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Type of Resource (primo): dissertations
Name: Personal
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Note: thesis: Thesis (Ph. D.)--Brown University, 2022
Genre (aat): theses
Abstract: In this thesis, we present studies of off-shell supergravity theories in ten- and eleven-dimensional superspaces, polytopic supersymmetry representation theory, and the SUSY holography conjecture. First, inspired by the history of how Einstein constructed General Relativity, we study the linearized Nordstr\"om supergravity in ten- and eleven-dimensional superspaces. We found no obstacles to applying the lessons we learned in 4D to higher dimensions. We also derive infinitesimal 10D superspace Weyl transformation laws. The identification of all off-shell ten-dimensional supergeometrical Weyl field strength tensors, constructed from respective torsions, is presented. Second, since the traditional approach requires a high computational price to be paid to elucidate the $\theta$-expansion for component fields residing within the superfields, we establish a novel approach founded by the polytopic SUSY representation theory. We realize that Lie Algebra techniques, in particular branching rules, Plethysm, and tensor product, provide the key to deciphering the \textit{complete} list of independent fields that describe a supersymmetric multiplet in \textit{arbitrary} spacetime dimensions efficiently. We show the explicit one-to-one correspondence between Lorentz irreps and field variables, leading to an ``adynkrafield'' formalism in which the traditional $\theta$-monomials are replaced by Young Tableaux. Further elaborations on the ``scan'' machinery and the consequent prepotential candidates for 10D, ${\cal N} =$1, 2A, and 2B superconformal supergravity theories are presented as well. Third, the SUSY holography conjecture proposes a way to study higher-dimensional supersymmetric models from their 1D counterparts. The problem of classifying off-shell representations of the $N$-extended 1D super Poincar\'e algebra is closely related to the classification of its graphical representations -- adinkras. We define the adinkra ``height yielding matrix number'' (HYMN) and study HYMN equivalence classes for all ${\cal GR}(4,4)$ valise adinkras. Another puzzle is that considering the projection from 4D, ${\cal N} = 1$ SUSY to 1D, $N = 4$ SUSY, a dissection of $\mathbb{S}_4$ is required to be consistent with SUSY. We show that this dissection can be seen from the symmetries of the permutahedron and the use of weak Bruhat ordering.
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