  Quenched disorder is very important but notoriously hard. In 1979, Parisi and
Sourlas proposed an interesting and powerful conjecture about the infrared
fixed points with random field type of disorder: such fixed points should
possess an unusual supersymmetry, by which they reduce in two less spatial
dimensions to usual non-supersymmetric non-disordered fixed points. This
conjecture however is known to fail in some simple cases, but there is no
consensus on why this happens. In this paper we give new non-perturbative
arguments for dimensional reduction. We recast the problem in the language of
Conformal Field Theory (CFT). We then exhibit a map of operators and
correlation functions from Parisi-Sourlas supersymmetric CFT in $d$ dimensions
to a $(d-2)$-dimensional ordinary CFT. The reduced theory is local, i.e. it has
a local conserved stress tensor operator. As required by reduction, we show a
perfect match between superconformal blocks and the usual conformal blocks in
two dimensions lower. This also leads to a new relation between conformal
blocks across dimensions. This paper concerns the second half of the
Parisi-Sourlas conjecture, while the first half (existence of a supersymmetric
fixed point) will be examined in a companion work.
