  Using results on topological line defects of 4D Chern-Simons theory and the
algebra/cycle homology correspondence in complex surfaces $\mathcal{S}$ with
ADE singularities, we study the graded properties of the $sl(m|n)$ chain and
its embedding in string theory. Because of the $\mathbb{Z}_{2}$-grading of $
sl(m|n)$, we show that the $\left( m+n\right) !/m!n!$ varieties of superspin
chains with underlying super geometries have different cycle homologies. We
investigate the algebraic and homological features of these integrable quantum
chains and give a link between graded 2-cycles and genus-g Rieman surfaces
$\Sigma _{g}$. Moreover, using homology language, we yield the brane
realisation of the $sl(m|n)$ chain in type IIA string and its uplift to
M-theory. Other \textrm{aspects} like graded complex surfaces with $sl(m|n)$
singularity as well as super magnons are also described.
