Sylvester James Gates

A reimagining of the supergravity prepotential formulation of 4D, [Formula: see text] = 1 supergravity and its Salam–Strathdee superfield superconformal gauge group is presented.

We review a variety of mathematical structures underlying supersymmetry, ranging from the description of gauge theories and gravity in Superspace to the combinatorial and graph-theoretic frameworks that capture the algebraic content of its representations. The discussion revisits how geometric notions commonly associated with manifestly supersymmetric gauge theories can emerge rather than being assumed a priori. This exploration is followed by an extensive treatment of supersymmetry representations through the lens of combinatorics, higher-dimensional polytopes and graph theory. The synthesis of these perspectives shows that supersymmetry, far from being confined to a single language, builds bridges to seemingly unrelated domains of mathematics and finds expression in multiple guises, each offering complementary insights into its structure.

The Supersymmetry Genomics project aims to classify supermultiplets by properties like adinkras and holoraumy. The project’s protocol is: (1) set up the SUSY transformation rules (and action) with unknown numerical coefficients, (2) solve those coefficients and (3) compute desired properties, e.g. non-closure geometry (the non-closure functions in the anticommutator of supercovariant derivatives) and holoraumy (the commutator of supercovariant derivatives). This paper provides the first broadly applicable computer algorithms for completing these computations, comprising a new Cadabra module we call “SusyPy”. We provide a significant extension of the available tensor arithmetic/canonicalization to include spinor-indexed expressions and NW–SE convention, and we provide a new Fierz expansion algorithm for spinor-indexed expressions. On top of this new tensor–spinor symbolic algebra, we have built algorithms for solving for the coefficients of a multiplet and its action, and for computing holoraumy, for both off-shell and on-shell [ ] multiplets of any dimension. We apply our tools to assimilate linearized 11D, [ ] supergravity into the Genomics project. We demonstrate solving the 11D, [ ] multiplet, as in Cremmer, Julia and Scherk, in a few lines of code, and we provide a comprehensive picture of the multiplet’s non-closure geometry. Further, we provide the first-ever computation of the holoraumy of 11D, [ ] supergravity.

We study algorithms for recursively creating arbitrary N-extended `supermultiplets' given minimal matrix representations of off-shell, N = 1 supermultiplet matrices. We observe connections between the color vertex problems in graph theory and the different supermultiplet sets locations in the permutahedron by using the concepts of truncation and chromatic number. The concept of `hopping operators' is introduced, constructed, and then used to partition the 8! vertices of the permutahedron. We explicitly partition these into 5,040 octets constrained in locations on the permutahedron by a magic number rule. Boolean factors in this recursive construction are shown to obey a doubly even binary flip rule. Although these hopping operators do not generally constitute normal subgroups of the permutation group, we find that `ab-normal cosets' exist where the same left- and right-hoppers appear as unordered sets. Finally, using computer simulations, we investigate the types of faces on higher-order permutahedron which may give rise to lower-order supermultiplets.