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A description, based on the notion of the super Drinfeld double, of non-Abelian duality for the (4, 4) supersymmetric sigma model is given. We show that this model has non-Abelian supercurrents that obey non-commutative conservation laws. The explicit formulae of general solutions to Poisson–Lie T duality with and without spectator superfields are given. The classical canonical equivalence to worldsheet (4, 4) supersymmetric sigma models related by Poisson–Lie T duality is explicitly formulated.

The quantum actions of the (4,4) supersymmetric non-linear sigma model and its dual in the Abelian case are constructed by using the background superfield method. The propagators of the quantum superfield and its dual and the gauge fixing actions of the original and dual (4,4) supersymmetric sigma models are determined. On the other hand, the BRST transformations are used to obtain the quantum dual action of the (4,4) supersymmetric non-linear sigma model in the sense of Poisson-Lie T-duality, Comment: 18 pages

The currents and the super-operators product expansions of theN = 4 super-Walgebra are obtained. The generators of this extended conformal algebra consistof the stress-tensor superfield J and the primary superfields Φ ± of integer orhalf-integer conformal weight Δ and Cartan–Weylcharge q. The algorithm forderiving the N = 4 super-W algebra is given for particular cases. Explicit formsof the operator product algebra ΦΦ for different values of Δ are given.

The conserved supercurrents and are deduced for the (4,0) supersymmetric sigma-model on harmonic superspace with arbitrary background gauge connection. These are introduced in the Lagrangian density of the model by their couplings to the analytic gauge superfields and . The T-duality transformations are obtained by integrating out the analytic gauge superfields. Finally, the (4,0) supersymmetric anomaly is derived.

The N = 4 super-Beltrami parametrization is formulated in the two-dimensional harmonic superspace. The N = 4 superconformal anomaly and its compensating action are constructed. Then the anomalous Ward identities are derived and the operator product expansions of the N = 4 super-stress--energy tensors are obtained.

The two-dimensional (4,4) super-Yang-Mills theory is formulated in the harmonic superspace and the corresponding gauge anomaly is given. The Ward identities for the local gauge symmetry are expressed in terms of the analytic Green-kernel operator and the super-Kac-Moody algebra is derived.

The authors study, in harmonic superspace, the N=2 SQED model based on the interaction of a massive Fayet-Sohnius hypermultiplet and a gauge multiplet. The Fayet-Iliopoulos mechanism is used to break the N=2 SQED model. A general discussion of symmetry breaking is given.

A manifestly covariant background field formalism for the (4,0) supersymmetric nonlinear sigma model in two dimensions is presented. The BRST argument is used in order to obtain Faddeev-Popov ghost terms.

The quantization ofN = 2 supersymmetric Yang-Mills theory coupled to a Fayet-Sohnius hypermuttiplet is performed in the harmonic superspace, by requiring BRS and anti-BRS invariances. The corresponding Ward identities and the renormalized BRS and anti-BRS symmetries are derived.