Supersymmetric Systems with Higher Order Symmetries: Classification, Integrability and Dark Equation

Source abstract

Supersymmetry, a cornerstone of modern theoretical physics, serves as a fundamental bridge between bosons and fermions, offering profound insights into the nature of particles with varying spins. This paper investigates the classification and integrability of supersymmetric systems, focusing on their higher order symmetries and implications for dark equations. Employing a combination of symmetry analysis and bosonization techniques, we aim to elucidate the unique characteristics inherent in these systems and their relevance to the broader realm of integrable system theory. Incorporating Grassmann variables, we formulate supersymmetry mathematically by transitioning from conventional space-time coordinates to a super-space formalism, thus generalizing well-known equations into their supersymmetric counterparts. The general model we propose is expressed through a novel equation, which is analyzed for its classification. Addressing the classification problem within integrable system theory, we employ symmetry analysis to identify equations exhibiting integrability. We systematically classify supersymmetric third-order evolution equations and explore various cases based on the presence or absence of supersymmetric derivatives. Furthermore, we investigate the bosonization of supersymmetric systems, offering a novel perspective on the construction of coupled bosonic systems known as dark equations.