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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 development of the N = 4 supersymmetric approach to quantum cosmology based on the non-compact global O(d,d) symmetries of the effective action is given. The N = 4 supersymmetric action whose bosonic sector is invariant under O(d,d) is determined. A representation for supercharges is obtained and the form of the zero and one-fermion quantum states leading to the Wheeler-DeWitt equation is found.The development of the N = 4 supersymmetric approach to quantum cosmology based on the non-compact global O(d,d) symmetries of the effective action is given. The N = 4 supersymmetric action whose bosonic sector is invariant under O(d,d) is determined. A representation for supercharges is obtained and the form of the zero and one-fermion quantum states leading to the Wheeler-DeWitt equation is found., Comment: 19 pages

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 N=2 superdiffeomorphism anomaly is defined on the compact N=2 super-Riemann surfaces by the inclusion of N=2 superprojective connection. The N=2 supercovariant operators which are superconformally covariant are constructed and the relations with the N=2 super-Poisson operators and the Krichever–Novikov algebra are given.

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 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 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 differential geometry formalism for N=2 conformal supergravity in harmonic superspace is developed. The N=2 superdiffeomorphisms and local super‐Weyl transformations and their BRST‐like symmetries are derived. These lead to the formulation of the cohomology problem and the N=2 superconformal anomaly.

We construct the Generalized Monodromy matrix $\mathcal{\hat{M}}(\omega)$ of two dimensional string effective action by introducing the T-duality group properties.The integrability conditions with general solutions depending on spectral parameter are given. This construction is investigated for the exactly solvable Wess, Zumino, Novikov and Witten (WZNW) model in pp-wave Limit when B=0., Comment: 13 pages, No figures

The superconformal structure of the (4,0) supergeometry of two-dimensional harmonic superspace is characterized by the super-Beltrami differentials. The BRST algebra including super-Beltrami variables and matter superfields of arbitrary U(1) charges is constructed. The (4,0) locally supersymmetric matter action in the super-Beltrami parametrization is constructed and the super-stress-energy tensors are obtained. Furthermore, the (4,0) superconformal anomaly is given and the super-operator-product expansion of the (4,0) super-stress-energy tensor [ital J]([ital Z]) is derived.

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.

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.