Sigma Models in (4,4) Harmonic Superspace

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.