Using results on topological band theory of phases of matter and discrete symmetries, we study topological properties of band structure of physical systems involving spin $\frac{ 1}{ 2}$ and $\frac{ 3}{ 2}$ fermions. We apply this approach to study partial breaking in 4D $\mathcal{ N}=2$ gauged supergravity in rigid limit and we describe the fermionic gapless mode in terms of chiral anomaly. 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 on the rigid limit of Andrianopoli et al obtained in [28] and propose an interpretation 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 with Nielson-Ninomiya theorem, are also studied.
Categories: hep-th