Diverse $N=(4,4)$ Twisted Multiplets in the $N=(2,2)$ Superspace

We describe four different types of $N=(4,4)$ twisted supermultiplets in the two-dimensional $N=(2,2)$ superspace $\mathbb{R}^{(1,1|2,2)}$. All these multiplets are represented by a pair of chiral and twisted chiral superfields and differ in the transformation properties under an extra hidden$ N=(2,2)$ supersymmetry. The sigma-model $N=(2,2)$ superfield Lagrangians for each type of the $N=(4,4)$ twisted supermultiplet are real functions subjected to some differential constraints implied by the hidden supersymmetry. We prove that the general sigma-model action including all types of $N=(4,4)$ twisted multiplets and invariant under the $N=(4,4)$ supersymmetry reduces to a sum of sigma-model actions for separate types. An interaction between the multiplets of different sorts is possible only through the appropriate mass terms and only for those multiplets that belong to the same “self-dual” pair.