$W_3$ strings, parafermions and the Ising model

We show that any covariant scattering amplitude of the $W_3$ string will contain, as part of its integrand, a factor that obeys the differential equations satisfied by an Ising model correlation function. This factor can thus be identified with such a correlation function, in agreement with a previous result of the authors. The $W_3$ string is also shown to contain an $N=2$ parafermion theory, and hence to contain in addition the non-linear infinite-dimensional $W$-algebra corresponding to this parafermion theory. The physical states form a representation of this algebra.