Advances in Theoretical and Mathematical Physics

Volume 22 (2018)

Number 3

Geometrization of $N$-extended $1$-dimensional supersymmetry algebras, II

Pages: 565 – 613

DOI: https://dx.doi.org/10.4310/ATMP.2018.v22.n3.a2

Authors

Charles Doran (Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada)

Kevin Iga (Natural Science Division, Pepperdine University, Malibu, California, U.S.A.)

Jordan Kostiuk (Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada)

Stefan Méndez-Diez (Mathematics Program, Bard College, Annandale-on-Hudson, New York, U.S.A.)

Abstract

The problem of classifying off-shell representations of the $N$-extended one-dimensional super Poincaré 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’enfants” construction of Grothendieck. The Adinkra graphs carry two additional structures: a selection of dashed edges and an assignment of integral heights 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.

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Published 22 October 2018