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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.

The quantum N=2 Einstein supergravity action is constructed by requiring BRS symmetry. This symmetry is derived by using the distorted horizontality conditions in the curved N=2 harmonic superspace.

We present the differential geometry formalism of the N = 2 conformal supergravity in harmonic superspace, which allows one to derive the BRST symmetry corresponding to N = 2 super-diffeomorphisms, local Lorentz transformations and local super-Weyl transformations. The extended transgression formula is used and the N = 2 superconformal anomalies are shown.

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.

The requirement of BRS and anti-BRS symmetries in N=1 supersymmetric theory leads to Ward-identities independent of the gauge parameter. The cohomology conditions are immediately satisfied. This yields a supersymmetric formula of the chiral anomaly.

The full set of extended BRS and anti-BRS symmetries are derived for components of super-connection and gauge superfield using the N=2 harmonic superspace. The quantization of N=2 supersymmetric theory is developed and the proof of its gauge invariance is presented.

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.