Extended Supersymmetric Quantum Mechanics Algebras in Scattering States of Fermions off Domain Walls

Abstract

We study the underlying extended supersymmetric structure in a system composed of fermions scattered off an infinitely extended static domain wall in the xz-plane. As we shall demonstrate, the fermionic scattered states are associated to two N = 2 one dimensional supersymmetric quantum mechanical algebras with zero central charge. These two symmetries are combined to form a non-trivial one dimensional N = 4 superalgebra with various central charges. In addition, we form higher dimensional irreducible representations of the two N = 2 algebras. Moreover, we study how the Witten index behaves under compact odd and even perturbations, coming from a background magnetic field and some non-renormalizable Yukawa mass terms for the fermions. As we shall demonstrate, the Witten index is invariant only when the magnetic field is taken into account and particularly when only the z-component of the field is taken into account. Finally, we study the impact of this supersymmetric structures on the Hilbert space of the fermionic states and also we present a deformed extension of the N = 2 supersymmetric structure.