Nonanticommutative deformations of $N=(1,1)$ supersymmetric theories

We discuss chirality-preserving nilpotent deformations of the four-dimensional $N=(1,1)$ Euclidean harmonic superspace and their implications in $N=(1,1)$ supersymmetric gauge and hypermultiplet theories. For the $SO(4)\times SU(2)$-invariant deformation, we present nonanticommutative Euclidean analogues of the $N=2$ gauge multiplet and hypermultiplet off-shell actions. As a new result, we consider a specific nonanticommutative hypermultiplet model with the $N=(1,0)$ supersymmetry. It involves free scalar fields and interacting right-handed spinor fields.