  We develop general counting formulae for primary fields in free four
dimensional (4D) scalar conformal field theory (CFT). Using a duality map
between primary operators in scalar field theory and multi-variable polynomial
functions subject to differential constraints, we identify a sector of
holomorphic primary fields corresponding to polynomial functions on a class of
permutation orbifolds. These orbifolds have palindromic Hilbert series, which
indicates they are Calabi-Yau. We construct the top-dimensional holomorphic
form expected from the Calabi-Yau property. This sector includes and extends
previous constructions of infinite families of primary fields. We sketch the
generalization of these results to free 4D vector and matrix CFTs.
