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 and discuss a 6D supersymmetric gauge theory involving four derivatives in the action. The theory involves a dimensionless coupling constant and is renormalizable. At the tree level, it enjoys N =(1, 0) superconformal symmetry, but the latter is broken by quantum anomaly. Our study should be considered as preparatory for seeking an extended version of this theory which would hopefully preserve conformal symmetry at the full quantum level and be ultraviolet-finite.

We present the analysis of all possible shortenings which occur for composite gauge invariant conformal primary superfields in SU(2,2/N) invariant gauge theories.These primaries have top-spin range y < J m ax < N with ^max = Ji + ^2) (^15^2) being the SL(2 J C) quantum numbers of the highest spin component of the superfield.In Harmonic superspace, analytic and chiral superfields give Jmax = y series while intermediate shortenings correspond to fusion of chiral with analytic in JV = 2, or analytic with different analytic structures in JV = 3,4.In the AdS/CFT language shortenings of UIR's correspond to all possible BPS conditions on bulk states.An application of this analysis to multitrace operators, corresponding to multiparticle supergravity states, is spelled out.

We study the SO(4)×SU(2) invariant Q-deformation of Euclidean N =(1, 1) gauge theories in the harmonic superspace formulation. This deformation preserves chirality and Grassmann harmonic analyticity but breaks N =(1, 1) to N =(1, 0) supersymmetry. The action of the deformed gauge theory is an integral over the chiral superspace, and only the purely chiral part of the covariant superfield strength contributes to it. We give the component form of the N =(1, 0) supersymmetric action for the gauge groups U(1) and U(n>1). In the U(1) and U(2) cases, we find the explicit nonlinear field redefinition (Seiberg-Witten map) relating the deformed N =(1, 1) gauge multiplet to the undeformed one. This map exists in the general U(n) case as well, and we use this fact to argue that the deformed U(n) gauge theory can be nonlinearly reduced to a theory with the gauge group SU(n).

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