  We explain that the supersymmetric $CP^{n-1}$ sigma model is directly related
to the level-zero chiral Gross-Neveu (cGN) model. In particular, beta functions
of the two theories should coincide. This is consistent with the
one-loop-exactness of the $CP^{n-1}$ beta function and a conjectured all-loop
beta function of cGN models. We perform an explicit four-loop calculation on
the cGN side and discuss the renormalization scheme dependence that arises.
