Subgroup Of SO Research Articles

CARTAN DECOMPOSITION OF THE DE SITTER GROUP SO(1,4). QUATERNION EULER ANGLES

Over the last twenty years, the de Sitter and anti-de Sitter spaces of various dimensions have become the focus of all theoretical high-energy physics. This is primarily due to the correspondence between supergravity in the five-dimensional anti-de Sitter space and supersymmetric field theory in four dimensions. The anti-de Sitter space turned out to be the most suitable manifold, on which nonperturbative results were obtained in the theory of superstrings and on which the theory of higher spin fields is naturally built. In turn, de Sitter's space is closely connected with the problems of modern cosmology, being essentially the theoretical basis of inflationary cosmology. On the other hand, de Sitter space-time and quantum field theory on this manifold are the subject of intensive study, mainly in connection with the task of constructing a quantum theory of gravitation in curved spaces. The central issue in the study of fields in the de Sitter space is a detailed analysis of a homogeneous group to this space — the de Sitter group. It is necessary to classify and describe homogeneous spaces of the group SO (1,4) up to subgroups SO (1,3) (Lorentz group), SO (4) (maximal compact subgroup of the group SO (1,4)), and SU (2). This problem is considered by determining all homogeneous spaces of the form M = SO (1,4) / H, where H is a stationary subgroup. The elements of the group are represented as the product of one-parameter matrices, which allows you to clearly see the relevance with the Euler angles in the classical three-dimensional case.

Wire billiards, the first steps

Wire billiard is defined by a smooth embedded closed curve of non-vanishing curvature k in Rn (a wire). For a class of curves, that we call nice wires, the wire billiard map is area preserving twist map of the cylinder. In this paper we are investigating whether the basic features of conventional planar billiards extend to this more general situation. In particular, we extend Lazutkin's KAM result, as well as Mather's converse KAM result, to wire billiards. We address the notion of caustics: for wire billiards, it corresponds to striction curve of the ruled surface spanned by the chords of the invariant curve. If the ruled surface is developable this is a genuine caustic. We found remarkable examples of the wires which are closed orbits of 1-parameter subgroup of SO(n). These wire billiards are totally integrable. Using the theory of interpolating Hamiltonians, we prove that the distribution of impact points of the wire becomes uniform with respect to the measure k2/3dx (where x is the arc length parameter), as the length of the chords tends to zero. Applying this result, we prove that the billiard transformation in an ellipsoid commutes with the reparameterized geodesic flow on a confocal ellipsoid: the speed of the foot point of the line tangent to a geodesic equals k−2/3, where k is the curvature of the geodesic in the ambient space. We also discuss perspectives and open problems of this new class of billiards.

Lie Group Based Type Synthesis Using Transformation Configuration Space for Reconfigurable Parallel Mechanisms With Bifurcation Between Spherical Motion and Planar Motion

Abstract This paper investigates novel reconfigurable parallel mechanisms with bifurcation between planar subgroup SE(2) and rotation subgroup SO(3) based on a transformation configuration space. Having recollected necessary theoretical fundamentals with regard to compositional submanifolds and kinematic bonds, the motion representation of the platform of reconfigurable parallel mechanisms are investigated. The transformation configuration space of a reconfigurable parallel mechanism with motion branches is proposed with respect to the fact that the intersection of Lie subgroups or submanifolds is the identity element or a non-identity element. The switch conditions of the transformation configuration space are discussed, leading to establishment of a theoretical foundation for realizing a switch of motion branches. The switch principle of reconfigurable parallel mechanisms is further investigated with respect to the common motion between SE(2) parallel-mechanism motion generators and SO(3) parallel-mechanism motion generators. Under this principle, the subchains with common motion generators are synthesized and divided into two types of generators. The first type of generators generates kinematic chains with a common intersection of three joint axes, and the second type of generators generates a common intersection of two joint axes. Following this, two types of reconfigurable parallel mechanisms with three identical subchains are synthesized, resulting in 11 varieties in which platforms can be switched between SE(2) and SO(3) after passing through the singularity configuration space.

Supersymmetric SU(5)×U(1)χ and the weak gravity conjecture

The gauge symmetry $SU(5) \times U(1)_\chi$ is the unique maximal subgroup of SO(10) which retains manifest unification at $M_{GUT}$ of the Standard Model gauge couplings, especially if low scale supersymmetry is present. The spontaneous breaking of $U(1)_\chi$ at some intermediate scale leaves unbroken a $Z_2$ symmetry which is precisely `matter' parity. This yields a stable supersymmetric dark matter particle as well as topologically stable cosmic strings. Motivated by the weak gravity conjecture we impose unification of $SU(5)$ and $U(1)_\chi$ at an ultraviolet cutoff $\Lambda \sim \alpha_\Lambda ^{1/2} M_{P} \approx 5 \times 10^{17}$ GeV, where $\alpha_\Lambda$ denotes the $SU(5)$ gauge coupling at $\Lambda$ and $M_P \approx 2.4 \times 10^{18}$ GeV is the reduced Planck Scale. The impact of dimension five operators suppressed by $\Lambda$ on gauge coupling unification, proton lifetime estimates and $b-\tau$ Yukawa unification is discussed. In particular, the gauge boson mediated proton decay into $e^+\pi^0$ can lie within the $2-\sigma$ sensitivity of HyperKamiokande. We also discuss how the intermediate scale strings may survive inflation while the $SU(5)$ monopoles are inflated away. The unbroken $Z_2$ symmetry provides an intriguing link between dark matter, black holes carrying `quantum hair' and cosmic strings.

Remarks on Texture Coefficients of Polycrystals with Improper Crystallite Symmetry

The orientation distribution function (ODF) in classical texture analysis is defined on the rotation group SO(3). For polycrystalline aggregates with crystallite symmetry defined by a crystallographic point group $G_{\mathrm{cr}}$ which is not a subgroup of SO(3), the improper group $G_{\mathrm{cr}}$ is routinely replaced by its proper peer (i.e., a subgroup of SO(3)) in the same Laue class. In this note we examine how the texture coefficients obtained from such a practice are related to their counterparts that pertain to the corresponding ODF defined on the orthogonal group O(3) as it should.

Topological Symmetries of ℝ3

If a finite group acts topologically, faithfully and orientation preservingly on R^3, then it is isomorphic to a subgroup of SO(3).

A unified leptoquark model confronted with lepton non-universality in B-meson decays

The anomalies in the B-meson sector, in particular RK(⁎) and RD(⁎), are often interpreted as hints for physics beyond the Standard Model. To this end, leptoquarks or a heavy Z′ represent the most popular SM extensions which can explain the observations. However, adding these fields by hand is not very satisfactory as it does not address the big questions like a possible embedding into a unified gauge theory. On the other hand, light leptoquarks within a unified framework are challenging due to additional constraints such as lepton flavor violation. The existing accounts typically deal with this issue by providing estimates on the relevant couplings. In this letter we consider a complete model based on the SU(4)C⊗SU(2)L⊗U(1)R gauge symmetry, a subgroup of SO(10), featuring both scalar and vector leptoquarks. We demonstrate that this setup has, in principle, all the potential to accommodate RK(⁎) and RD(⁎) while respecting bounds from other sectors usually checked in this context. However, it turns out that KL→e±μ∓ severely constraints not only the vector but also the scalar leptoquarks and, consequently, also the room for any sizeable deviations of RK(⁎) from 1. We briefly comment on the options for extending the model in order to conform this constraint. Moreover, we present a simple criterion for all-orders proton stability within this class of models.

Tractor calculus, BGG complexes, and the cohomology of cocompact Kleinian groups

For a compact, oriented, hyperbolic n-manifold (M,g), realised as M=Γ\\Hn where Γ is a torsion-free cocompact subgroup of SO(n,1), we establish and study a relationship between differential geometric cohomology on M and algebraic invariants of the group Γ. In particular for F an irreducible SO(n,1)-module, we show that the group cohomology with coefficients H•(Γ,F) arises from the cohomology of an appropriate projective BGG complex on M. This yields the geometric interpretation that H•(Γ,F) parameterises solutions to certain distinguished natural PDEs of Riemannian geometry, modulo the range of suitable differential coboundary operators. Viewed in another direction, the construction shows one way that non-trivial cohomology can arise in a BGG complex, and sheds considerable light on its geometric meaning. We also use the tools developed to give a new proof that H1(Γ,S0kRn+1)≠0 whenever M contains a compact, orientable, totally geodesic hypersurface. All constructions use another result that we establish, namely that the canonical flat connection on a hyperbolic manifold coincides with the tractor connection of projective differential geometry.

Holographic RG flows in $$N=4$$ N = 4 SCFTs from half-maximal gauged supergravity

We study four-dimensional N=4 gauged supergravity coupled to six vector multiplets with semisimple gauge groups SO(4)times SO(4), SO(3,1)times SO(3,1) and SO(4)times SO(3,1). All of these gauge groups are dyonically embedded in the global symmetry group SO(6, 6) via its maximal subgroup SO(3,3)times SO(3,3). For SO(4)times SO(4) gauge group, there are four N=4 supersymmetric AdS_4 vacua with SO(4)times SO(4), SO(4)times SO(3), SO(3)times SO(4) and SO(3)times SO(3) symmetries, respectively. These AdS_4 vacua correspond to N=4 SCFTs in three dimensions with SO(4) R-symmetry and different flavor symmetries. We explicitly compute the full scalar mass spectra at all these vacua. Holographic RG flows interpolating between these conformal fixed points are also given. The solutions describe supersymmetric deformations of N=4 SCFTs by relevant operators of dimensions Delta =1,2. A number of these solutions can be found analytically although some of them can only be obtained numerically. These results provide a rich and novel class of N=4 fixed points in three-dimensional Chern–Simons-Matter theories and possible RG flows between them in the framework of N=4 gauged supergravity in four dimensions. Similar studies are carried out for non-compact gauge groups, but the SO(4)times SO(4) gauge group exhibits a much richer structure.

Two remarks on Wall's D2 problem

AbstractIf a finite group G is isomorphic to a subgroup of SO(3), then G has the D2-property. Let X be a finite complex satisfying Wall's D2-conditions. If π1(X) = G is finite, and χ(X) ≥ 1 − def(G), then X ∨ S2 is simple homotopy equivalent to a finite 2-complex, whose simple homotopy type depends only on G and χ(X).

On topological actions of finite groups on S3

We consider orientation-preserving actions of a finite group G on the 3-sphere S3 (and also on Euclidean space R3). By the geometrization of finite group actions on 3-manifolds, if such an action is smooth then it is conjugate to an orthogonal action, and in particular G is isomorphic to a subgroup of the orthogonal group SO(4) (or of SO(3) in the case of R3). On the other hand, there are topological actions with wildly embedded fixed point sets; such actions are not conjugate to smooth actions but one would still expect that the corresponding groups G are isomorphic to subgroups of the orthogonal groups SO(4) (or of SO(3), resp.). In the present paper, we obtain some results in this direction; we prove that the only finite, nonabelian simple group with a topological action on S3, or on any homology 3-sphere, is the alternating or dodecahedral group A5 (the only finite, nonabelian simple subgroup of SO(4)), and that every finite group with a topological, orientation-preserving action on Euclidean space R3 is in fact isomorphic to a subgroup of SO(3).

ASYMPTOTIC SYMMETRIES IN 3-DIM GENERAL RELATIVITY: THE B(2,1) CASE

The ordinary Bondi—Metzner—Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian space—times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). Strongly continuous unitary irreducible representations (IRs) of B(2, 1), the analogue of B in three space—time dimensions, are analysed in the Hilbert topology. It is proved that all IRs of B(2, 1) are induced from IRs of compact ‘little groups’, which are the closed subgroups of SO(2). It is proved that all IRs of B(2, 1) are obtained by Wigner—Mackey’s inducing construction notwithstanding the fact that B(2, 1) is not locally compact in the employed Hilbert topology.

Projections of Patterson-Sullivan measures and the dichotomy of Mohammadi-Oh

Let Γ be some discrete subgroup of SO°(n + 1, R) with finite Bowen-Margulis-Sullivan measure. We study the dynamics of the Bowen-Margulis-Sullivan measure with respect to closed connected subspaces of the N component in some Iwasawa decomposition SO°(n+1, R) = KAN. We also study the dimension of projected Patterson-Sullivan measures along some fixed direction.

Counting and construction of holomorphic primary fields in free CFT4 from rings of functions on Calabi-Yau orbifolds

Counting formulae for general primary fields in free four dimensional conformal field theories of scalars, vectors and matrices are derived. These are specialised to count primaries which obey extremality conditions defined in terms of the dimensions and left or right spins (i.e. in terms of relations between the charges under the Cartan subgroup of SO(4, 2)). The construction of primary fields for scalar field theory is mapped to a problem of determining multi-variable polynomials subject to a system of symmetry and differential constraints. For the extremal primaries, we give a construction in terms of holomorphic polynomial functions on permutation orbifolds, which are shown to be Calabi-Yau spaces.

Correspondence between Einstein-Yang-Mills-Lorentz systems and dynamical torsion models

In the framework of Einstein-Yang-Mills theories, we study the gauge Lorentz group and establish a particular correspondence between this case and a certain class of theories with torsion within Riemann-Cartan space-times. This relation is specially useful in order to simplify the problem of finding exact solutions to the Einstein-Yang-Mills equations. The applicability of the method is divided into two approaches: one associated with the Lorentz group SO(1,n-1) of the space-time rotations and another one with its subgroup SO(n-2). Solutions for both cases are presented by the explicit use of this correspondence and, interestingly, for the last one by imposing on our ansatz the same kind of rotation and reflection symmetry properties as for a nonvanishing space-time torsion. Although these solutions were found in previous literature by a different approach, our method provides an alternative way to obtain them and it may be used in future research to find other exact solutions within this theory.

SO*(2N) coherent states for loop quantum gravity

A SU(2) intertwiner with N legs can be interpreted as the quantum state of a convex polyhedron with N faces (when working in 3D). We show that the intertwiner Hilbert space carries a representation of the non-compact group SO*(2N). This group can be viewed as the subgroup of the symplectic group Sp(4N,R) which preserves the SU(2) invariance. We construct the associated Perelomov coherent states and discuss the notion of semi-classical limit, which is more subtle than we could expect. Our work completes the work by Freidel and Livine [J. Math. Phys. 51, 082502 (2010) and J. Math. Phys. 52, 052502 (2011)], which focused on the U(N) subgroup of SO*(2N).

Sub-Riemannian distance on the Lie group $\operatorname {SO}_0(2,1)$

We find the distances between arbitrary elements of the Lie group SL(2) for the left invariant sub-Riemannian metric also invariant with respect to the right shifts by elements of the Lie subgroup SO(2) ⊂ SL(2), in other words, the invariant sub-Riemannian metric on the weakly symmetric space (SL(2) × SO(2))/ SO(2) of Selberg.

SO(8) fermion dynamical symmetry and strongly correlated quantum Hall states in monolayer graphene

A formalism is presented for treating strongly-correlated graphene quantum Hall states in terms of an SO(8) fermion dynamical symmetry that includes pairing as well as particle--hole generators. The graphene SO(8) algebra is isomorphic to an SO(8) algebra that has found broad application in nuclear physics, albeit with physically very different generators, and exhibits a strong formal similarity to SU(4) symmetries that have been proposed to describe high-temperature superconductors. The well-known SU(4) symmetry of quantum Hall ferromagnetism for single-layer graphene is recovered as one subgroup of SO(8), but the dynamical symmetry structure associated with the full set of SO(8) subgroup chains extends quantum Hall ferromagnetism and allows analytical many-body solutions for a rich set of collective states exhibiting spontaneously-broken symmetry that may be important for the low-energy physics of graphene in strong magnetic fields. The SO(8) symmetry permits a natural definition of generalized coherent states that correspond to symmetry-constrained Hartree--Fock--Bogoliubov solutions, or equivalently a microscopically-derived Ginzburg--Landau formalism, exhibiting the interplay between competing spontaneously broken symmetries in determining the ground state.

Asymmetric dark matter models in SO(10)

We systematically study the possibilities for asymmetric dark matter in the context of non-supersymmetric SO(10) models of grand unification. Dark matter stability in SO(10) is guaranteed by a remnant Z2 symmetry which is preserved when the intermediate scale gauge subgroup of SO(10) is broken by a {\\bf 126} dimensional representation. The asymmetry in the dark matter states is directly generated through the out-of-equilibrium decay of particles around the intermediate scale, or transferred from the baryon/lepton asymmetry generated in the Standard Model sector by leptogenesis. We systematically classify possible asymmetric dark matter candidates in terms of their quantum numbers, and derive the conditions for each case that the observed dark matter density is (mostly) explained by the asymmetry of dark matter particles.

Sub-Riemannian distance on the Lie group SL(2)

We find the distances between arbitrary elements of the Lie group SL(2) for the left invariant sub-Riemannian metric also invariant with respect to the right shifts by elements of the Lie subgroup SO(2) ⊂ SL(2), in other words, the invariant sub-Riemannian metric on the weakly symmetric space (SL(2) × SO(2))/ SO(2) of Selberg.