Direct approach to operator conformal constructions: From fermions to primary fields

I discuss the direct solution of Sugawara and coset constructions, including a path to construction of the primary fields. The basic tools are (1) a construction of affine-conformal highest-weight states, pretensors and tensors from quantum-irreducible representations of the currents of affine g, and (2) construction of primary fields by factorization and boosting of the pretensors. Large classes of pretensors are easily obtained in fermionic constructions, and guesswork is minimized with factorization of bosonized fermionic pretensors: The simplest case constructs conformal-weights h g = mN(n − N) 2n of SU m ( n) and h K = mN(n − N) n of SU m(n) ⊗ SU m(n) SU 2m(n) and extension to simply-laced g is clear. More general cases are left for future study.