  Starting with a conformal Quantum Field Theory on the real line, we show that
the dual net is still conformal with respect to a new representation of the
Moebius group. We infer from this that every conformal net is normal and
conormal, namely the local von Neumann algebra associated with an interval
coincides with its double relative commutant inside the local von Neumann
algebra associated with any larger interval. The net and the dual net give
together rise to an infinite dimensional symmetry group, of which we study a
class of positive energy irreducible representations. We mention how
superselsection sectors extend to the dual net and we illustrate by examples
how, in general, this process generates solitonic sectors. We describe the free
theories associated with the lowest weight n representations of PSL(2,R),
showing that they violate 3-regularity for n>2. When n>1, we obtain examples of
non Moebius-covariant sectors of a 3-regular (non 4-regular) net.
