Nilpotent (anti-)BRST and (anti-)co-BRST symmetries in 2D non-Abelian gauge theory: Some novel observations

We discuss the nilpotent Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST and (anti-)co-BRST symmetry transformations and derive their corresponding conserved charges in the case of a two (1[Formula: see text]+[Formula: see text]1)-dimensional (2D) self-interacting non-Abelian gauge theory (without any interaction with matter fields). We point out a set of novel features that emerge out in the BRST and co-BRST analysis of the above 2D gauge theory. The algebraic structures of the symmetry operators (and corresponding conserved charges) and their relationship with the cohomological operators of differential geometry are established too. To be more precise, we demonstrate the existence of a single Lagrangian density that respects the continuous symmetries which obey proper algebraic structure of the cohomological operators of differential geometry. In the literature, such observations have been made for the coupled (but equivalent) Lagrangian densities of the 4D non-Abelian gauge theory. We lay emphasis on the existence and properties of the Curci–Ferrari (CF)-type restrictions in the context of (anti-)BRST and (anti-)co-BRST symmetry transformations and pinpoint their key differences and similarities. All the observations, connected with the (anti-)co-BRST symmetries, are completely novel.

Polytypism of Compounds with the General Formula Cs{Al2[TP6O20]} (T = B, Al): OD (Order-Disorder) Description, Topological Features, and DFT-Calculations

The crystal structures of compounds with the general formula Cs{[6]Al2[[4]TP6O20]} (where T = Al, B) display order−disorder (OD) character and can be described using the same OD groupoid family. Their structures are built up by two kinds of nonpolar layers, with the layer symmetries Pc(n)2 (L2n+1-type) and Pc(a)m (L2n-type) (category IV). Layers of both types (L2n and L2n+1) alternate along the b direction and have common translation vectors a and c (a ~ 10.0 Å, c ~ 12.0 Å). All ordered polytypes as well as disordered structures can be obtained using the following partial symmetry operators that may be active in the L2n type layer: the 21 screw axis parallel to c [– – 21] or inversion centers and the 21 screw axis parallel to a [21 – –]. Different sequences of operators active in the L2n type layer ([– – 21] screw axes or inversion centers and [21 – –] screw axes) define the formation of multilayered structures with the increased b parameter, which are considered as non-MDO polytypes. The microporous heteropolyhedral MT-frameworks are suitable for the migration of small cations such as Li+, Na+ Ag+. Compounds with the general formula Rb{[6]M3+[[4]T3+P6O20]} (M = Al, Ga; T = Al, Ga) are based on heteropolyhedral MT-frameworks with the same stoichiometry as in Cs{[6]Al2[[4]TP6O20]} (where T = Al, B). It was found that all the frameworks have common natural tilings, which indicate the close relationships of the two families of compounds. The conclusions are supported by the DFT calculation data.

Conformal bridge in a cosmic string background

Hidden symmetries of non-relativistic $$ \mathfrak{so}\left(2,1\right)\cong \mathfrak{sl}\left(2,\mathrm{\mathbb{R}}\right) $$ so 2 1 ≅ sl 2 ℝ invariant systems in a cosmic string background are studied using the conformal bridge transformation. Geometric properties of this background are analogous to those of a conical surface with a deficiency/excess angle encoded in the “geometrical parameter” α, determined by the linear positive/negative mass density of the string. The free particle and the harmonic oscillator on this background are shown to be related by the conformal bridge transformation. To identify the integrals of the free system, we employ a local canonical transformation that relates the model with its planar version. The conformal bridge transformation is then used to map the obtained integrals to those of the harmonic oscillator on the cone. Well-defined classical integrals in both models exist only at α = q/k with q, k = 1, 2, . . ., which for q > 1 are higher-order generators of finite nonlinear algebras. The systems are quantized for arbitrary values of α; however, the well-defined hidden symmetry operators associated with spectral degeneracies only exist when α is an integer, that reveals a quantum anomaly.

Algebra of Symmetry Operators for Klein-Gordon-Fock Equation

All external electromagnetic fields in which the Klein-Gordon-Fock equation admits the first-order symmetry operators are found, provided that in the space-time V4 a group of motion G3 acts simply transitively on a non-null subspace of transitivity V3. It is shown that in the case of a Riemannian space Vn, in which the group Gr acts simply transitively, the algebra of symmetry operators of the n-dimensional Klein-Gordon-Fock equation in an external admissible electromagnetic field coincides with the algebra of operators of the group Gr.

Z2 Topological Order and Topological Protection of Majorana Fermion Qubits

The Kitaev chain model exhibits topological order that manifests as topological degeneracy, Majorana edge modes and Z2 topological invariant of the bulk spectrum. This model can be obtained from a transverse field Ising model(TFIM) using the Jordan–Wigner transformation. TFIM has neither topological degeneracy nor any edge modes. Topological degeneracy associated with topological order is central to topological quantum computation. In this paper, we explore topological protection of the ground state manifold in the case of Majorana fermion models which exhibit Z2 topological order. We show that there are at least two different ways to understand this topological protection of Majorana fermion qubits: one way is based on fermionic mode operators and the other is based on anti-commuting symmetry operators. We also show how these two different ways are related to each other. We provide a very general approach to understanding the topological protection of Majorana fermion qubits in the case of lattice Hamiltonians. We then show how in topological phases in Majorana fermion models gives rise to new braid group representations. So, we give a unifying and broad perspective of topological phases in Majorana fermion models based on anti-commuting symmetry operators and braid group representations of Majorana fermions as anyons.

Extension of Hall symbols of crystallographic space groups to magnetic space groups

The Hall symbols for describing unambiguously the generators of space groups have been extended to describe any setting of the 1651 types of magnetic space groups (Shubnikov groups). A computer program called MHall has been developed for parsing the Hall symbols, generating the full list of symmetry operators and identifying the transformation to the standard setting.

Lie Symmetry Analysis, Exact Solutions, and Conservation Laws for the Generalized Time-Fractional KdV-Like Equation

In this paper, the problem of constructing the Lie point symmetries group of the nonlinear partial differential equation appeared in mathematical physics known as the generalized KdV-Like equation is discussed. By using the Lie symmetry method for the generalized KdV-Like equation, the point symmetry operators are constructed and are used to reduce the equation to another fractional ordinary differential equation based on Erdélyi-Kober differential operator. The symmetries of this equation are also used to construct the conservation Laws by applying the new conservation theorem introduced by Ibragimov. Furthermore, another type of solutions is given by means of power series method and the convergence of the solutions is provided; also, some graphics of solutions are plotted in 3D.