  It is shown how twisted N=2 (k=1) provides for the first time a complete
conformal field theory description of the usual geometrical phase transitions
in two dimensions, like polymers, percolation or brownian motion. In
particular, four point functions of operators with half integer Kac labels are
computed, together with geometrical operator products. In addition to Ramond
and Neveu Schwartz, a sector with quarter twists has to be introduced. The role
of fermions and their various sectors is geometrically interpreted, modular
invariant partition functions are built. The presence of twisted N=2 is traced
back to the Parisi Sourlas supersymmetry. It is shown that N=2 leads also to
new non trivial predictions; for instance the fractal dimension of the
percolation backbone in two dimensions is conjectured to be D=25/16, in good
agreement with numerical studies.
