Flat currents of the Green-Schwarz superstrings inAdS5×S1andAdS3×S3backgrounds

We construct a one-parameter family of flat currents in $Ad{S}_5\times S^1$ and $Ad{S}_3\times S^3$ Green-Schwarz superstrings, which would naturally lead to a hierarchy of classical conserved nonlocal charges. In the former case we rewrite the $Ad{S}_5\times S^1$ string using a new ${\mathbb{Z}}_4$-graded base of the superalgebra $su(2,2|2)$. In both cases the existence of the ${\mathbb{Z}}_4$ grading in the superalgebras plays a key role in the construction. As a result, we find that the flat currents, when formally written in terms of the ${\mathcal{G}}_0$-gauge invariant lowercase 1-forms, take the same form as the one in the $Ad{S}_5\times S^5$ case.