The D=2 (4, 0) and (4, 4) harmonic superspace are constructed. Nonlinear supersymmetric σ-models for both real and complex (4, 0) matter multiplets are obtained. Other features are also discussed.

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The (4,4) supersymmetric dual gauge action of the two-dimensional (4,4) super-Yang–Mills theory is constructed and the corresponding gauge anomaly is expressed. The Abelian and non-Abelian T duality of the two-dimensional (4,4) supersymmetric sigma model are discussed.

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We construct the HST type hyperkahler N = 2 supersymmetric nonlinear σ model. We show that for 4n dimensional hyperkahler manifolds, one needs in general [Formula: see text] scalar functions to specify the metric. The n = 1 and n = 2 are studied in detail. The SU (2) × U (1) Taub-Nut model in the HST hypermultiplet representation is discussed as an illustrating example. The corresponding Taub-Nut metric is worked out explicitly. Some general features of these models are discussed.