An abelian gauge-theoretic variant of the Seiberg-Witten equations for multiple-spinors

We consider a variant of the Seiberg-Witten equations for multiple-spinors. The moduli space of solutions to our generalized Seiberg-Witten equations in the setting of Kähler surfaces has a direct relation with ASD connections of holomorphic vector bundle. Also in Kähler setting, we construct a numerical invariant from the equations that detects a notion of $\phi -$stability of $SU(n)-$holomorphic vector bundles where $\phi$ is some prescribed non-trivial holomorphic section.