Modern tensor–spinor symbolic algebra algorithms and computing non-closure geometry and holoraumy in 11D, 𝒩=1 supergravity
Abstract
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 𝒩=1 multiplets of any dimension. We apply our tools to assimilate linearized 11D, 𝒩=1 supergravity into the Genomics project. We demonstrate solving the 11D, 𝒩=1 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, 𝒩=1 supergravity.
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