  Counting formulae for general primary fields in free four dimensional
conformal field theories of scalars, vectors and matrices are derived. These
are specialised to count primaries which obey extremality conditions defined in
terms of the dimensions and left or right spins (i.e. in terms of relations
between the charges under the Cartan subgroup of $SO(4,2)$). The construction
of primary fields for scalar field theory is mapped to a problem of determining
multi-variable polynomials subject to a system of symmetry and differential
constraints. For the extremal primaries, we give a construction in terms of
holomorphic polynomial functions on permutation orbifolds, which are shown to
be Calabi-Yau spaces.
