  Higher spin fields in four dimensions, and more generally conformal fields in
arbitrary dimensions, can be described by spinning particle models with a
gauged SO(N) extended supergravity on the worldline. We consider here the
one-loop quantization of these models by studying the corresponding partition
function on the one-dimensional torus. After gauge fixing the supergravity
multiplet, the partition function reduces to an integral over the corresponding
moduli space which is computed using orthogonal polynomial techniques. We
obtain a compact formula which gives the number of physical degrees of freedom
for all N in all dimensions. As an aside we compute the physical degrees of
freedom of the SO(4) = SU(2)xSU(2) model with only a SU(2) factor gauged, which
has attracted some interest in the literature.
