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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.

This is a pedagogical introduction to the harmonic superspace method in extended supersymmetry. Inspired by exciting developments in superstring theory, it provides a systematic treatment of the quantum field theories with N=2 and N=3 supersymmetry in harmonic superspace. The authors present the harmonic superspace approach as a means of providing an off-shell description of the N=2 supersymmetric theories, both at the classical and quantum levels. Furthermore, they show how it offers a unique way to construct an off-shell formulation of a theory with higher supersymmetry, namely the N=3 supersymmetric Yang-Mills theory. Harmonic Superspace makes manifest many remarkable geometric properties of the N=2 theories, for example, the one-to-one correspondence between N=2 supersymmetric matter, and hyper-Kähler and quaternionic manifolds. This book will be of interest to researchers and graduate students working in the areas of supersymmetric quantum field theory, string theory and complex geometries.

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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.