On unitary representations of the exceptional non-linear N=7 and N=8 superconformal algebras in terms of free fields

The simplest free-field realizations of the exceptional non-linear (quadratically generated, or W-type) N = 8 and N = 7 superconformal algebras with Spin(7) and G₂ affine currents, respectively, are investigated. Both the N = 8 and N = 7 algebras are found to admit unitary and highest-weight irreducible representations in terms of a single free boson and free fermions in 8 of Spin(7) or 7 of G₂, respectively, at level k = 1 and the corresponding central charges c₈ = 26/5 and c₇ = 5.