arXiv subjects
Publications and source records attributed to S. James Gates, Jr..
A reimagining of the supergravity prepotential formulation of 4D, ${\mathcal N}$ = 1 supergravity and its Salam-Strathdee superfield superconformal gauge group is presented.
Motivated by the search for embedded on-shell supermultiplets in higher dimensional off-shell theories, we investigate several 16-color supermultiplets and their topology. An Adinkra's topology is known to be equivalent to $(\mathbb{Z}_2)$-quotients of an N-cube. This is revisited with the focus of closing the off-shell problem for the 4D $\mathcal{N} = 4$ Maxwell supermultiplet.
Adinkras are graphical representations of the gauge invariant field components in supersymmetric theories and their orbits under the action of supersymmetry (SUSY) generators in the context of supermultiplets. A discussion is given that provides a thorough review of the concepts of holoraumy, permutahedra, and gadgets. One consequence of these concepts, the additional concept of hopper operators, is discussed. These play a particularly important role that ignites the processes needed for the study of adinkras related to minimal 4D, $\cal N$ = 1 supermultiplets (chiral, vector, tensor and complex linear supermultiplets) through the prism of very simple cellular automata following Wolfram's `New Kind of Science' paradigm.
Adinkra networks arise in the Carroll limit of supersymmetric QFT. Extensions of adinkras that are infinite dimensional graphs have never previously been discussed in the literature. We call these "infinite unfolded'' adinkras and study the properties of their realization on familiar 4D, $\cal N$ = 1 supermultiplets. A new feature in "unfolded'' adinkras is the appearance of quantities whose actions resemble BRST operators within Verma-like modules. New "net-centric" quantities ${\widetilde χ}_{(1)}$ and ${\widetilde χ}_{(2)}$ are introduced, which along with quantity $χ_{\rm o}$, describe distinctions between familiar supermultiplets in 4D, $\cal N $ = 1 theories. A previously unobserved property in all adinkras that we call "adinkra vorticity" is noted.
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We explore the relationship between holoraumy and Hodge duality beyond four dimensions. We find this relationship to be ephemeral beyond six dimensions: it is not demanded by the structure of such supersymmetrical theories. In four dimensions for the case of the vector-tensor $\cal N$ = 4 multiplet, however, we show that such a linkage is present. Reduction to 1D theories presents evidence for a linkage from higher-dimensional supersymmetry to an infinite-dimensional algebra extending $\mathfrak{Spin}(N)$.
We continue the search for rules that govern when off-shell 4D, $\cal N$ = 1 supermultiplets can be combined to form off-shell 4D, $\cal N$ = 2 supermultiplets. We study the ${\mathbb S}_8$ permutations and Height Yielding Matrix Numbers (HYMN) embedded within the adinkras that correspond to these putative 4D, $\cal N$ = 2 supermultiplets off-shell supermultiplets. Even though the HYMN definition was designed to distinguish between the raising and lowering of nodes in one dimensional valises supermultiplets, they are shown to accurately select out which combinations of off-shell 4D, $\cal N$ = 1 supermultiplets correspond to off-shell 4D, $\cal N$ = 2 supermultiplets. Only the combinations of the chiral + vector and chiral + tensor are found to have valises in the same class. This is consistent with the well known structure of 4D, $\cal N$ = 2 supermultiplets.
We take some initial steps to explore physical applications of the cluster superalgebras recently defined by Ovsienko and Shapiro. Our primary example is a fermionic extension of the $A_2$ cluster algebra, having fifteen cluster supervariables instead of the usual five. We also explore an alternate definition of cluster superalgebras based on the promotion of cluster variables to superfields.
This paper is an extension of our last 1D, N = 4 supersymmetric SYK paper [arXiv:2103.11899]. In this paper we introduced the complex linear supermultiplet (CLS), which is "usefully inequivalent" to the chiral supermultiplet. We construct three types of models based on the complex linear supermultiplet containing quartic interactions from modified CLS kinetic term, quartic interactions from 3-pt vertices integrated over the whole superspace, and 2(q-1)-pt interactions generated via superpotentials respectively. A strong evidence for the inevitability of dynamical bosons for 1D, N = 4 SYK is also presented.
We present Adynkra Libraries that can be used to explore the embedding of multiplets of component field (whether on-shell or partial on-shell) within Salam-Strathdee superfields for theories in dimension nine through four.
Proposals are made to describe 1D, N = 4 supersymmetrical systems that extend SYK models by compactifying from 4D, N = 1 supersymmetric Lagrangians involving chiral, vector, and tensor supermultiplets. Quartic fermionic vertices are generated via integrals over the whole superspace, while 2(q-1)-point fermionic vertices are generated via superpotentials. The coupling constants in the superfield Lagrangians are arbitrary, and can be chosen to be Gaussian random. In that case, these 1D, N = 4 supersymmetric SYK models would exhibit Wishart-Laguerre randomness, which share the same feature among other 1D supersymmetric SYK models in literature. One difference with 1D, N = 1 and N = 2 models though, is our models contain dynamical bosons, but this is consistent with other 1D, N = 4 and 2D, N = 2 models in literature. Added conjectures on duality and possible mirror symmetry realizations on these models is noted.
Proposals are made to describe the Weyl scaling transformation laws of supercovariant derivatives $\nabla{}_{\underline A}$, the torsion supertensors $T{}_{{\underline A} \, {\underline B}}{}^{\underline C}$, and curvature supertensors $R{}_{{\underline A} \, {\underline B}}{}_{\, \underline c} {}^{\underline d}$ in 10D superspaces. Starting from the proposal that an unconstrained supergravity prepotential for the 11D, $\mathcal{N}$ = 1 theory is described by a scalar superfield, considerations for supergravity prepotentials in the 10D theories are enumerated. We derive infinitesimal 10D superspace Weyl transformation laws and discover ten possible 10D, $\mathcal{N}$ = 1 superfield supergravity prepotentials. The first identification of all off-shell ten dimensional supergeometrical Weyl field strength tensors, constructed from respective torsions, is presented.
The mathematical concept of a "Banchoff index" associated with discrete Morse functions for oriented triangular meshes has been shown to correspond to the height assignments of nodes in adinkras. In recent work there has been introduced the concept of "Banchoff matrices" leading to HYMNs - height yielding matrix numbers. HYMNs map the shape of an adinkra to a set of eigenvalues derived from Banchoff matrices. In the context of some examples of four-color, minimal five-color, and minimal six-color adinkras, properties of the HYMNs are explored.
Starting from higher dimensional adinkras constructed with nodes referenced by Dynkin Labels, we define "adynkras." These suggest a computationally direct way to describe the component fields contained within supermultiplets in all superspaces. We explicitly discuss the cases of ten dimensional superspaces. We show this is possible by replacing conventional $θ$-expansions by expansions over Young Tableaux and component fields by Dynkin Labels. Without the need to introduce $σ$-matrices, this permits rapid passages from Adynkras $\to$ Young Tableaux $\to$ Component Field Index Structures for both bosonic and fermionic fields while increasing computational efficiency compared to the starting point that uses superfields. In order to reach our goal, this work introduces a new graphical method, "tying rules," that provides an alternative to Littlewood's 1950 mathematical results which proved branching rules result from using a specific Schur function series. The ultimate point of this line of reasoning is the introduction of mathematical expansions based on Young Tableaux and that are algorithmically superior to superfields. The expansions are given the name of "adynkrafields" as they combine the concepts of adinkras and Dynkin Labels.
For the first time in the physics literature, the Lorentz representations of all 2,147,483,648 bosonic degrees of freedom and 2,147,483,648 fermionic degrees of freedom in an unconstrained eleven dimensional scalar superfield are presented. Comparisons of the conceptual bases for this advance in terms of component field, superfield, and adinkra arguments, respectively, are made. These highlight the computational efficiency of the adinkra-based approach over the others. It is noted at level sixteen in the 11D, N = 1 scalar superfield, the {65} representation of SO(1,10), the conformal graviton, is present. Thus, Adinkra-based arguments suggest the surprising possibility that the 11D, N = 1 scalar superfield alone might describe a Poincare supergravity prepotential in analogy to one of the off-shell versions of 4D, N = 1 superfield supergravity.
Holoraumy is a tool being developed for dimensional enhancement (supersymmetry holography) where the goal is to build higher dimensional supersymmetric multiplets from lower dimensional supersymmetric multiplets. In this paper, for the first time we investigate holoraumy for on-shell supersymmetry. Specifically, the holoraumy tensors for a number of familiar 4D, $\mathcal{N}=1$ multiplets are calculated. It is shown in all of these cases of on-shell theories, the holoraumy is of the form of an electromagnetic duality charge multiplying a composite transformation involving an electromagnetic duality rotation through an angle of $π/2$ times a space time translation. The details of our calculations can be found at the HEPTHools Data Repository at https://hepthools.github.io/Data/.
As the full off-shell theories of supergravity in the important dimensions of eleven and ten dimensions are currently unknown, we introduce a superfield formalism as a foundation and experimental laboratory to explore the possibility that the scalar versions of the higher dimensional supergravitation theory can be constructed.
An adinkra is a graph-theoretic representation of spacetime supersymmetry. Minimal four-color valise adinkras have been extensively studied due to their relations to minimal 4D, $\cal N$ = 1 supermultiplets. Valise adinkras, although an important subclass, do not encode all the information present when a 4D supermultiplet is reduced to 1D. Eigenvalue equivalence classes for valise adinkra matrices exist, known as $χ_{\rm o}$ equivalence classes, where valise adinkras within the same $χ_{\rm o}$ equivalence class are isomorphic in the sense that adinkras within a $χ_{\rm o}$-equivalence class can be transformed into each other via field redefinitions of the nodes. We extend this to non-valise adinkras, via Python code, providing a complete eigenvalue classification of "node-lifting" for all 36,864 valise adinkras associated with the Coxeter group $BC{}_4$. We term the eigenvalues associated with these node-lifted adinkras Height Yielding Matrix Numbers (HYMNs) and introduce HYMN equivalence classes. These findings have been summarized in a $Mathematica$ notebook that can found at the HEPTHools Data Repository (https://hepthools.github.io/Data/) on GitHub.