector fields, six self-dual and six anti-self-dual anti-symmetric tensor fields. The fields of the massless multiplet are expected to be those of a gauged N = 8 theory in D = 5 with a local gauge group SU(4).Recently there has been a surge of interest in Kaluza-Klein type theories in the context of supergravity. The main object of study so far has been the eleven-dimensional simple supergravity of Cremmer er a1 [ 11 and various compactifications of this theory down to four dimensions have been studied. A compactification is called spontaneous if, for example, the eleven-dimensional theory admits a vacuum solution leading to a four-dimensional spacetime M4 and a seven-dimensional compact internal manifold M , parametrised by the extra seven coordinates. Then from a four-dimensional point of view the spectrum of the theory consists of massless modes and an infinite tower of massive modes at the Planck scale corresponding to the Fourier expansion on M,.In this letter we shall study the spectrum of the chiral N = 2, D = 10 supergravity theory [2] compactified on S5 down to five spacetime dimensions. This theory cannot be obtained from the eleven-dimensional supergravity by dimensional reduction in contrast to the non-chiral N = 2, D = 10 theory which can be. The non-chiral and chiral N = 2 supergravity theories can be regarded as the massless sector of type IIa and type IIb superstring theories in D = 1011. Both ofthese theories when dimensionally reduced to D = 4 give the N = 8 theory.The fields of the chiral N = 2, D = 10 supergravity are: a complex scalar cp, a complex Weyl spinor A, a complex anti-symmetric second rank tensor A,,, a complex Weyl gravitino (cIw, a real graviton ('zehnbein') e; and a real fourth-rank anti-symmetric tensor APupA. The field strength FpvpAm of the field A,,,, is self-dual in the free theory. The complex scalar field cp parametrises the non-compact coset space SU( 1, l)/U( 1)[4]. The covariant field equations of chiral N = 2, D = 10 supergravity were obtained t

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We present the alternative topological twisting of N = 4 Yang-Mills, in which the path integral is dominated not by instantons, but by flat connections of the complexified gauge group. The theory is nontrivial on compact orientable four-manifolds with nonpositive Euler number, which are necessarily not simply connected. On such manifolds, one finds a single topological invariant, analogous to the Casson invariant of three-manifolds.

It is argued that the vectors of N-extended supergravity theories should be thought of as self-dual tensor fields, and that they contribute to the anomalies of the (S)U(N) currents. Some implications of these anomalies are discussed. The total anomaly is found to vanish for N > 4, so the composite gauging scenario is consistent in these cases.