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 biharmonic 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. † On leave from the Laboratory of Theoretical Physics, JINR, Dubna, Head Post Office,

Exploiting the gauging procedure developed by us in hep-th/0605211, we study the relationships between the models of N =4 mechanics based on the off-shell multiplets (4, 4, 0) and (1, 4, 3) . We make use of the off-shell N =4, d = 1 harmonic superspace approach as most adequate for treating this circle of problems. We show that the most general sigma-model type superfield action of the multiplet (1, 4, 3) can be obtained in a few non-equivalent ways from the (4, 4, 0) actions invariant under certain three-parameter symmetries, through gauging the latter by the appropriate non-propagating gauge multiplets. We discuss in detail the gauging of both the Pauli-Gürsey SU (2) symmetry and the abelian three-generator shift symmetry. We reveal the (4, 4, 0) origin of the known mechanisms of generating potential terms for the multiplet (1, 4, 3) , as well as of its superconformal properties. A new description of this multiplet in terms of unconstrained harmonic analytic gauge superfield is proposed. It suggests, in particular, a novel mechanism of generating the (1, 4, 3) potential terms via coupling to the fermionic off-shell N =4 multiplet (0, 4, 4) .

We define the worldline harmonic SU(2|1) superspace and its analytic subspace as a deformation of the flat N = 4, d = 1 harmonic superspace. The harmonic superfield description of the two mutually mirror off-shell (4, 4, 0) SU(2|1) supermultiplets is developed and the corresponding invariant actions are presented, as well as the relevant classical and quantum supercharges. Whereas the σ-model actions exist for both types of the (4, 4, 0) multiplet, the invariant Wess-Zumino term can be defined only for one of them, thus demonstrating non-equivalence of these multiplets in the SU(2|1) case as opposed to the flat N = 4, d = 1 supersymmetry. A superconformal subclass of general SU(2|1) actions invariant under the trigonometric-type realizations of the supergroup D(2, 1; α) is singled out. The superconformal Wess-Zumino actions possess an infinite-dimensional supersymmetry forming the centerless N = 4 super Virasoro algebra. We solve a few simple instructive examples of the SU(2|1) supersymmetric quantum mechanics of the (4, 4, 0) multiplets and reveal the SU(2|1) representation contents of the corresponding sets of the quantum states.

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