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Superconducting Cosmic Strings and One Dimensional Extended Supersymmetric Algebras

In this article we study in detail the supersymmetric structures that underlie the system of fermionic zero modes around a superconducting cosmic string. Particularly, we extend the analysis existing in the literature on the one dimensional $N=2$ supersymmetry and we find multiple $N=2$, $d=1$ supersymmetries. In addition, compact perturbations of the Witten index of the system are performed and we find to which physical situations these perturbations correspond. More importantly, we demonstrate that there exists a much more rich supersymmetric structure underlying the system of fermions with $N_f$ flavors and these are $N$-extended supersymmetric structures with non-trivial topological charges, with "$N$" depending on the fermion flavors.