  Holomorphic fields play an important role in 2d conformal field theory. We
generalize them to d>2 by introducing the notion of Cauchy conformal fields,
which satisfy a first order differential equation such that they are determined
everywhere once we know their value on a codimension 1 surface. We classify all
the unitary Cauchy fields. By analyzing the mode expansion on the unit sphere,
we show that all unitary Cauchy fields are free in the sense that their
correlation functions factorize on the 2-point function. We also discuss the
possibility of non-unitary Cauchy fields and classify them in d=3 and 4.
