$\mathcal{N}$-Extended $D=4$ Supergravity, Unconventional SUSY and Graphene

We derive a $2+1$ dimensional model with unconventional supersymmetry at the boundary of an ${\mathrm{A}\mathrm{d}\mathrm{S}}_4$ $\mathcal{N}$-extended supergravity, generalizing previous results. The (unconventional) extended supersymmetry of the boundary model is instrumental in describing, within a top-down approach, the electronic properties of graphene-like 2D materials at the two Dirac points, $\mathbf{K}$ and ${\mathbf{K}}^{\mathrm{\prime }}$. The two valleys correspond to the two independent sectors of the $\mathrm{O}\mathrm{S}\mathrm{p}(p\mathrm{\mid }2)\times \mathrm{O}\mathrm{S}\mathrm{p}(q\mathrm{\mid }2)$ boundary model in the $p=q$ case, which are related by a parity transformation. The Semenoff and Haldane-type masses entering the corresponding Dirac equations are identified with the torsion parameters of the substrate in the model.