Superspace Approach Research Articles

The incommensurate composite YxOs4B4 (x = 1.161).

YxOs4B4 (x = 1.161) crystallizes as a tetragonal incommensurate composite of columns of Y atoms extending along [001] in an Os4B4 framework. The structure was refined using the superspace approach. The basic structure of the Y subsystem can be idealized as having I4/mmm symmetry, with a crystallographically unique Y atom located on the 4/mmm position. The actual superspace symmetry is P42/nmc(00σ3)s0s0. The Y atoms feature only subtle positional modulation in the [001] direction. The Os4B4 subsystem [P42/ncm(00σ3)00ss superspace symmetry] is built of columns of edge-sharing Os4 tetrahedra extending along [001] and B2 dumbbells. The Os4 tetrahedra feature pronounced positional modulation with a distinct variation of the Os-Os bond lengths. Modulation of the B2 dumbbells is best described as a rotation about the [001] axis.

Off-shell invariants of linearized 4D,N=2 supergravity in the harmonic approach

Using the harmonic superspace approach, we construct, at the linearized level, N=2 supersymmetric curvatures generalizing scalar curvature, Ricci curvature and Weyl tensor. These supercurvatures are the building blocks of various linearized 4D,N=2 Einstein supergravity invariants. The supercurvatures involving the scalar and Ricci curvatures are analytic harmonic N=2 superfields, while the Weyl supertensor is a chiral N=2 superfield. As the basic distinguished feature of our construction, all these objects are expressed through the fundamental analytic gauge prepotentials h++M,M=(αα˙,α+,α˙+,5). The related characteristic features are the heavy use of harmonic derivatives and harmonic zero-curvature equations. On a number of instructive examples, we describe the component reduction of the superfield invariants constructed. Published by the American Physical Society 2024

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The supertwistor and bi-supertwistor formulations for N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document}-extended anti-de Sitter (AdS) superspace in four dimensions, AdS4∣4N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ Ad{S}^{4\\mid 4\\mathcal{N}} $$\\end{document}, were derived two years ago in [1]. In the present paper, we introduce a novel realisation of the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document}-extended AdS supergroup OSp(N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document}|4; ℝ) and apply it to develop a coset construction for AdS4∣4N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\ extrm{AdS}}^{4\\mid 4\\mathcal{N}} $$\\end{document} and the corresponding differential geometry. This realisation naturally leads to an atlas on AdS4∣4N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\ extrm{AdS}}^{4\\mid 4\\mathcal{N}} $$\\end{document} (that is a generalisation of the stereographic projection for a sphere) that consists of two charts with chiral transition functions for N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} > 0. A manifestly OSp(N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document}|4; ℝ) invariant model for a superparticle in AdS4∣4N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\ extrm{AdS}}^{4\\mid 4\\mathcal{N}} $$\\end{document} is proposed. Additionally, by employing a conformal superspace approach, we describe the most general conformally flat N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document}-extended supergeometry. This construction is then specialised to the case of AdS4∣4N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\ extrm{AdS}}^{4\\mid 4\\mathcal{N}} $$\\end{document}.

(0, 4) Projective superspaces. Part I. Interacting linear sigma models

We describe the projective superspace approach to supersymmetric models with off-shell (0, 4) supersymmetry in two dimensions. In addition to the usual superspace coordinates, projective superspace has extra bosonic variables — one doublet for each SU(2) in the R-symmetry SU(2) × SU(2) which are interpreted as homogeneous coordinates on CP1 × CP1. The superfields are analytic in the CP1 coordinates and this analyticity plays an important role in our description. For instance, it leads to stringent constraints on the interactions one can write down for a given superfield content of the model. As an example, we describe in projective superspace Witten’s ADHM sigma model — a linear sigma model with non-derivative interactions whose target is R4 with a Yang-Mills instanton solution. The hyperkähler nature of target space and the twistor description of instantons by Ward, and Atiyah, Hitchin, Drinfeld and Manin are natural outputs of our construction.

On the Role of Torsion and Higher Forms in Off‐Shell Supergravity

AbstractThe presence of a nonvanishing totally antisymmetric (super)torsion, equivalent to an axial vector, and higher forms in the “new minimal” and “old minimal” off‐shell formulations of , supergravity is elaborated. The geometric superspace approach and study both the geometric Lagrangian and the off‐shell closure of the Bianchi identities in this framework, showing how the aforementioned axial vector torsion contributes to both the new and the old minimal set of auxiliary fields is adopted. In particular, to reproduce (a variant of) the old minimal set within the geometric setup, two real auxiliary 3‐form potentials.

Structural peculiarities? Aperiodic crystals, modulated phases, composite structures

Abstract According to a general understanding, a crystal structure is defined by a lattice and the content of the unit cell of this lattice. As consequence a crystal exhibits three-dimensional periodicity with respect to the atoms. However, an increasing number of known crystal structures does not follow this idea of periodicity, but shows an aperiodic arrangement of its atoms. This group of so-called “aperiodic crystals” contains quasicrystals, modulated phases and composite structures. The latter two can be properly described within the higher-dimensional superspace approach to enable an accurate crystal-chemical analysis. Here the superspace is a mathematical tool, in which periodicity can be recovered in a higher-dimensional space. In the first part of this review the basic concept of periodic and aperiodic crystals is presented and similarities and differences of modulated phases, composite structures and quasicrystals are discussed. In a second part the higher-dimensional superspace approach is introduced in reciprocal and in direct space and the implementation of symmetry in superspace is reviewed. In the last part representative examples and the origin of aperiodicity in the crystal structures are discussed.

Superspace approach helps: determination of proton dynamics in the phase transition of modulated supramolecular ferroelectrics: 5,5′-dimethyl-2,2′-bipyridine and bromanilic acid

For the first time, the superspace approach describes the relation between four ferroelectric-paraelectric phases, by using a unique superspace group and analyzing intermolecular interactions in terms of t-plots. The proton dynamic has a dominant role in phase transitions.

Synthesis and crystal structure of Fe[SeO4]OH and prediction of polytypes in the extended R[MO4]Z family

The newly synthesized compound Fe[SeO4]OH belongs to the kieserite-like structure. Using the superspace approach, possible polytypes are predicted for R[MO4]Z, where R = Mg, Fe, Ni, Co, Zn, Mn, Al, V, Sb, Y; M = S, Se, P, As; Z = OH, H2O, F, O.

Force-Extension Curve of a Polymer Chain Entangled with a Static Ring-Shaped Obstacle.

The way to theoretically approach dynamic and static topological constraints of polymer entanglements still presents a great challenge in polymer physics. So far, only the problem of static entanglement with multiple simple objects has been solved in theory by a superspace approach in our previous work. This work is devoted to extending the superspace approach to study a polymer chain entangled with a relatively complicated object-a ring-shaped object with genus one. Taking advantage of the axial symmetry of the model setup, the 3D diffusion equations in the superspace can be numerically solved within the 2D coordinates using a specially designed alternating-direction implicit (ADI) scheme. A series of numerical calculations reveal that the topological entanglement effect of the ring will exert a topological entropy attractive force on the linear chain, which can be used to explain the viscosity-increase phenomenon observed in recent simulations and experiments. Furthermore, the influences of the ring size and the entangling modes on the topological entropy force are also investigated by examining the corresponding force-extension curves. This work, together with our previous work, might pave the path toward the complete formulation of static topological constraints.

M5-brane in the superspace approach

Motivated by Sen's spacetime prescription for the construction of theories with self-dual field strengths, we present a rigid superspace Lagrangian describing noninteracting tensor multiplets living on a stack of M5-branes and containing all the physical constraints on the fields, yielding the on-shell matching of the degrees of freedom. The geometric superspace approach adopted here offers a natural realization of superdiffeomorphisms and is particularly well suited for the coupling to supergravity. However, within this formulation the (anti-)self-duality property of the 3-form field strengths is lost when the superspace Lagrangian is trivially restricted to spacetime. We propose two main paths to address this issue: a first-order superspace extension of Sen's spacetime results, which, once trivially restricted to spacetime, yields all the dynamical equations including the (anti-)self-duality constraint on the 3-form field strengths, and a possible way to obtain a full superspace description of the theory, based on integral forms.

Unconstrained off-shell superfield formulation of 4D, mathcal{N} = 2 supersymmetric higher spins

We present, for the first time, the complete off-shell 4D, mathcal{N} = 2 superfield actions for any free massless integer spin s ≥ 2 fields, using the mathcal{N} = 2 harmonic super-space approach. The relevant gauge supermultiplet is accommodated by two real analytic bosonic superfields {h}_{alpha left(s-1right)dot{alpha}left(s-1right)}^{++} , {h}_{alpha left(s-2right)dot{alpha}left(s-2right)}^{++} and two conjugated complex analytic spinor superfields {h}_{alpha left(s-1right)dot{alpha}left(s-1right)}^{+3} , {h}_{alpha left(s-2right)dot{alpha}left(s-1right)}^{+3} , where α(s) := (α1. . . αs), dot{alpha} (s) := ( dot{alpha} 1. . . dot{alpha} s). Like in the harmonic superspace formulations of mathcal{N} = 2 Maxwell and supergravity theories, an infinite number of original off-shell degrees of freedom is reduced to the finite set (in WZ-type gauge) due to an infinite number of the component gauge parameters in the analytic superfield parameters. On shell, the standard spin content (s,s−1/2,s−1/2,s−1) is restored. For s = 2 the action describes the linearized version of “minimal” mathcal{N} = 2 Einstein supergravity.

On the Geometric Approach to the Boundary Problem in Supergravity

We review the geometric superspace approach to the boundary problem in supergravity, retracing the geometric construction of four-dimensional supergravity Lagrangians in the presence of a non-trivial boundary of spacetime. We first focus on pure N=1 and N=2 theories with negative cosmological constant. Here, the supersymmetry invariance of the action requires the addition of topological (boundary) contributions which generalize at the supersymmetric level the Euler-Gauss-Bonnet term. Moreover, one finds that the boundary values of the super field-strengths are dynamically fixed to constant values, corresponding to the vanishing of the OSp(N|4)-covariant supercurvatures at the boundary. We then consider the case of vanishing cosmological constant where, in the presence of a non-trivial boundary, the inclusion of boundary terms involving additional fields, which behave as auxiliary fields for the bulk theory, allows to restore supersymmetry. In all the cases listed above, the full, supersymmetric Lagrangian can be recast in a MacDowell-Mansouri(-like) form. We then report on the application of the results to specific problems regarding cases where the boundary is located asymptotically, relevant for a holographic analysis.

Neutron Diffraction Study of the Martensitic Transformation of Ni2.07Mn0.93Ga Heusler Alloy

The martensitic transition featuring the ternary Heusler alloy Ni2.09Mn0.91Ga was investigated by neutron diffraction. Differential scanning calorimetry indicated that structural transition starts at 230 K on cooling with a significant increase in the martensitic transformation onset compared to the classical Ni2MnGa. The low-temperature martensite presents the 5M type of modulated structure, and the structural analysis was performed by the application of the superspace approach. As already observed in Mn-rich modulated martensites, the periodical distortion corresponds to an incommensurate wave-like shift of the atomic layers. The symmetry of the modulated martensite at 220 K is orthorhombic with unit cell constants a = 4.2172(3) Å, b = 5.5482(2) Å, and c = 4.1899(2) Å; space group Immm(00γ)s00; and modulation vector q = γc* = 0.4226(5)c*. Considering the different neutron scattering lengths of the elements involved in this alloy, it was possible to ascertain that the chemical composition was Ni2.07Mn0.93Ga, close to the nominal formula. In order to characterize the martensitic transformation upon increasing the temperature, a series of neutron diffraction patterns was collected at different temperatures. The structural analysis indicated that the progressive change of the martensitic lattice is characterized by the exponential change of the c/a parameter approaching the limit value c/a = 1 of the cubic austenite.

Higher-dimensional invariants in 6D super Yang-Mills theory

We exploit the 6D, mathcal{N} = (1, 0) and mathcal{N} = (1, 1) harmonic superspace approaches to construct the full set of the maximally supersymmetric on-shell invariants of the canonical dimension d = 12 in 6D, mathcal{N} = (1, 1) supersymmetric Yang-Mills (SYM) theory. Both single- and double-trace invariants are derived. Only four single-trace and two double-trace invariants prove to be independent. The invariants constructed can provide the possible counterterms of mathcal{N} = (1, 1) SYM theory at four-loop order, where the first double-trace divergences are expected to appear. We explicitly exhibit the gauge sector of all invariants in terms of mathcal{N} = (1, 0) gauge superfields and find the absence of mathcal{N} = (1, 1) supercompletion of the F6 term in the abelian limit.

Are the St John's wort Hyp-1 superstructures different?

Two commensurately modulated structures (PDB entries 4n3e and 6sjj) were solved using translational noncrystallographic symmetry (tNCS). The data required the use of large supercells, sevenfold and ninefold, respectively, to properly index the reflections. Commensurately modulated structures can be challenging to solve. Molecular-replacement software such as Phaser can detect tNCS and either handle it automatically or, for more challenging situations, allow the user to enter a tNCS vector, which the software then uses to place the components. Although this approach has been successful in solving these types of challenging structures, it does not make it easy to understand the underlying modulation in the structure or how these two structures are related. An alternate view of this problem is that the atoms and associated parameters are following periodic atomic modulation functions (AMFs) in higher dimensional space, and what is being observed in these supercells are the points where these higher dimensional AMFs intersect physical 3D space. In this case, the two 3D structures, with a sevenfold and a ninefold superstructure, seem to be quite different. However, describing those structures within the higher dimensional superspace approach makes a strong case that they are closely related, as they show very similar AMFs and can be described with one unique (3+1)D structure, i.e. they are two different 3D intersections of the same (3+1)D structure.

Incommensurately modulated crystal structure of α'(O'3)-type sodium cobalt oxide NaxCoO2 (x ∼ 0.78).

A single-phase sample of α' (O'3)-type layered sodium cobalt oxide NaxCoO2 (x ∼ 0.78) was prepared and its incommensurately modulated crystal structure was analyzed using the (3+1)-dimensional superspace approach to the powder neutron diffraction data. The crystal structure of the cobaltate is accurately described based on the superspace group C2/m(α0γ)00, wherein the positions of Na atoms are most significantly modulated in the monoclinic a direction to form an ordered arrangement. Such a displacive modulation causes a quasi-periodic shift of Na atoms from the centers of the NaO6 polyhedra between undulated CoO2 sheets, changing the form of the NaO6 polyhedron from an octahedral coordination (O) to a trigonal prismatic (P) one, via an intermediate capped trigonal prismatic NaO7 coordination (C). At the positions where the Na atoms are most significantly shifted, the neighboring Na atoms are located at almost touching distances. However, the occupation factor of Na atoms becomes zero at such positions, yielding Na-deficient sites VNa, sandwiched either between C and P, or C and C-type polyhedra.

A pedagogical discussion of N = 1 four-dimensional supergravity in superspace

A short introduction to N = 1 supergravity in four dimensions in the superspace approach is given emphasizing on all steps to obtain the final Lagrangian. In particular, starting from geometrical principles and the introduction of superfields in curved superspace, the action coupling matter and gauge fields to supergravity is derived. This paper is based on the book [M. Rausch de Traubenberg and M. Valenzuela, A Supergravity Primer: From Geometrical Principles to the Final Lagrangian (World Scientific, 2020)] and on several lectures given at the doctoral school of Strasbourg.

Toward Understanding High-Z′ Organic Molecular Crystals through the Superspace Method: The Example of Glycyl-l-valine

The high-Z′ (Z′ = 7) structure of glycyl-l-valine has been redetermined at 298 K, using synchrotron radiation and exploiting the superspace approach. The analysis of the diffraction data reveals that the structure can be described as a commensurately modulated crystal structure with superspace group P212121(0σ20)000 and modulation wave vector q = (0, 2/7, 0). A new Z′ = 1 phase has been discovered for this compound at 323 K that is related to the known Z′ = 7 phase. The origin of the modulated phase has been explored by analyzing intermolecular interactions and molecular conformations in terms of t-plots and through a comparison to (1) the newly discovered high-temperature phase, (2) a hypothetical Z′ = 1 structure at 298 K, and (3) a similar Z′ = 1 structure of the related compound glycyl-l-leucine. The results show that the conformational flexibility of the glycyl-l-valine molecule ensures optimization of the hydrogen-bond network which causes the modulation and thus a high Z′ value of the supercell. This study highlights the elegance and convenience of the superspace approach to explain the occurrence of at least part of the rarely occurring high-Z′ structures of molecular compounds.

Influence of Stoichiometry and Aging at Operating Temperature on Thermoelectric Higher Manganese Silicides

Thermoelectric higher manganese silicides, MnSix, were synthesized by magnesioreduction followed by spark plasma sintering with different nominal compositions (x = 1.65, 1.74, and 1.80) and various postsynthesis annealing durations (0, 48, 96, and 336 h). The composite Nowotny chimney-ladder crystal structures of the resulting samples were investigated by synchrotron X-ray powder diffraction. The modulation vector component gamma, generally considered corresponding to the stoichiometry (x) of the material, was accurately determined by Rietveld refinement using a (3 + 1)D superspace approach. Regardless of the initial nominal composition, all the samples have a similar gamma similar to 1.736 after 48 h of annealing at 900 K. This result suggests that MnSix, at a temperature of 900 K, is better described as a defined compound with x close to 1.736, rather than intermediate solid-solution phases with 1.725 < x < 1.75 as predicted by the commonly accepted phase diagram. At the fixed nominal composition MnSi1.74, gamma increases significantly from 1.7313(2) to 1.7411(1) after 336 h of annealing, indicating that the thermal history influences the Si stoichiometry. The evolution of gamma with time is concomitant with a power factor drop (-19%), attributed to a decrease in charge carrier concentration. The drop of the power factor, partially compensated by a decrease in thermal conductivity, results in a -12% reduction of the maximum figure-of-merit ZT, after prolonged annealing under realistic application conditions.

Quantifying closeness between black hole spacetimes: a superspace approach

The set of all metrics that can be placed on a given manifold defines an infinite-dimensional ‘superspace’ that can itself be imbued with the structure of a Riemannian manifold. Geodesic distances between points on Met(M) measure how close two different metrics over M are to one another. Restricting our attention to only those metrics that describe physical black holes, these distances may therefore be thought of as measuring the level of geometric similarity between different black hole structures. This allows for a systematic quantification of the extent to which a black hole, possibly arising as an exact solution to a theory of gravity extending general relativity in some way, might be ‘non-Kerr’. In this paper, a detailed construction of a superspace for stationary black holes with an arbitrary number of hairs is carried out. As an example application, we are able to strengthen a recent claim made by Konoplya and Zhidenko about which deviation parameters describing a hypothetical, non-Schwarzschild black hole are likely to be most relevant for astrophysical observables.