Modern Tensor-Spinor Symbolic Algebra Algorithms and Computing Non-Closure Geometry & Holoraumy in 11D, N = 1 Supergravity

TL;DR

This paper introduces new symbolic algebra algorithms and a software module for analyzing supersymmetric theories, enabling efficient computation of supermultiplet properties like non-closure geometry and holoraumy, demonstrated on 11D, N=1 supergravity.

Contribution

The paper presents the first broadly applicable algorithms and software for tensor-spinor symbolic algebra, solving supermultiplet coefficients, and computing holoraumy in high-dimensional supergravity.

Findings

Successfully applied to 11D, N=1 supergravity multiplet

Provided the first computation of holoraumy in 11D supergravity

Extended tensor algebra to include spinor-indexed expressions

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.