Adinkras for mathematicians

Abstract

Adinkras are graphical tools created to study representations of supersymmetry algebras. Besides having inherent interest for physicists, the study of adinkras has already shown non-trivial connections with coding theory and Clifford algebras. Furthermore, adinkras offer many easy-to-state and accessible mathematical problems of algebraic, combinatorial, and computational nature. We survey these topics for a mathematical audience, make new connections to other areas (homological algebra and poset theory), and solve some of these said problems, including the enumeration of all hypercube adinkras up through dimension 5 and the enumeration of odd dashings of adinkras for any dimension.

Adinkras from ordered quartets of BC4 Coxeter group elements and regarding 1,358,954,496 matrix elements of the Gadget

“…The adinkra concept has generated at least one publication purely in the mathematical literature [23] and uncovered other surprising structures [24,25].…”

“…This suggests a notation for the matrices M in the form of (234) as we take each column from left to right. We can apply the same logic to N to suggest a notation name (1342) and for O notation name (23).…”

Adinkras and SUSY holography: Some explicit examples

“…the discovery that adinkras (and therefore off-shell supersymmetry representation theory) describe spaces of 'marked cubical topology' [5], (b.) the irreducible representations of adinkras are determined by self-dual block linear error-correcting codes [6], and (c.) that there exist a relation to mathematical structures called posets [7]. The L-matrices have a well defined role for one dimensional quantum mechanical systems.…”

Think different: applying the old macintosh mantra to the computability of the SUSY auxiliary field problem

“…The matrices (given the designation of L-matrices and R-matrices) would later become recognized as the adjacency matrices of adinkra networks [10]. This latter identification became critical in providing a definition of these matrices independent of field theory models and opening a path to totally unexpected connections to subjects such as cubical cohomology [11], error-correcting codes [12][13][14], ranked poset [15], Coxeter Groups [17] and most recently Riemann surfaces [18].…”