  We define basics of $(4,4)\;\; 2D$ harmonic superspace with two independent
sets of $SU(2)$ harmonic variables and apply it to construct new superfield
actions of $(4,4)$ supersymmetric two-dimensional sigma models with torsion and
mutually commuting left and right complex structures, as well as of their
massive deformations. We show that the generic off-shell sigma model action is
the general action of constrained analytic superfields $q^{(1,1)}$ representing
twisted $N=4$ multiplets in $(4,4)$ harmonic superspace. The massive term of
$q^{(1,1)}$ is shown to be unique; it generates a scalar potential the form of
which is determined by the metric on the target bosonic manifold. We discuss in
detail $(4,4)$ supersymmetric group manifold $SU(2)\times U(1)$ WZNW sigma
model and its Liouville deformation. A deep analogy of the relevant
superconformally invariant analytic superfield action to that of the improved
tensor $N=2\;\;4D$ multiplet is found. We define $(4,4)$ duality transformation
and find new off-shell dual representations of the previously constructed
actions via {\it unconstrained} analytic $(4,4)$ superfields. The dual
representation suggests some hints of how to describe $(4,4)$ models with
non-commuting complex structures in the harmonic superspace.

High Energy Physics - Theory