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The concept of analyticity of Grassmann spinor variables is introduced. It is shown that this concept makes it possible to realize N = 2 supersymmetry on the ordinary, complex, N = 1 superpole outside the mass shell.

The harmonic N=3 superspace with the even part M4×[SU(3)U(1)×U(1)] is used to build up an unconstrained off-shell superfield formulation of N=3 super Yang-Mills theory. It is defined in an analytic subspace of this N=3 superspace and is described by three analytic gauge connections entering into the harmonic derivatives. Jumping over the “N=3 barrier” becomes possible due to the presence of an infinite set of auxiliary fields.

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Witten's linear sigma model for ADHM instantons possesses a natural (0, 4) supersymmetry. We study generalizations of the infrared limit of the model that are invariant under (4, 4) supersymmetry. In the case of four spacetime dimensions a background with a conformally flat metric and torsion is required. The geometry is specified by a single real scalar function satisfying Laplace's equation. It gives rise to 't Hooft instantons for the gauge group SU(2), instead of the general ADHM instantons for an SO(n) gauge group in the case (0, 4).

We present the (0,4) superspace version of Witten's sigma model construction for ADHM instantons. We use the harmonic superspace formalism, which exploits the three complex structures common to both (0,4) supersymmetry and self-dual Yang-Mills theory. A novel feature of the superspace formulation is the manifest interplay between the ADHM construction and its twistor counterpart.

We find a principle of harmonic analyticity underlying the quaternionic (quaternion-Kähler) geometry and solve the differential constraints which define this geometry. To this end the original 4n-dimensional quaternionic manifold is extended to a bi-harmonic space. The latter includes additional harmonic coordinates associated with both the tangent local Sp(1) group and an extra rigid SU(2) group rotating the complex structures. Then the constraints can be rewritten as integrability conditions for the existence of an analytic subspace in the bi-harmonic space and solved in terms of two unconstrained potentials on the analytic subspace. Geometrically, the potentials have the meaning of vielbeins associated with the harmonic coordinates. We also establish a one-to-one correspondence between the quaternionic spaces and off-shell N = 2 supersymmetric sigma-models coupled to N = 2 supergravity. The general N = 2 sigma-model Lagrangian when written in the harmonic superspace is composed of the quaternionic potentials. Coordinates of the analytic subspace are identified with superfields describing N = 2 matter hypermultiplets and a compensating hypermultiplet of N = 2 supergravity. As an illustration we present the potentials for the symmetric quaternionic spaces.

We show how many superfield equations of physical interest, such as those of linearized ten-dimensional supergravity, may be solved by a twistor-like transform of corresponding (off-shell) worldline superfields.

We construct N = 2 superspace lagrangians for quaternionic symmetric sigma-models G/H x Sp(1), or equivalently, quaternionic potentials for these symmetric spaces. They are homogeneous H invariant polynomials of order 4 which are similar to the quadratic Casimir operator of H. The construction is based on an identity for the structure constants specific for quaternionic symmetric spaces.

We present a unified group-theoretical framework for superparticle theories. This explains the origin of the ``twistor-like'' variables that have been used in trading the superparticle's $\kappa$-symmetry for worldline supersymmetry. We show that these twistor-like variables naturally parametrise the coset space ${\cal G}/{\cal H}$, where $\cal G$ is the Lorentz group $SO^\uparrow(1,d-1)$ and $\cal H$ is its maximal subgroup. This space is a compact manifold, the sphere $S^{d-2}$. Our group-theoretical construction gives the proper covariantisation of a fixed light-cone frame and clarifies the relation between target-space and worldline supersymmetries.

We study conditions for the existence of extended supersymmetry in topological Yang-Mills theory. These conditions are most conveniently formulated in terms of the holonomy group of the underlying manifold, on which the topological Yang-Mills theory is defined. For irreducible manifolds we find that extended supersymmetries are in 1–1 correspondence with covariantly constant complex structures. Therefore, the topological Yang-Mills theory on any Kähler manifold possesses one additional supersymmetry and on any hyper Kähler manifold there are three additional supersymmetries. The Donaldson map, which plays a crucial role in the construction of the topological invariants, is generalized for Kähler manifolds, thus providing candidates for new invariants of complex manifolds.

A deep similarity is established between the Hamiltonian mechanics of point particles and supersymmetric N=2, D=4 sigma -models formulated within harmonic superspace. An essential part of the latter, the sphere S2, comes out as a counterpart of the role of the time variable.

Extended supersymmetries in D=4 topological Yang-Mills theory are studied. Their existence imposes constraints on the metric of the manifold, on which the topological Yang-Mills theory is defined. For irreducible manifolds we establish a 1-1 correspondence between extended supersymmetries and covariantly constant complex structures. In particular, the theory possesses one additional supersymmetry on a Kähler manifold. By analogy with a general riemannian case, this gives a way for the construction of invariants of complex structures. Ingredients, needed for this construction, such as the Donaldson map, are generalized for Kähler manifolds.