Grassmann Coordinates Research Articles

A foray on SCFT3 via super spinor-helicity and Grassmann twistor variables

In this paper, we develop a momentum super space formalism 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} = 1, 2 superconformal field theories in three dimensions. First, we solve for super-correlators in the usual momentum superspace variables. However, we found that expressing quantities in super space spinor helicity variables greatly simplifies the analysis. Further, by performing a “half” Fourier transform of the Grassmann coordinates which is analogous to the Twistor transform, an even more remarkable simplification occurs. Using this formalism, we first compute all three point correlation functions involving conserved super-currents with arbitrary spins in 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} = 1, 2 theories. We discover interesting double copy relations in 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} = 1 super-correlators. Further, we discovered super double copy relations that take us from 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} = 1 to 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} = 2 super-correlators. We also extend our results to correlators involving scalars as well as to higher points. Further, we comment on the connection of our results with the flat space super amplitudes in one higher dimension.

Grassmann 공간에서 2-입력, 2-출력, 4차 시스템의 정적출력궤한 극배치

In this paper, a parametric study of pole placement of two-input, two-output, 4SUPth/SUP order systems with strictly proper transfer functions is investigated in real Grassmann space. To characterize the features of pole placement, the well-known Plücker matrix formula is introduced as a real constant matrix formula constructed from the static output feedback (SOF) variable’s coefficient functions of characteristic polynomial under Grassmann coordinates. It is shown that the SOF pole placement entirely depends upon the columns’ condition of a real Grassmann parameter. The results presented in this paper shows that this real Grassmannian parametrization method can generalize the existing pole placement methods for two-input, two-output, 4SUPth/SUP order systems.

Extended BRST–anti-BRST transformations in non-Abelian self-dual Chern–Simons coupling theory

We establish the extended BRST and anti-BRST transformations for non-Abelian relativistic self-dual Chern–Simons theory coupled to scalar fields within the Batalin–Vilkovisky framework. Based on these constructions, we describe the total effective extended BRST-invariant Lagrangian density of the coupling theory by means of superfields with one Grassmann coordinates. Moreover, we need two fermionic variables to reexpress the extended BRST–anti-BRST-invariant Lagrangian in the superspace.

A novel approach to quantum gravity in the presence of matter without the problem of time

An approach to the quantization of gravity in the presence of matter is examined which starts from the classical Einstein–Hilbert action and matter approximated by “point” particles minimally coupled to the metric. Upon quantization, the Hamilton constraint assumes the form of the Schrödinger equation: it contains the usual Wheeler–DeWitt term and the term with the time derivative of the wave function. In addition, the wave function also satisfies the Klein–Gordon equation, which arises as the quantum counterpart of the constraint among particles’ momenta. Comparison of the novel approach with the usual one in which matter is represented by scalar fields is performed, and shown that those approaches do not exclude, but complement each other. In final discussion it is pointed out that the classical matter could consist of superparticles or spinning particles, described by the commuting and anticommuting Grassmann coordinates, in which case spinor fields would occur after quantization.

Constructing phase space distributions with internal symmetries

We discuss an ab initio world-line approach to constructing phase space distributions in systems with internal symmetries. Starting from the Schwinger-Keldysh real time path integral in quantum field theory, we derive the most general extension of the Wigner phase space distribution to include color and spin degrees of freedom in terms of dynamical Grassmann variables. The corresponding Liouville distribution for colored particles, which obey Wong's equation, has only singlet and octet components, while higher moments are fully constrained by the Grassmann algebra. The extension of phase space dynamics to spin is represented by a generalization of the Pauli-Lubanski vector; its time evolution via the Bargmann-Michel-Telegdi equation also follows from the phase space trajectories of the underlying Grassmann coordinates. Our results for the Liouville phase space distribution in systems with both spin and color are of interest in fields as diverse as chiral fluids, finite temperature field theory and polarized parton distribution functions. We also comment on the role of the chiral anomaly in the phase space dynamics of spinning particles.

R-current three-point functions in 4d mathcal{N} = 1 superconformal theories

In 4d mathcal{N} = 1 superconformal field theories (SCFTs) the R-symmetry current, the stress-energy tensor, and the supersymmetry currents are grouped into a single object, the Ferrara-Zumino multiplet. In this work we study the most general form of three-point functions involving two Ferrara-Zumino multiplets and a third generic multiplet. We solve the constraints imposed by conservation in superspace and show that non-trivial solutions can only be found if the third multiplet is R-neutral and transforms in suitable Lorentz representations. In the process we give a prescription for counting independent tensor structures in superconformal three-point functions. Finally, we set the Grassmann coordinates of the Ferrara-Zumino multiplets to zero and extract all three-point functions involving two R-currents and a third conformal primary. Our results pave the way for bootstrapping the correlation function of four R-currents in 4d mathcal{N} = 1 SCFTs.

Nambu–Poisson bracket on superspace

We propose an extension of [Formula: see text]-ary Nambu–Poisson bracket to superspace [Formula: see text] and construct by means of superdeterminant a family of Nambu–Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace [Formula: see text]. We prove in the case of the superspaces [Formula: see text] and [Formula: see text] that our [Formula: see text]-ary bracket, defined with the help of superdeterminant, satisfies the conditions for [Formula: see text]-ary Nambu–Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov–Jacobi identity (fundamental identity). We study the structure of [Formula: see text]-ary bracket defined with the help of superdeterminant in the case of superspace [Formula: see text] and show that it is the sum of usual [Formula: see text]-ary Nambu–Poisson bracket and a new [Formula: see text]-ary bracket, which we call [Formula: see text]-bracket, where [Formula: see text] is the product of two odd degree smooth functions.

Generalization of Nambu–Hamilton Equation and Extension of Nambu–Poisson Bracket to Superspace

We propose a generalization of the Nambu–Hamilton equation in superspace R 3 | 2 with three real and two Grassmann coordinates. We construct the even degree vector field in the superspace R 3 | 2 by means of the right-hand sides of the proposed generalization of the Nambu–Hamilton equation and show that this vector field is divergenceless in superspace. Then we show that our generalization of the Nambu–Hamilton equation in superspace leads to a family of ternary brackets of even degree functions defined with the help of a Berezinian. This family of ternary brackets is parametrized by the infinite dimensional group of invertible second order matrices, whose entries are differentiable functions on the space R 3 . We study the structure of the ternary bracket in a more general case of a superspace R n | 2 with n real and two Grassmann coordinates and show that for any invertible second order functional matrix it splits into the sum of two ternary brackets, where one is the usual Nambu–Poisson bracket, extended in a natural way to even degree functions in a superspace R n | 2 , and the second is a new ternary bracket, which we call the Ψ -bracket, where Ψ can be identified with an invertible second order functional matrix. We prove that the ternary Ψ -bracket as well as the whole ternary bracket (the sum of the Ψ -bracket with the usual Nambu–Poisson bracket) is totally skew-symmetric, and satisfies the Leibniz rule and the Filippov–Jacobi identity ( Fundamental Identity).

Abelian 1-Form Gauge Theory in Very Special Relativity

We study an Abelian 1-form gauge theory in the context of very special relativity. The very special relativity regards subgroup of Lorentz group as fundamental symmetry of the nature than the full Lorentz group. The rotational symmetry of the Lorentz symmetry breaks down by chosing a privileged direction in spacetime. We discuss the BRST invariant and anti-BRST invariant superspace formulations for such BV actions. The VSR modified BRST invariant BV action corresponding to the Abelian 1-form gauge theory can be written manifestly covariant manner in a superspace with one Grassmann coordinate. Moreover, we can write down the the VSR modified extended BRST and extended anti-BRST invariant BV actions in a superspace with two Grassmann coordinates. The results thus obtained is in complete agreement with the Lorentz invariant theory.

Lorentz violating p-form gauge theories in superspace

Very special relativity (VSR) keeps the main features of special relativity but breaks rotational invariance due to an intrinsic preferred direction. We study the VSR-modified extended BRST and anti-BRST symmetry of the Batalin–Vilkovisky (BV) actions corresponding to the p=1,2,3-form gauge theories. Within the VSR framework, we discuss the extended BRST invariant and extended BRST and anti-BRST invariant superspace formulations for these BV actions. Here we observe that the VSR-modified extended BRST invariant BV actions corresponding to the p=1,2,3-form gauge theories can be written in a manifestly covariant manner in a superspace with one Grassmann coordinate. Moreover, two Grassmann coordinates are required to describe the VSR-modified extended BRST and extended anti-BRST invariant BV actions in a superspace. These results are consistent with the Lorentz-invariant (special relativity) formulation.

New approach to curved projective superspace

We present a new formulation of curved projective superspace. The 4D N=2 supermanifold M^{4|8} (four bosonic and eight Grassmann coordinates) is extended by an auxiliary SU(2) manifold, which involves introducing a vielbein and related connections on the full M^{7|8} = M^{4|8} x SU(2). Constraints are chosen so that it is always possible to return to the central gauge where the auxiliary SU(2) manifold largely decouples from the curved manifold M^{4|8} describing 4D N=2 conformal supergravity. We introduce the relevant projective superspace action principle in the analytic subspace of M^{7|8} and construct its component reduction in terms of a five-form J living on M^4 x C, with C a contour in SU(2). This approach is inspired by and generalizes the original approach taken in arXiv:0805.4683 and related works, which can be identified with a complexified version of the central gauge of the formulation presented here.

New formulation of the type IIB superstring action in AdS5 × S 5

Previous studies of the type IIB superstring in an AdS_5×S^5 background are based on a description of the superspace geometry as the quotient space PSU(2,2|4)/SO(4,1)×SO(5). This paper develops an alternative approach in which the Grassmann coordinates provide a nonlinear realization of PSU(2,2|4) based on the quotient space PSU(2,2|4)/SU(2,2)×SU(4), and the bosonic coordinates are described as a submanifold of SU(2,2)×SU(4). This formulation keeps all bosonic symmetries manifest, and it provides the complete dependence on the Grassmann coordinates in terms of simple analytic expressions. It is used to construct the superstring world-sheet action in a form in which the PSU(2,2|4) symmetry is manifest and kappa symmetry can be established. This formulation might have some advantages compared to previous ones, but this remains to be demonstrated.

A Superspace Description of Chern-Simons Theory in Batalin-Vilkovisky Formulation

We discuss the extended BRST and anti-BRST symmetry (including shift symmetry) in the Batalin-Vilkovisky (BV) formulation for the Chern-Simons (CS) theories in (2 + 1) spacetime dimensions. Further we develop the superspace description of BV formulation for such theories. Interestingly, the extended BRST invariant CS theories can be described in superspace in covariant manner with the help of one more (Grassmann) coordinate. However, a superspace with two Grassmann coordinates are required for a manifestly covariant formulation of the extended BRST and extended anti-BRST invariant actions for these theories.

Perturbative quantum gravity in Batalin–Vilkovisky formalism

In this Letter we consider the perturbative quantum gravity on the super-manifold which remains invariant under absolutely anticommuting BRST and anti-BRST transformations. In addition to that the theory posses one more symmetry known as shift symmetry. The BRST invariant Batalin–Vilkovisky (BV) action for perturbative quantum gravity is realized as a translation in Grassmann coordinate. Further we show that the quantum master equation of the BV quantization method at one-loop order can be translated to have a superfield structure for the action. However, the BRST as well as anti-BRST invariant BV action is constructed in superspace with the help of two Grassmann coordinates.

The BV Formalization of Chern—Simons Theory on Deformed Superspace

In this paper we will study non-abelian Chern—Simons theory on a deformed superspace. We will deform the superspace in such a way that it includes the noncommutativity between bosonic and fermionic coordinates. We will first analyse the BRST and the anti-BRST symmetries of the Chern—Simons theory on this deformed superspace. Then we will analyse the extended BRST and the extended anti-BRST symmetries of this theory in the Batalin—Vilkovisky (BV) formalism. Finally, we will express these extended BRST and extended anti-BRST symmetries in extended superspace formalism by introducing new Grassmann coordinates.

BV formulation of higher form gauge theories in a superspace

We discuss the extended BRST and anti-BRST symmetry (including shift symmetry) in the Batalin-Vilkovisky (BV) formulation for two and three form gauge theories. Further we develop the superspace formulation for the BV actions for these theories. We show that the extended BRST invariant BV action for these theories can be written manifestly covariant manner in a superspace with one Grassmann coordinate. On the hand a superspace with two Grassmann coordinates are required for a manifestly covariant formulation of the extended BRST and extended anti-BRST invariant BV actions for higher form gauge theories.

Conformal anomaly of super Wilson loop

Classically supersymmetric Wilson loop on a null polygonal contour possesses all symmetries required to match it onto non-MHV amplitudes in maximally supersymmetric Yang–Mills theory. However, to define it quantum mechanically, one is forced to regularize it since perturbative loop diagrams are not well defined due to presence of ultraviolet divergences stemming from integration in the vicinity of the cusps. A regularization that is adopted by practitioners by allowing one to use spinor helicity formalism, on the one hand, and systematically go to higher orders of perturbation theory is based on a version of dimensional regularization, known as Four-Dimensional Helicity scheme. Recently it was demonstrated that its use for the super Wilson loop at one loop breaks both conformal symmetry and Poincaré supersymmetry. Presently, we exhibit the origin for these effects and demonstrate how one can undo this breaking. The phenomenon is alike the one emerging in renormalization group mixing of conformal operators in conformal theories when one uses dimensional regularization. The rotation matrix to the diagonal basis is found by means of computing the anomaly in the Ward identity for the conformal boost. Presently, we apply this ideology to the super Wilson loop. We compute the one-loop conformal anomaly for the super Wilson loop and find that the anomaly depends on its Grassmann coordinates. By subtracting this anomalous contribution from the super Wilson loop we restore its interpretation as a dual description for reduced non-MHV amplitudes which are expressed in terms of superconformal invariants.

Generating higher‐derivative couplings in 𝒩 = 2 supergravity

AbstractUsing a recently developed off‐shell formulation for general 4D 𝒩 = 2 supergravity‐matter systems, we propose a construction to generate higher derivative couplings. We address here mainly the interactions of tensor and vector multiplets, but the construction is quite general. For a certain subclass of terms, the action is naturally written as an integral over 3/4 of the Grassmann coordinates of superspace.

An Invariant for Hypersurfaces in Prime Characteristic

A hypersurface of order $(n+1)(p^h-1)$ in projective space of dimension $n$ of prime characteristic $p$ has an invariant monomial. This implies that a hypersurface of order $(n+1)(p^h-1)-1$ determines an invariant point. A hypersurface of order $d<n+1$ in a projective space of dimension $n$ of characteristic two has an invariant set of subspaces of dimension $d-1$ determined by one linear condition on the Grassmann coordinates of the dual subspaces.

Singularity analysis of 5-RPUR parallel mechanisms (3T2R)

Singularities of parallel mechanisms are related to the regularity of certain Jacobian matrices whose rank deficiency makes the mechanisms lose their inherent rigidity. This paper presents the results of a detailed investigation of the singular configurations of 5-degree-of-freedom parallel mechanisms performing all three translations and two independent rotations. The general architecture of the mechanism—arising from the type synthesis of symmetrical 5-degree-of-freedom parallel mechanisms—is comprised of a mobile platform attached to a base through five identical revolute–prismatic–universal revolute-jointed serial kinematic chains. From the results of screw theory, it is demonstrated that the Jacobian matrix of the above mechanism is constituted by six Plucker lines, a special case of Grassmann coordinates. This channels us to use the so-called Grassmann line geometry instead of applying a classical linear algebra approach. Grassmann line geometry can be regarded as a classification of the degeneration of the Plucker line set. Moreover, six simplified designs are proposed for which their singular configurations can be predicted by means of the Grassmann line geometry. The principles of this study can also be applied to other types of symmetrical 5-degree-of-freedom parallel mechanisms.