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

TL;DR

This paper explores the supersymmetric algebraic structures underlying fermionic scattering states off a domain wall, revealing complex supersymmetry relations and invariance properties under certain perturbations.

Contribution

It demonstrates the presence of extended supersymmetric quantum mechanical algebras in fermion-domain wall scattering and analyzes their invariance under magnetic and Yukawa perturbations.

Findings

Fermionic states are associated with two N=2 supersymmetric algebras.

These algebras combine into a non-trivial N=4 superalgebra with central charges.

The Witten index remains invariant under specific magnetic field configurations.

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.