Geometrization of $N$-Extended $1$-Dimensional Supersymmetry Algebras II

TL;DR

This paper explores the geometric interpretation of $N$-extended 1D supersymmetry algebras by linking graph structures called Adinkras to super Riemann surfaces, revealing new geometric and topological insights.

Contribution

It establishes a correspondence between Adinkra graph structures and super Riemann surfaces, completing the geometric classification of supersymmetry representations.

Findings

Dashings correspond to special spin structures on Riemann surfaces

Height assignments relate to Morse functions and divisors

Complete geometric framework for classifying supersymmetry representations

Abstract

The problem of classifying off-shell representations of the $N$ -extended one-dimensional super Poincar\'e algebra is closely related to the study of a class of decorated $N$-regular, $N$-edge colored bipartite graphs known as Adinkras. In previous work we canonically embedded these graphs into explicitly uniformized Riemann surfaces via the "dessins d'enfant" construction of Grothendieck. The Adinkra graphs carry two additional structures: a selection of dashed edges and an assignment of integral helghts to the vertices. In this paper, we complete the passage from algebra, through discrete structures, to geometry. We show that the dashings correspond to special spin structures on the Riemann surface, defining thereby super Riemann surfaces. Height assignments determine discrete Morse functions, from which we produce a set of Morse divisors which capture the topological properties of the height assignments.