  Kadanoff's "correlations along a line" in the critical two-dimensional Ising
model (1969) are reconsidered. They are the analytical aspect of a
representation of abelian chiral vertex operators as quadratic polynomials, in
the sense of operator valued distributions, in non-abelian exchange fields.
This basic result has interesting applications to conformal coset models. It
also gives a new explanation for the remarkable relation between the "doubled"
critical Ising model and the free massless Dirac theory. As a consequence,
analogous properties as for the Ising model order/disorder fields with respect
both to doubling and to restriction along a line are established for the
two-dimensional local fields with chiral level 2 SU(2) symmetry.
