A new approach to N = 2 supersymmetry based on the concept of harmonic superspace is proposed and is used to give an unconstrained superfield geometric description of N = 2 super Yang-Mills and supergravity theories as well as of matter N = 2 hypermultiplets. The harmonic N = 2 superspace has as independent coordinates, in addition to the usual ones, the isospinor harmonics u : on the sphere SU(2)/U(l). The role of u : is to relate the SU(2) group realised on the component fields to a U ( l ) group acting on the relevant superfields. Their introduction makes it possible to SU(2)-covariantise the notion of Grassmann analyticity. Crucial for our construction is the existence of an analytic subspace of the general harmonic N = 2 superspace. The hypermultiplet superfields and the true prepotentials (pre-prepotentials) of N = 2 super Yang-Mills and supergravity are unconstrained superfunctions over this analytic subspace. The pre-prepotentials have a clear geometric interpretation as gauge connections with respect to the internal SU(2)/U( 1) directions. A radically new feature arises: the number of gauge and auxiliary degrees of freedom becomes infinite while the number of physical degrees of freedom remains finite. Other new results are the massive N = 2 Yang-Mills theory and various off-shell selfinteractions of hypermultiplets. The propagators for matter and Yang-Mills superfields are given.

The partial spontaneous breaking of rigid N = 2 supersymmetry implies the existence of a massless N = 1 Goldstone multiplet. In this paper we show that the spin-(1/2, 1) Maxwell multiplet can play this role. We construct its full nonlinear transformation law and find the invariant Goldstone action. The spin-1 piece of the action turns out to be of BornInfeld type, and the full superfield action is duality invariant. This leads us to conclude that the Goldstone multiplet can be associated with a D-brane solution of superstring theory for p = 3. In addition, we find that N = 1 chirality is preserved in the presence of the Goldstone-Maxwell multiplet. This allows us to couple it to N = 1 chiral and gauge field multiplets. We find that arbitrary Kähler and superpotentials are consistent with partially broken N = 2 supersymmetry.To the memory of Viktor I. Ogievetsky

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The quantisation procedure in the harmonic superspace approach is worked out. Harmonic distributions are introduced and are used to construct the analytic superspace delta functions and the Green functions for the hypermultiplet and the N=2 Yang-Mills superfields. The gauge fixing is described and the relevant Faddeev-Popov ghosts are defined. The corresponding BRST transformations are found. The harmonic superspace quantisation of the N=2 gauge theory turns out to be rather simple and has many parallels with that for the standard (N=0) Yang-Mills theory. In particular, no ghosts-for-ghosts are needed.

Harmonic superspace is used to build up an unconstrained off-shell formulation of N=3 supersymmetry Yang-Mills theory. The theory is defined in an analytic N=3 superspace having M4(X)SU(3)/(U(1)(X)U(1)) as an even part. The basic objects are the analytic potentials which serve as gauge connections entering harmonic derivatives. The action is an integral over analytic superspace. The Lagrange density is surprisingly simple and it is gauge invariant up to total harmonic derivative. The equations of motion are integrability conditions on the internal space SU(3)/U(1)(X)U(1). It is the infinite set of auxiliary fields that allows the authors to overcome the 'N=3 barrier'.