We describe two constructions of hyperkahler manifolds, one based on a Legendre transform, and one on a symplectic quotient. These constructions arose in the context of supersymmetric nonlinear σ-models, but can be described entirely geometrically. In this general setting, we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry, along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience.

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We study the generalization of R → 1/R duality to arbitrary conformally invariant sigma models with an isometry. We show that any pair of dual sigma models can be represented as quotients of a self-dual sigma model obtained by gauging different combinations of chiral currents. This observation is used to clarify the interpretation of the generalized duality as a symmetry of conformal field theory. We extend these results to N = 2 supersymmetric sigma models. * Permanent address: ITP, SUNY at Stony Brook, Stony Brook NY 11794-3840.

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We show that the connection between partial breaking of supersymmetry and nonlinear actions is not accidental and has to do with constraints that lead directly to nonlinear actions of the Born-Infeld type. We develop a constrained superfield approach that gives a universal way of deriving and using these actions. In particular, we find the manifestly supersymmetric form of the action of the 3-brane in 6-dimensional space in terms of Nϭ1 superfields by using the tensor multiplet as a tool. We explain the relation between the Born-Infeld action and the model of partial Nϭ2 supersymmetry breaking by a dual D term. We represent the Born-Infeld action in a novel form quadratic in the gauge field strengths by introducing two auxiliary complex scalar fields; this makes duality covariance and the connection with the Nϭ1 supersymmetric extension of the action very transparent. We also suggest a general procedure for deriving manifestly duality symmetric actions, explaining in a systematic way relations between previously discussed Lorentz-covariant and noncovariant actions. ͓S0556-2821͑99͒05508-3͔

We present new constructions of hyperkahler metrics along with the three new classes of N = 4 supermultiplets that they derive from. Further, we provide a general setting for understanding the constructions and give a detailed description of the multiplets in N = 2 and N = 4 superspace.

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