We establish combinatorial and numerical criteria for determining when affine type A bow diagrams generate non-empty bow varieties. Specifically, we show that these diagrams correspond to supersymmetric brane configurations. In this context, we provide a characterization of supersymmetry in affine type A brane systems through Hanany–Witten
transitions and introduce a finite-step algorithm to determine whether a given affine type A bow or brane diagram is supersymmetric. This algorithm
reduces the problem of verifying supersymmetry to checking a finite number of inequalities.