We define N =4, d=1 harmonic superspace HR 1+2|4 with an SU(2)/U(1) harmonic part, SU(2) being one of two factors of the R-symmetry group SU(2)× SU(2) of N =4, d=1 Poincaré supersymmetry. We reformulate, in this new setting, the models of N =4 supersymmetric quantum mechanics associated with the off-shell multiplets (3, 4, 1) and (4, 4, 0). The latter admit a natural description as constrained superfields living in an analytic subspace of HR 1+2|4 . We construct the relevant superfield actions consisting of a sigma-model as well as a superpotential parts and demonstrate that the superpotentials can be written off shell in a manifestly N =4 supersymmetric form only in the analytic superspace. The constraints implied by N =4 supersymmetry for the component bosonic target-space metrics, scalar potentials and background one-forms automatically follow from the harmonic superspace description. The analytic superspace is shown to be closed under the most general N =4, d=1 superconformal group D(2, 1; α). We give its action on the analytic superfields comprising the (3, 4, 1) and (4, 4, 0) multiplets, reveal a surprising relation between the latter and present the corresponding superconformally invariant actions. The harmonic superspace approach suggests a natural generalization of these multiplets, with a [2(n+1), 4n, 2(n−1)] off-shell content for n>2.

Proceeding from a nonlinear realization of the most general N =4, d=1 superconformal symmetry, associated with the supergroup D(2, 1; α), we construct a new model of nonrelativistic N =4 superconformal mechanics. In the bosonic sector it combines the worldline dilaton with the fields parametrizing the R-symmetry coset S 2 ∼ SU (2)/U (1). We present invariant off-shell N =4 and N =2 superfield actions for this system and show the existence of an independent N =4 superconformal invariant which extends the dilaton potential. The extended supersymmetry requires this potential to be accompanied by a d=1 WZW term on S 2 . We study the classical dynamics of the bosonic action and the geometry of its sigma-model part. It turns out that the relevant target space is a cone over S 2 for any non-zero α = ± 1 2 . The constructed model is expected to be related to the 'relativistic' N =4 mechanics of the AdS 2 × S 2 superparticle via a nonlinear transformation of the fields and the time variable.

Proceeding from nonlinear realizations of the most general N = 4, d = 1 superconformal symmetry associated with the supergroup D(2, 1; α), we construct all known and two new off-shell N = 4, d = 1 supermultiplets as properly constrained N = 4 superfields. We find plenty of nonlinear interrelations between the multiplets constructed and present a few examples of invariant superfield actions for them. The superconformal transformation properties of these multiplets are explicit within our method.

We investigate deformations of four-dimensional N =(1, 1) euclidean superspace induced by nonanticommuting fermionic coordinates. We essentially use the harmonic superspace approach and consider nilpotent bi-differential Poisson operators only. One variant of such deformations (termed chiral nilpotent) directly generalizes the recently studied chiral deformation of N =( 1 2 , 1 2 ) superspace. It preserves chirality and harmonic analyticity but generically breaks N =(1, 1) to N =(1, 0) supersymmetry. Yet, for degenerate choices of the constant deformation matrix N =(1, 1 2 ) supersymmetry can be retained, i.e. a fraction of 3/4. An alternative version (termed analytic nilpotent) imposes minimal nonanticommutativity on the analytic coordinates of harmonic superspace. It does not affect the analytic subspace and respects all supersymmetries, at the expense of chirality however. For a chiral nilpotent deformation, we present non(anti)commutative euclidean analogs of N =2 Maxwell and hypermultiplet off-shell actions.

We construct a variety of off-shell N =8, d=1 supermultiplets with finite numbers of component fields as direct sums of properly constrained N =4, d=1 superfields. We also show how these multiplets can be described in N =8, d=1 superspace where the whole amount of supersymmetry is manifest. Some of these multiplets can be obtained by dimensional reduction from N =2 multiplets in d=4, whereas others cannot. We give examples of invariant superfield actions for the multiplets constructed, including N =8 superconformally invariant ones.