$T\bar{T}$ Deformation of $\mathcal{N}=(1,1)$ Off-Shell Supersymmetry and Partially Broken Supersymmetry

We construct the superaction for the $T\bar{T}$ deformation of 2D free $\mathcal{N}=(1,1)$ supersymmetric model with a deformed superfield. We show that the $\mathcal{N}=(1,1)$ off-shell supersymmetry in the free theory is deformed under the $T\bar{T}$ deformation, which is incorporated in the deformed superfield. We interpret this superaction as an effective action of the Goldstone superfield for the partial spontaneous breaking of $\mathcal{N}=(2,2)$ supersymmetry to $\mathcal{N}=(1,1)$. We show that the unbroken and broken supersymmetry of the effective superaction corresponds to the off-shell $\mathcal{N}=(1,1)$ supersymmetry and the off-shell fermi global non-linear symmetry in the $T\bar{T}$-deformed theory, respectively. We demonstrate that this effective superaction can be obtained by the non-linear realization of the partially broken global supersymmetry~(PBGS) from the coset superspace. Furthermore, we reproduce the superaction by the constrained superfield method accompanied by a field redefinition.