A reimagining of the supergravity prepotential formulation of 4D, 𝒩 = 1 supergravity and its Salam-Strathdee superfield superconformal gauge group is presented.

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.

We investigate the contribution of higher spin particles in the signal of direct detection searches for dark matter. We consider a bosonic or fermionic higher spin dark matter (HSDM) candidate which interacts with the Standard Model via a dark U(1) mediator. For a particular subclass of interactions, spin-polarized targets may be used for spin determination: The angular dependence of scatterings can distinguish integer (spin-s) vs. half-integer (spin-s+1/2), while the recoil energy dependence of the signal determines s. We consider also the signal of a supersymmetric higher spin dark sector, which suggests a characteristic signal (“SUSY Rilles”) for directional direct detection.

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.

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 ${\widetilde \chi}_{(1)}$ and ${\widetilde \chi}_{(2)}$ are introduced, which along with quantity $\chi_{\rm o}$, 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 consider the superspace BRST and BV description of 4D,N=1 super-Maxwell theory and its non-abelian generalization Super Yang–Mills. By fermionizing the superspace gauge transformation of the gauge superfields, we define the nilpotent superspace BRST symmetry transformation (𝓈). After introducing an appropriate set of anti-superfields and defining the superspace antibracket, we use it to construct the BV-BRST nilpotent differential operator (s) in terms of superspace covariant derivatives. The anti-superfield independent terms of s provide a superspace generalization of the Koszul–Tate resolution (δ). In the linearized limit, the set of superspace differential operators that appear in s satisfy a nonlinear algebra which can be used to construct a BRST charge Q, without requiring pure spinor variables. Q acts on the Hilbert space of superfield states, and its cohomology generates the expected superspace equations of motion.

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 $\mathfrak{Spin}(N)$.

A bstract We continue the search for rules that govern when off-shell 4D, $$ \mathcal{N} $$ N = ∞ supermultiplets can be combined to form off-shell 4D, $$ \mathcal{N} $$ N = ∈ supermultiplets. We study the 𝕊 8 permutations and Height Yielding Matrix Numbers (HYMN) embedded within the adinkras that correspond to these putative off-shell 4D, $$ \mathcal{N} $$ N = 2 supermultiplets. Even though the HYMN definition was designed to distinguish between the raising and lowering of nodes in one dimensional valise supermultiplets, they are shown to accurately select out which combinations of off-shell 4D, $$ \mathcal{N} $$ N = 1 supermultiplets correspond to off-shell 4D, $$ \mathcal{N} $$ 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, $$ \mathcal{N} $$ N = 2 supermultiplets.

Abstract We introduce a first order description of linearized non-minimal (n = −1) supergravity in superspace, using the unconstrained prepotential superfield instead of the conventionally constrained super one forms. In this description, after integrating out the connection-like auxiliary superfield of first-order formalism, the superspace action is expressed in terms of a single superfield which combines the prepotential and compensator superfields. We use this description to construct the supersymmetric cubic interaction vertex 3/2 − 3/2 − 1/2 which describes the electromagnetic interaction between two non-minimal supergravity multiplets (superspin Y = 3/2 which contains a spin 2 and a spin 3/2 particles) and a vector multiplet (superspin Y = 1/2 contains a spin 1 and a spin 1/2 particles). Exploring the trivial symmetries emerging between the two Y = 3/2 supermultiplets, we show that this cubic vertex must depend on the vector multiplet superfield strength. This result generalize previous results for non-supersymmetric electromagnetic interactions of spin 2 particles. The constructed cubic interaction generates non-trivial deformations of the gauge transformations.

Proposals are made to describe the Weyl scaling transformation laws of supercovariant derivatives ∇ A , the torsion supertensors T AB C , and curvature supertensors R ABc d in 10D superspaces. Starting from the proposal that an unconstrained supergravity prepotential for the 11D, 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, 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.

A conjecture is made that the weight space for 4D, $$ \mathcal{N} $$ -extended supersymmetrical representations is embedded within the permutahedra associated with permutation groups 𝕊d. Adinkras and Coxeter Groups associated with minimal representations of 4D, $$ \mathcal{N} $$ = 1 supersymmetry provide evidence supporting this conjecture. It is shown that the appearance of the mathematics of 4D, $$ \mathcal{N} $$ = 1 minimal off-shell supersymmetry representations is equivalent to solving a four color problem on the truncated octahedron. This observation suggest an entirely new way to approach the off-shell SUSY auxiliary field problem based on IT algorithms probing the properties of 𝕊d.

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 $\theta$-expansions by expansions over Young Tableaux and component fields by Dynkin Labels. Without the need to introduce $\sigma$-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.

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.

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.

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 intergrals 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 1D, N = 1 and N = 2 models in literature. One difference though, is our models contain dynamical bosons, but this is consistent with other 1D, N = 4 and 2D, N = 2 models in literature. PACS: 11.30.Pb, 12.60.Jv

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.

We introduce a superspace frame-like description of linearized non-minimal supergravity. In this description the action can be written in terms of a single superfield, which combines the unconstrained prepotential and compensator superfields. We use it to construct a manifestly supersymmetric cubic interaction vertex $3/2-3/2-1/2$ which describes the electromagnetic interaction between two non-minimal supergravity multiplets ($\textsf{Y}=3/2$) and a vector multiplet ($\textsf{Y}=1/2$). Exploring the trivial symmetries emerging between the two $\textsf{Y}=3/2$ supermultiplets, we show that the cubic interaction must depend on the vector multiplet superfield strength. This result generalize previous results for non-supersymmetric electromagnetic interactions of spin 2 particles. The constructed cubic interaction generates non-trivial deformations of the gauge transformations.

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.

On the basis of comparing eigenvalues of an operator ${\hat {\mathbf{\cal C}}}{}^{({\cal R})}$, that proved useful in distinguishing how off-shell 4D, $\cal N$ = 1 supermultiplets become off-shell 4D, $\cal N$ = 2 supermultiplets, the double tensor supermultiplet is shown to be radically different for other known multiplets. This suggests difficulties, if not impossibilities, to embed it into an off-shell structure.