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Extended Supersymmetry in Gapped and Superconducting Graphene

In view of the many quantum field theoretical descriptions of graphene in $2+1$ dimensions, we present another field theoretical feature of graphene, in the presence of defects. Particularly, we shall be interested in gapped graphene in the presence of a domain wall and also for superconducting graphene in the presence of a vortex. As we explicitly demonstrate, the gapped graphene electrons that are localized on the domain wall are associated with four $N=2$ one dimensional supersymmetries, with each pair combining to form an extended $N=4$ supersymmetry with non-trivial topological charges. The case of superconducting graphene is more involved, with the electrons localized on the vortex being associated with $n$ one dimensional supersymmetries, which in turn combine to form an $N=2n$ extended supersymmetry with no-trivial topological charges. As we shall prove, all supersymmetries are unbroken, a feature closely related to the number of the localized fermions and also to the exact form of the associated operators. In addition, the corresponding Witten index is invariant under compact and odd perturbations.