Search

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 $$ \mathcal{N} $$ 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 $$ \mathfrak{Spin} $$ Spin (N).

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., 39 pages, 9 figures, added background material on coadjoint orbits of Virasoro algebra that demonstrates the string-inspired nature of the model, made various minor edits and corrections

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 $\Pi^{a}_{\,\,\,bc}$, and the diffeomorphism field $\mathcal D_{ab}$., Comment: 52 pages. Made revisions for acceptance to the journal Physical Review D

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., Comment: 49 pages. Made revisions required for acceptance to the journal Classical and Quantum Gravity

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., Comment: Added references, made small text changes, 39 pages, 5 figures

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., Comment: HEPTHools Data Repository available open-source at https://hepthools.github.io/Data/, added references and related content, corrected group from SU(2) to GL(2,R)

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., 31 pages, 12 figures, figure 12 corrected

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 B C 3 , the signed permutation group of three elements, and B C 4 , the signed permutation group of four elements. It is shown how all 36,864 adinkras can be generated via B C 4 boson &times, B C 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 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 B C 3 and B C 4 . We also comment on the importance of the gadget as it relates to separating out dynamics in terms of K&auml, hler-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 B C 4 non-linear &sigma, model.

Superstring theory is one current, promising attempt at unifying gravity with the other three known forces: the electromagnetic force, and the weak and strong nuclear forces. Though this is still a work in progress, much effort has been put toward this goal. A set of specific tools which are used in this effort are gauge/gravity dualities. This thesis consists of a specific implementation of gauge/gravity dualities to describe k-strings of strongly coupled gauge theories as objects dual to Dp-branes embedded in confining supergravity backgrounds from low energy superstring field theory. Along with superstring theory, k-strings are also commonly investigated with lattice gauge theory and Hamiltonian methods. A k-string is a colorless combination of quark-antiquark source pairs, between which a color flux tube develops. The two most notable terms of the k-string energy are, for large quark anti-quark separation L, the tension term, proportional to L, and the Coulombic 1/L correction, known as the Luscher term. This thesis provides an overview of superstring theories and how gauge/gravity dualities emerge from them. It shows in detail how these dualities can be used for the specific problem of calculating the k-string energy in 2 + 1 and 3 + 1 space-time dimensions as the energy of Dp-branes in the dual gravitational theory. A detailed review of k-string tension calculations is given where good agreement is found with lattice gauge theory and Hamiltonian methods. In reviewing the k-string tension, we also touch on how different representations of k-strings can be described with Dp-branes through gauge/gravity dualities. The main result of this thesis is how the Luscher term is found to emerge as the one loop quantum corrections to the Dp-brane energy. In 2+1 space-time dimensions, we have Luscher term data to compare with from lattice gauge theory, where we find good agreement.

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/., Comment: Latex twice, 21 pages, final version includes one additional reference, teo citation correction, and 2-form clarifying language, link to the HEPTHools Data Repository included

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., Comment: 43 pages, 9 figures, LaTeX twice (revision includes analogy between d-coefficients and holoraumy tensors)

Evidence is presented in some examples that an adinkra quantum number, $\chi_{\rm o}$ (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., Comment: 37 pages

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.

We investigate a holographic description of k-strings in higher representations via D5 branes with worldvolume fluxes. The D-brane configurations are embedded in supergravity backgrounds dual to confining field theories in 3 and 4 dimensions. We compute the tensions and find qualitative agreement for the totally symmetric and totally anti-symmetric representations with the results of other methods such as lattice as well as the Hamiltonian approach of Karabali and Nair. A one-loop computation on the D-brane configurations yields the Luescher term and allows us to confirm a previously proposed universal expression following from holography., 32 pages, 2 figures, 1 table. v2 references added; comparison with other methods enhanced

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.

Recent work on classicication of off-shell representations of N-extended worldline supersymmetry without central charges has uncovered an unexpectedly vast number--trillions of even just (chromo)topology types--of so called adinkraic supermultiplets. Herein, we show by explicit analysis that a long-known but rarely used representation, the complex linear supermultiplet, is not adinkraic, cannot be decomposed locally, but may be reduced by means of a Wess-Zumino type gauge. This then indicates that the already unexpectedly vast number of adinkraic off-shell supersymmetry representations is but the proverbial tip of the iceberg., 21 pages, 4 figures

This paper investigates the d = 4, N = 4 Abelian, global Super-Yang Mills system (SUSY-YM). It is shown how the N = 2 Fayet Hypermultiplet (FH) and N = 2 vector multiplet (VM) are embedded within. The central charges provide a plethora of information as to further symmetries of the Lagrangian. Several of these symmetries are calculated to second order. It is hoped that investigations such as these may yield avenues to help solve the auxiliary field closure problem for d = 4, N = 4, SUSY-YM and the d = 4, N = 2 Fayet-Hypermultiplet, without using an infinite number of auxiliary fields., Comment: 47 pages, reworded text relating to central charges, renumbered pages, updated title page and references

The leading term for the energy of a bound state of k-quarks and k-antiquarks is proportional to its separation L. These k-string configurations have a Luscher term associated with their quantum fluctuations which is typically a 1/L correction to the energy. We review the status of tensions and Luscher terms in the context of lattice gauge theory, Hamiltonian methods, and gauge/gravity correspondence. Furthermore we explore how different representations of the k-string manifest themselves in the gauge/gravity duality. We calculate the Luscher term for a strongly coupled SU(N) gauge theory in (2+1) dimensions using the gauge/gravity correspondence. Namely, we compute one-loop corrections to a probe D4-brane embedded in the Cvetic, Gibbons, Lu, and Pope supergravity background. We investigate quantum fluctuations of both the bosonic and the fermionic sectors., 39 pages, reference added, same version to be published in JHEP

We perform a systematic analysis of k-strings in the framework of the gauge/gravity correspondence. We discuss the Klebanov-Strassler supergravity background which is known to be dual to a confining supersymmetric gauge theory with chiral symmetry breaking. We obtain the k-string tension in agreement with expectations of field theory. Our main new result is the study of one-loop corrections on the string theoretic side. We explicitly find the frequency spectrum for both the bosons and the fermions for quadratic fluctuations about the classical supergravity solution. Further we use the massless modes to compute 1/L contributions to the one loop corrections to the k-string energy. This corresponds to the Luscher term contribution to the k-string potential on the gauge theoretic side of the correspondence., Comment: 39 pages, 3 figures. New Calculation showing explicit k -> M - k symmetry of Energy utilizing the new figure. Discussion of non-k-dependence of Luscher term at end of last section right before Conclusion. Same version to be published in JHEP

We analyze the group of maximal automorphisms of the N-extended worldline supersymmetry algebra, and its action on off-shell supermultiplets. This defines a concept of "holoraumy" that extends the notions of holonomy and curvature in a novel way and provides information about the geometry of the supermultiplet field-space. In turn, the "holoraumy" transformations of 0-brane dimensionally reduced supermultiplets provide information about Lorentz transformations in the higher-dimensional space–time from which the 0-brane supermultiplets are descended. Specifically, Spin(3) generators are encoded within 0-brane "holoraumy" tensors. Worldline supermultiplets are thus able to holographically encrypt information about higher-dimensional space–time geometry.

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., 17 pages, corrected reference, added note at the end, various typo corrections

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, [Formula: see text] = 1 representations: the chiral, vector and tensor supermultiplets.

We present an expanded and detailed discussion of the mathematical tools required to cull and filter representations of the Coxeter Group BC 4 into providing bases for the construction of minimal off-shell representations of the 4D, $$ \mathcal{N} $$ = 1 spacetime supersymmetry algebra.