arXiv subjects

Kory Stiffler

Publications and source records attributed to Kory Stiffler.

At least 19 recordsLinked to original sources

arXiv2024

Covariant and Manifestly Projective Invariant Formulation of Thomas-Whitehead Gravity

Thomas-Whitehead (TW) gravity is a recently formulated projectively invariant extension of Einstein-Hilbert gravity. Projective geometry was used long ago by Thomas et. al. to succinctly package equivalent paths encoded by the geodesic equation. Projective invariance in gravity has further origins in string theory through a geometric action constructed from the method of coadjoint orbits using the Virasoro Algebra. A projectively invariant connection arises from this construction, a part of which is known as the diffeomorphism field. TW gravity exploits projective Gauss-Bonnet terms in the action functional to endow the diffeomorphism field with dynamics, while allowing the theory to collapse to general relativity in the limit that the diffeomorphism field vanishes and the connection becomes Levi-Civita. In the original formulation of TW gravity, the diffeomorphism field is projectively invariant but not tensorial and the connection is projectively invariant but not affine. In this paper we reformulate TW gravity in terms of projectively invariant tensor fields and a projectively invariant covariant derivative, derive field equations respecting these symmetries, and show that the field equations obtained are classically equivalent across formulations. This provides a 'Rosetta Stone' between this newly constructed covariant and projective invariant formulation of TW gravity and the original formulation that was manifestly projective invariant, but not covariant.

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arXiv2022

Inflation from Dynamical Projective Connections

We show how the recently developed string-inspired, projectively-invariant gravitational model Thomas-Whitehead gravity (TW gravity) naturally gives rise to a field acting as the inflaton. In the formulation of TW gravity, a field $\mathcal{D}_{ab}$ is introduced into the projective connection components and is related to a rank-two tensor field $\mathcal{P}_{ab}$. Through the dynamical action of TW gravity, in terms of projective curvature, the tensor field $\mathcal{P}_{ab}$ acquires dynamics. By decomposing $\mathcal{P}_{ab}$ into its trace and traceless degrees of freedom, and choosing the connection to be Levi-Civita, we demonstrate that TW gravity contains a non-minimally coupled scalar field with a specific potential. Considering only the trace degrees of freedom, we demonstrate that the scalar field acts as an inflaton in the slow roll approximation. We find a range of values for the parameters introduced by TW gravity that fit the experimental constraints of the most recent cosmological data.

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arXiv2021

General structure of Thomas$-$Whitehead gravity

Thomas-Whitehead (TW) gravity is a projectively invariant model of gravity over a d-dimensional manifold that is intimately related to string theory through reparameterization invariance. Unparameterized geodesics are the ubiquitous structure that ties together string theory and higher dimensional gravitation. This is realized through the projective geometry of Tracy Thomas. The projective connection, due to Thomas and later Whitehead, admits a component that in one dimension is in one-to-one correspondence with the coadjoint elements of the Virasoro algebra. This component is called the diffeomorphism field $\mathcal{D}_{ab }$ in the literature. It also has been shown that in four dimensions, the TW\ action collapses to the Einstein-Hilbert action with cosmological constant when $\mathcal{D}_{ab}$ is proportional to the Einstein metric. These previous results have been restricted to either particular metrics, such as the Polyakov 2D\ metric, or were restricted to coordinates that were volume preserving. In this paper, we review TW gravity and derive the gauge invariant TW action that is explicitly projectively invariant and general coordinate invariant. We derive the covariant field equations for the TW action and show how fermionic fields couple to the gauge invariant theory. The independent fields are the metric tensor $g_{ab}$, the fundamental projective invariant $Π^{a}_{\,\,\,bc}$, and the diffeomorphism field $\mathcal D_{ab}$.

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arXiv2020

Properties of HYMNs in Examples of Four-Color, Five-Color, and Six-Color Adinkras

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.

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arXiv2020

Infinite-Dimensional Algebraic $\mathfrak{Spin}$($N$) Structure in Extended/Higher Dimensional SUSY Holoraumy for Valise and On-Shell Supermultiplet Representations

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 $\mathfrak{Spin}(N)$.

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arXiv2019

The Signed Monodromy Group of an Adinkra

An ordering of colours in an Adinkra leads to an embedding of this Adinkra into a Riemann surface $X$, and a branched covering map $\beta_X:X\to\mathbb{CP}^1$. This paper shows how the dashing of edges in an Adinkra determines a signed permutation version of the monodromy group, and shows that it is isomorphic to a Salingaros Vee group.

math.CO↗

arXiv2019

Dark Energy From Dynamical Projective Connections

We further develop the gravitational model, Thomas-Whitehead Gravity (TW Gravity), that arises when projective connections become dynamical fields. TW Gravity has its origins in geometric actions from string theory where the TW projective connection appears as a rank two tensor, $\mathcal{D}_{ab}$, on the spacetime manifold. Using a Gauss-Bonnet (GB) action built from the $(\mathrm{d}+1)$-dimensional TW connection, and applying the tensor decomposition $\mathcal{D}_{ab} = D_{ab} + 4\Lambda /(\mathrm{d}(\mathrm{d}-1)) g_{ab}$, we arrive at a gravitational model made up of a $\mathrm{d}$-dimensional Einstein-Hilbert + GB action sourced by $D_{ab}$ and with cosmological constant $\Lambda$. The $\mathrm{d}=4$ action is studied and we find that $\Lambda \propto 1/J_0$, with $J_0$ the coupling constant for $D_{ab}$. For $\Lambda$ equal to the current measured value, $J_0$ is on the order of the measured angular momentum of the observable Universe. We view this as $\Lambda$ controlling the scale of patches of the Universe that acquire angular momentum, with the net angular momentum of multiple patches vanishing, as required by the cosmological principle. We further find a universal axial scalar coupling to all fermions where the trace, $\mathcal{D} = \mathcal{D}_{ab}g^{ab}$ acts as the scalar. This suggests that $\mathcal{D}$ is also a dark matter portal for non-standard model fermions.

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arXiv2019

On the Ubiquity Of Electromagnetic-Duality Rotations in 4D, N = 1 Holoraumy Tensors for On-Shell 4D Supermultiplets

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 $\pi/2$ times a space time translation. The details of our calculations can be found at the HEPTHools Data Repository at https://hepthools.github.io/Data/.

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arXiv2019

Adinkra Height Yielding Matrix Numbers: Eigenvalue Equivalence Classes for Minimal Four-Color Adinkras

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 $\chi_{\rm o}$ equivalence classes, where valise adinkras within the same $\chi_{\rm o}$ equivalence class are isomorphic in the sense that adinkras within a $\chi_{\rm o}$-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.

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arXiv2019

Generating all 36,864 Four-Color Adinkras via Signed Permutations and Organizing into $\ell$- and $\tilde{\ell}$-Equivalence Classes

Recently, all 1,358,954,496 values of the gadget between the 36,864 adinkras with four colors, four bosons, and four fermions have been computed. In this paper, we further analyze these results in terms of $BC_3$, the signed permutation group of three elements, and $BC_4$, the signed permutation group of four elements. It is shown how all 36,864 adinkras can be generated via $BC_4$ boson $\times$ $BC_3$ color transformations of two quaternion adinkras that satisfy the quaternion algebra. An adinkra inner product has been used for some time, known as the \emph{gadget}, which is used to distinguish adinkras. We~show how 96 equivalence classes of adinkras that are based on the gadget emerge in terms of $BC_3$ and $BC_4$. We also comment on the importance of the gadget as it relates to separating out dynamics in terms of Kähler-like potentials. Thus, on the basis of the complete analysis of the supersymmetrical representations achieved in the preparatory first four sections, the final comprehensive achievement of this work is the construction of the universal $BC_4$ non-linear $σ$-model.

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arXiv2019

4D, $\mathcal{N}=1$ Matter Gravitino Genomics

Adinkras are graphs that encode a supersymmetric representation's transformation laws that have been reduced to one dimension, that of time. A goal of the supersymmetry ``genomics'' project is to classify all 4D, $\mathcal{N}=1$ off-shell supermultiplets in terms of their adinkras. In~previous works, the genomics project uncovered two fundamental isomer adinkras, the cis- and trans-adinkras, into which all multiplets investigated to date can be decomposed. The number of cis- and trans-adinkras describing a given multiplet define the isomer-equivalence class to which the multiplet belongs. A further refining classification is that of a supersymmetric multiplet's holoraumy: the commutator of the supercharges acting on the representation. The one-dimensionally reduced, matrix representation of a multiplet's holoraumy defines the multiplet's holoraumy-equivalence class. Together, a multiplet's isomer-equivalence and holoraumy-equivalence classes are two of the main characteristics used to distinguish the adinkras associated with different supersymmetry multiplets in higher dimensions. This paper focuses on two matter gravitino formulations, each with 20 bosonic and 20 fermionic off-shell degrees of freedom, analyzes them in terms of their isomer- and holoraumy-equivalence classes, and compares with non-minimal supergravity which is also a 20x20 multiplet. This analysis fills a missing piece in the supersymmetry genomics project, as now the isomer-equivalence and holoraumy-equivalence for representations up to spin two in component fields have been analyzed for 4D, $\mathcal{N}=1$ supersymmetry. To handle the calculations of this research effort, we have used the Mathematica software package called Adinkra.m. This package is open-source and available for download at a GitHub Repository. Data files associated with this paper are also published open-source at a Data Repository also on GitHub.

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arXiv2018

Exploring the Abelian 4D, $\mathcal{N}$ = 4 Vector-Tensor Supermultiplet and Its Off-Shell Central Charge Structure

An abelian 4D, $\mathcal{N}$ = 4 vector supermultiplet allows for a duality transformation to be applied to one of its spin-0 states. The resulting theory can be described as an abelian 4D, $\mathcal{N}$ = 4 vector-tensor supermultiplet. It is seen to decompose into a direct sum of an off-shell 4D, $\mathcal{N}$ = 2 vector supermultiplet and an off-shell 4D, $\mathcal{N}$ = 2 tensor supermultiplet. The commutator algebra of the other two supersymmetries are still found to be on-shell. However, the central charge structure in the resulting 4D, $\mathcal{N}$ = 4 vector-tensor supermultiplet is considerably simpler that that of the parent abelian 4D, $\mathcal{N}$ = 4 vector supermultiplet. This appears to be due to the replacement of the usual SO(4) symmetry associated with the abelian 4D, $\mathcal{N}$ = 4 vector supermultiplet being replaced by a GL(2,$\mathbb{R}$)$\otimes$GL(2,$\mathbb{R}$) symmetry in the 4D, $\mathcal{N}$ = 4 vector-tensor supermultiplet. The $Mathematica$ code detailing the calculations is available open-source at the HEPTHools Data Repository on GitHub.

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arXiv2015

Adinkras, 0-branes, Holoraumy and the SUSY QFT/QM Correspondence

We propose the recently defined "Holoraumy Tensor" to play a critical role in defining a metric to establish a correspondence between 4D, N-extended 0-brane-based valise supermultiplet representations and, correspondingly via "SUSY Holography," on the space of 1D, 4N-extended network-based adinkras. Using an analogy with the su(3) algebra, it is argued the 0-brane holoraumy tensors plays the role of the "d-coefficients" and provides a newly established tool for investigating supersymmetric representation theory.

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arXiv2014

Adinkra `Color' Confinement In Exemplary Off-Shell Constructions Of 4D, ${\cal N}$ = 2 Supersymmetry Representations

Evidence is presented in some examples that an adinkra quantum number, $χ_{\rm o}$ (arXiv:\ 0902.3830 [hep-th]), seems to play a role with regard to off-shell 4D, $\cal N$ = 2 SUSY similar to the role of color in QCD. The vanishing of this adinkra quantum number appears to be a condition required for when two off-shell 4D, $\cal N$ = 1 supermultiplets form an off-shell 4D, $\cal N$ = 2 supermultiplet. We also explicitly comment on a deformation of the Lie bracket and anti-commutator operators that has been extensively and implicitly used in our work on "Garden Algebras" adinkras, and codes.

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arXiv2013

Reduction Redux of Adinkras

We show performing general "0-brane reduction" along an arbitrary fixed direction in spacetime and applied to the starting point of minimal, off-shell 4D, $\cal N$ $=$ 1 irreducible supermultiplets, yields adinkras whose adjacency matrices are among some of the special cases proposed by Kuznetsova, Rojas, and Toppan. However, these more general reductions also can lead to `Garden Algebra' structures beyond those described in their work. It is also shown that for light-like directions, reduction to the 0-brane breaks the equality in the number of fermions and bosons for dynamical theories. This implies that light-like reductions should be done instead to the space of 1-branes or equivalently to the worldsheet.

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arXiv2013

4D, N=1 Supergravity Genomics

The off-shell representation theory of 4D, $\mathcal{N}=1$ supermultiplets can be categorized in terms of distinct irreducible graphical representations called adinkras as part of a larger effort we call supersymmetry `genomics.' Recent evidence has emerged pointing to the existence of three such fundamental adinkras associated with distinct equivalence classes of a Coxeter group. A partial description of these adinkras is given in terms of two types, termed cis-and trans-adinkras (the latter being a degenerate doublet) in analogy to cis/trans isomers in chemistry. Through a new and simple procedure that uses adinkras, we find the irreducible off-shell adinkra representations of 4D, $\mathcal{N}=1$ supergravity, in the old-minimal, non-minimal, and conformal formulations. This procedure uncovers what appears to be a selection rule useful to reverse engineer adinkras to higher dimensions. We categorize the supergravity representations in terms of the number of cis-($n_c$) and trans-($n_t$) adinkras in the representation and synthesize our new results with our previous supersymmetry genomics results into a group theoretic framework.

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arXiv2012

Adinkras and SUSY Holography

We discuss the mechanism by which adinkras holographically store the required information for the Spin(1,3) Clifford Algebra fiber bundle in the cases of three 4D, N=1 representations: the chiral, vector and tensor supermultiplets.

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arXiv2012

4D, N = 1 Supersymmetry Genomics (II)

We continue the development of a theory of off-shell supersymmetric representations analogous to that of compact Lie algebras such as SU(3). For off-shell 4D, N = 1 systems, quark-like representations have been identified [1] in terms of cis-Adinkras and trans-Adinkras and it has been conjectured that arbitrary representations are composites of $n_c$-cis and $n_t$-trans representations. Analyzing the real scalar and complex linear superfield multiplets, these "chemical enantiomer" numbers are found to be $n_c$ = $n_t$ = 1 and $n_c$ = 1, $n_t$ = 2, respectively.

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