  In the conformal field theory (CFT) approach to the quantum Hall effect, the
multi-electron wave functions are expressed as correlation functions in certain
rational CFTs. While this approach has led to a well-understood description of
the fractionally charged quasihole excitations, the quasielectrons have turned
out to be much harder to handle. In particular, forming quasielectron states
requires non-local operators, in sharp contrast to quasiholes that can be
created by local chiral vertex operators. In both cases, the operators are
strongly constrained by general requirements of symmetry, braiding and fusion.
Here we construct a quasielectron operator satisfying these demands and show
that it reproduces known good quasiparticle wave functions, as well as predicts
new ones. In particular we propose explicit wave functions for quasielectron
excitations of the Moore-Read Pfaffian state. Further, this operator allows us
to explicitly express the composite fermion wave functions in the positive Jain
series in hierarchical form, thus settling a longtime controversy. We also
critically discuss the status of the fractional statistics of quasiparticles in
the Abelian hierarchical quantum Hall states, and argue that our construction
of localized quasielectron states sheds new light on their statistics. At the
technical level we introduce a generalized normal ordering, that allows us to
"fuse" an electron operator with the inverse of an hole operator, and also an
alternative approach to the background charge needed to neutralize CFT
correlators. As a result we get a fully holomorphic CFT representation of a
large set of quantum Hall wave functions.
