Geometric and Graphical Study of 1-Dimensional N-Extended Supersymmetry Algebras

Abstract

In this thesis, we explore the relationship between graphical representations of the 1-dimensional N-extended supersymmetry algebra called Adinkras, compact Riemann surfaces, and quivers. An Adinkra is a graph which was originated in physics to study off-shell representations of the supersymmetry algebra. We focus on N = 4 Adinkras in this thesis. From a mathematical point of view, Adinkras are dessins d'enfants. Using this fact, we explain Adinkras as branched covering spaces for partricular dessins. We also demonstrate how to generate a quiver Q_A from an Adinkra A and relate Q_A to the noncommutative generalization of Calabi-Yau varieties called Calabi-Yau algebras. More precisely, we construct a Jacobi algebra of Q_A with a superpotential. However, in general, a Jacobi algebra generated in this way need not be a Calabi-Yau algebra. We show that Jacobi algebras of quivers constructed from Adinkras are Calabi-Yau algebras of dimension 3. We also discuss Jacobians of the Riemann surfaces in which Adinkras are embedded using isogenous decompositions of Jacobians of the Adinkra Riemann surfaces.