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Using results on topological band theory of phases of matter and discretesymmetries, we study topological properties of band structure of physicalsystems involving spin $\frac{1}{2}$ and $\frac{3}{2}$ fermions. We apply thisapproach to study partial breaking in 4D $\mathcal{N}=2$ gauged supergravity inrigid limit and we describe the fermionic gapless mode in terms of chiralanomaly. We study as well the homologue of the usual spin-orbit coupling$\vec{L}.\vec{S}$, that opens the vanishing band gap for free $s=\frac{1}{2}$fermions; and show that is precisely given by the central extension of the$\mathcal{N}=2$ supercurrent algebra in 4D spacetime. We also give comments onthe rigid limit of Andrianopoli et al obtained in [28] and propose aninterpretation of energy bands in terms of a chiral gapless isospin$\frac{1}{2}$-particle (iso-particle). Other features, such as discrete T-symmetry in FI coupling space, effect of quantum fluctuations and the link withNielson-Ninomiya theorem, are also studied.

Gapped gravitinos, isospin $\frac{\mathbf{1}}{2}$ particles and $\mathcal{N}=2$ partial breaking