${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ generalizations of $\mathcal{N}=2$ super Schr\"odinger algebras and their representations

We generalize the real and chiral $\mathcal{N}=2$ super Schr"odinger algebras to ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie superalgebras. This is done by $D$-module presentation and as a consequence, the $D$-module presentations of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded superalgebras are identical to the ones of super Schr"odinger algebras. We then generalize the calculus over Grassmann number to ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ setting. Using it and the standard technique of Lie theory, we obtain a vector field realization of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded superalgebras. A vector field realization of the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ generalization of $\mathcal{N}=1$ super Schr"odinger algebra is also presented.