Subgroup Of SO Research Articles

Embedding space approach to Lorentzian CFT amplitudes and causal spherical functions

Conformal field theory in a Minkowski setting is discussed in an embedding space approach, paying special attention to causality constraints for four-point amplitudes. The physics of dilatation and Lorentz boost is emphasized in specifying the noncompact maximal Abelian subgroup of SO(d,2). Reduction of a conformal field theory four-point amplitudes as functions of cross ratios is shown to be equivalent to enforcing H bi-invariance, i.e., F(hgh′)=F(g), with g∈SO(d,2) and H an appropriate subgroup. Causality is imposed by introducing appropriate semigroups. Causal zonal spherical functions are constructed, making contact with Minkowski conformal blocks introduced previously. Published by the American Physical Society 2024

Topological structures, dark matter and gravitational waves in E6

We discuss the appearance of topological structures from the spontaneous breaking of E6 to the Standard Model via its maximal subgroup SO(10) × U(1)ψ. They include dumbbells, metastable strings, as well as domain walls bounded by necklaces. We provide a novel scenario for producing metastable strings based on the symmetry breaking U(1)ψ ⟶ ℤ8 ⟶ ℤ4. The metastable string arises from the merger of ℤ8 strings that bound a domain wall. An unbroken gauge ℤ2 symmetry from SO(10) breaking yields viable stable dark matter candidates as well as topologically stable strings. We discuss the gravitational wave emission from two varieties of cosmic strings, namely the superheavy metastable ones and the intermediate scale topologically stable cosmic strings.

On the Geometry of Quantum Spheres and Hyperboloids

We study two classes of quantum spheres and hyperboloids, one class consisting of homogeneous spaces, which are ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}-quantum spaces for the quantum orthogonal group O(SOq(3))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(SO_q(3))$$\\end{document}. We construct line bundles over the quantum homogeneous space associated with the quantum subgroup SO(2) of SOq(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SO_q(3)$$\\end{document}. The line bundles are associated to the quantum principal bundle via representations of SO(2) and are described dually by finitely-generated projective modules En\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {E}_n$$\\end{document} of rank 1 and of degree computed to be an even integer -2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-2n$$\\end{document}. The corresponding idempotents, that represent classes in the K-theory of the base space, are explicitly worked out and are paired with two suitable Fredhom modules that compute the rank and the degree of the bundles. For q real, we show how to diagonalise the action (on the base space algebra) of the Casimir operator of the Hopf algebra Uq1/2(sl2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {U}_{q^{1/2}}(sl_2)}$$\\end{document} which is dual to O(SOq(3))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(SO_q(3))$$\\end{document}.

Solving the Matrix Exponential Function for Special Orthogonal Groups SO(n) up to n = 9 and the Exceptional Lie Group G2

In this work the matrix exponential function is solved analytically for the special orthogonal groups SO(n) up to n=9. The number of occurring k-th matrix powers gets limited to 0≤k≤n−1 by exploiting the Cayley–Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm V and the roots of a polynomial equation that depends on a few specific invariants. Besides the well-known case of SO(3), a quadratic equation needs to be solved for n=4,5, a cubic equation for n=6,7, and a quartic equation for n=8,9. As an interesting subgroup of SO(7), the exceptional Lie group G2 of dimension 14 is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, ξ=1. The traces of the SO(n)-matrices arising from the exponential function are sums of cosines of several angles. This feature confirms that the employed method is equivalent to exponentiation after diagonalization, but avoids complex eigenvalues and eigenvectors and operates only with real-valued quantities.

Spontaneous Emergence of a Causal Time Axis in Euclidean Space from a Gauged Rotational Symmetry Theory

We demonstrate the emergence of an effective “time” axis in the ground state of a gauged rotational symmetry theory in four-dimensional Euclidean space. In so doing, we remove the necessity of Wick rotation to Lorentz spacetime, an arbitrary and sometimes ill-defined procedure, especially for gravity-related theories. We begin by adapting the Cho-Duan-Ge decomposition to the gauge theory of the four-dimensional rotational symmetry group SO(4), where it identifies the maximal Abelian subgroup SO(2)⊗SO(2) in a gauge covariant manner. We then find the one-loop effective theory to have a stable condensate of monopoles corresponding to the reduction of SO(4) symmetry to SO(2)⊗SO(2). The construction of the condensate ensures that the four-dimensional spatial direction of its field strength must coincide with that of this embedding, and that a magnetic potential must be worked against to divert a trajectory away from this direction. Indeed, movement along this direction represents minimal potential energy. We take it to be the time direction. The gauge-dependent nature of the condensate is such that different gauge choices may lead to different time axes and we show on very general grounds that these different coordinate systems must be relatable by transformations of Lorentz form.

Tensorial Fourier expansion oforientation distribution function defined on the orthogonal group O(3).

Recently, Man and Du, in the context of classical texture analysis where the orientation distribution function (ODF) is defined on the rotation group SO(3), presented a systematic procedure by which the classical expansion of an ODF truncated at the order [Formula: see text] can be directly rewritten as a tensorial Fourier expansion truncated at the same order. In classical texture analysis, the groups of crystallite and texture symmetries are assumed to be subgroups of SO(3), which is unreasonable, say, for aggregates of crystallites in a crystal class defined by an improper point group. In this paper, we consider ODFs defined on the orthogonal group O(3) and extend the aforementioned procedure to write down tensorial Fourier expansions for polycrystals with crystallite symmetry defined by any of the 21 improper point groups. Examples that illustrate the general procedure are given. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.

Convex Hulls of Random Order Types

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3): (a) The number of extreme points in an n -point order type, chosen uniformly at random from all such order types, is on average 4+ o (1). For labeled order types, this number has average \(4- \mbox{$\frac{8}{n^2 - n +2}$}\) and variance at most 3. (b) The (labeled) order types read off a set of n points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size. Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points 2 d + o (1) and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods show the following relative of the Erdős-Szekeres theorem: for any fixed k , as n → ∞, a proportion 1 - O (1/ n ) of the n -point simple order types contain a triangle enclosing a convex k -chain over an edge. For the unlabeled case in (a), we prove that for any antipodal, finite subset of the two-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral, or one of A 4 , S 4 , or A 5 (and each case is possible). These are the finite subgroups of SO (3) and our proof follows the lines of their characterization by Felix Klein.

Homotopy classification of 4–manifolds whose fundamental group is dihedral

We show that the homotopy type of a finite oriented Poincar\'{e} 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. By combining with results of Hambleton-Kreck and Bauer, this applies in the case of smooth oriented 4-manifolds whose fundamental group is a finite subgroup of SO(3). An important class of examples are elliptic surfaces with finite fundamental group.

Geometric limits of cyclic subgroups of SO0(1,k + 1) and SU(1,k + 1)

We study geometric limits of convex-cocompact cyclic subgroups of the rank 1 groups SO_0(1, k+1) and SU(1, k+1). We construct examples of sequences of subgroups of such groups G that converge algebraically and whose geometric limit strictly contains the algebraic limit, thus generalizing the example first described by Jorgensen for subgroups of SO_0(1,3). We also give necessary and sufficient conditions for a subgroup of SO_0(1, k+1) to arise as geometric limit of a sequence of cyclic subgroups. We then discuss generalizations of such examples to sequence of representations of free groups, and applications of our constructions in that setting.

Generalization of Selberg's 3/16 theorem for convex cocompact thin subgroups of SO(n,1)

Let Γ be a convex cocompact thin subgroup of an arithmetic lattice in SO(n,1). We generalize Selberg's 316 theorem in this setting, i.e., we prove uniform exponential mixing of the frame flow and obtain a uniform resonance-free half plane for the congruence covers of the hyperbolic manifold Γ﹨Hn. This extends the work of Oh–Winter who established the n=2 case. The theorem follows from uniform spectral bounds for the congruence transfer operators with holonomy. We employ Sarkar–Winter's frame flow version of Dolgopyat's method uniformly over the congruence covers as well as Golsefidy–Varjú's generalization of Bourgain–Gamburd–Sarnak's expansion machinery by using the properties that the return trajectory subgroups are Zariski dense and have trace fields which coincide with that of Γ. These properties follow by proving that the return trajectory subgroups are of finite index in Γ.

Heisenberg Parabolic Subgroup of SO∗(10) and Invariant Differential Operators

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra so∗(10). We use the maximal Heisenberg parabolic subalgebra P=M⊕A⊕N with M=su(3,1)⊕su(2)≅so∗(6)⊕so(3). We give the main and the reduced multiplets of indecomposable elementary representations. This includes the explicit parametrization of the intertwining differential operators between the ERS. Due to the recently established parabolic relations the multiplet classification results are valid also for the algebras so(p,q) (with p+q=10, p≥q≥2) with maximal Heisenberg parabolic subalgebra: P′=M′⊕A′⊕N′, M′=so(p−2,q−2)⊕sl(2,IR), M′C≅MC.

Non‐commutative amoebas

The group of isometries of the hyperbolic space H 3 $ \mathbb {H}^3$ is the 3-dimensional group PSL 2 ( C ) $\operatorname{PSL}_2(\mathbb {C})$ , which is one of the simplest non-commutative complex Lie groups. Its quotient by the subgroup SO ( 3 ) ⊂ PSL 2 ( C ) $\operatorname{SO}(3)\subset \operatorname{PSL}_2(\mathbb {C})$ naturally maps it back to H 3 $ \mathbb {H}^3$ . Each fiber of this map is diffeomorphic to the real projective 3-space R P 3 ${\mathbb {R}}{\mathbb {P}}^3$ . The resulting map PSL 2 ( C ) → H 3 $\operatorname{PSL}_2(\mathbb {C})\rightarrow \mathbb {H}^3$ can be viewed as the simplest non-commutative counterpart of the map Log : ( C × ) n → R n $\operatorname{Log}:(\mathbb {C}^\times )^n\rightarrow \mathbb {R}^n$ from the commutative complex Lie group ( C × ) n $(\mathbb {C}^\times )^n$ with the Lagrangian torus fibers that can be considered as a Liouville–Arnold type integrable system. Gelfand, Kapranov and Zelevinsky have introduced amoebas of algebraic varieties V ⊂ ( C × ) n $V\subset (\mathbb {C}^\times )^n$ as images Log ( V ) ⊂ R n $\operatorname{Log}(V)\subset \mathbb {R}^n$ . We define the amoeba of an algebraic subvariety of PSL 2 ( C ) $\operatorname{PSL}_2(\mathbb {C})$ as its image in H 3 $ \mathbb {H}^3$ . The paper surveys basic properties of the resulting hyperbolic amoebas and compares them against the commutative amoebas R n $\mathbb {R}^n$ .

Maximal discrete subgroups of 𝑆𝑂⁺(2,𝑛+2)

We characterize the maximal discrete subgroups ofSO+(2,n+2)SO^+(2,n+2), which contain the discriminant kernel of an even lattice with two hyperbolic planes overZ\mathbb {Z}. They coincide with the normalizers inSO+(2,n+2)SO^+(2,n+2)and are given by the group of all integral matrices insideSO+(2,n+2)SO^+(2,n+2), whenever the underlying lattice is maximal even. Finally we deal with the irreducible root lattices as examples.

Relative scale separation in orbifolds of S2 and S5

In orbifold vacua containing an Sq/Γ factor, we compute the relative order of scale separation, r, defined as the ratio of the eigenvalue of the lowest-lying Γ-invariant state of the scalar Laplacian on Sq, to the eigenvalue of the lowest-lying state. For q = 2 and Γ finite subgroup of SO(3), or q = 5 and Γ finite subgroup of SU(3), the maximal relative order of scale separation that can be achieved is r = 21 or r = 12, respectively. For smooth S5 orbifolds, the maximal relative scale separation is r = 4.2. Methods from invariant theory are very efficient in constructing Γ-invariant spherical harmonics, and can be readily generalized to other orbifolds.

Towards top-down holographic composite Higgs: minimal coset from maximal supergravity

Within the context of top-down holography, we study a one-parameter family of regular background solutions of maximal gauged supergravity in seven dimensions, dimensionally reduced on a 2-torus. The dual, four-dimensional confining field theory realises the global (spontaneous as well as explicit) symmetry breaking pattern SO(5) → SO(4). We compute the complete mass spectrum for the fluctuations of the 128 bosonic degrees of freedom of the five-dimensional gravity theory, which correspond to scalar, pseudoscalar, vector, axial-vector, and tensor bound states of the dual field theory, and includes particles with exotic SO(4) quantum numbers. We confirm the existence of tachyonic instabilities near the boundaries of the parameter space.We discuss the interplay between explicit and spontaneous symmetry breaking. The SO(5)/SO(4) coset might provide a first step towards the realisation of a calculable framework and ultraviolet completion of minimal composite Higgs models, if the four pseudo-Nambu-Goldstone bosons are identified with the real components of the Higgs doublet in the standard model (SM), and a subgroup of SO(4) with the SU(2) × U(1) SM gauge group. We exhibit an example with an additional localised boundary term that mimics the effect of a weakly-coupled external sector.

Robust adaptive sliding mode tracking control for a rigid body based on Lie subgroups of SO(3)

<p style='text-indent:20px;'>This paper considers the attitude tracking control problem for a rigid body. In order to avoid the complexity and ambiguity associated with other attitude representations (such as Euler angles or quaternions), the attitude dynamics and the proposed control system are represented globally on special orthogonal groups. An adaptive controller based on a Lie subgroup of SO(3) is developed such that the rigid body can track any given attitude command asymptotically without requiring the exact knowledge of the inertia moment. In the presence of external disturbances, the adaptive controller is enhanced with an additional robust sliding mode term by following the same idea within the framework of SO(3). Finally, simulation results are presented to demonstrate efficiency of the proposed controllers.</p>

Spectral gap and exponential mixing on geometrically finite hyperbolic manifolds

Let M=Γ∖Hd+1 be a geometrically finite hyperbolic manifold with critical exponent exceeding d∕2. We obtain a precise asymptotic expansion of the matrix coefficients for the geodesic flow in L2(T1(M)), with exponential error term essentially as good as the one given by the spectral gap for the Laplace operator on L2(M) due to Lax and Phillips. Combined with the work of Bourgain, Gamburd, and Sarnak and its generalization by Golsefidy and Varjú on expanders, this implies uniform exponential mixing for congruence covers of M when Γ is a Zariski-dense subgroup contained in an arithmetic subgroup of SO∘(d,1).

The Geometrical Meaning of Spinors Lights the Way to Make Sense of Quantum Mechanics

This paper aims at explaining that a key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R3. In the three-dimensional rotation group, the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding that is acquired to derive the free-space Dirac equation from scratch, proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further, we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.

Searching for gauge theories with the conformal bootstrap

Infrared fixed points of gauge theories provide intriguing targets for the modern conformal bootstrap program. In this work we provide some preliminary evidence that a family of gauged fermionic CFTs saturate bootstrap bounds and can potentially be solved with the conformal bootstrap. We start by considering the bootstrap for SO(N) vector 4-point functions in general dimension D. In the large N limit, upper bounds on the scaling dimensions of the lowest SO(N) singlet and traceless symmetric scalars interpolate between two solutions at ∆ = D/2 − 1 and ∆ = D − 1 via generalized free field theory. In 3D the critical O(N) vector models are known to saturate the bootstrap bounds and correspond to the kinks approaching ∆ = 1/2 at large N. We show that the bootstrap bounds also admit another infinite family of kinks {mathcal{T}}_D , which at large N approach solutions containing free fermion bilinears at ∆ = D − 1 from below. The kinks {mathcal{T}}_D appear in general dimensions with a D-dependent critical N* below which the kink disappears. We also study relations between the bounds obtained from the bootstrap with SO(N) vectors, SU(N) fundamentals, and SU(N) × SU(N) bi-fundamentals. We provide a proof for the coincidence between bootstrap bounds with different global symmetries. We show evidence that the proper symmetries of the underlying theories of {mathcal{T}}_D are subgroups of SO(N), and we speculate that the kinks {mathcal{T}}_D relate to the fixed points of gauge theories coupled to fermions.

SO(9) characterization of the standard model gauge group

A recent series of works characterized the Standard Model (SM) gauge group GSM as the subgroup of SO(9) that, in the octonionic model of the latter, preserves the split O=C⊕C3 of the space of octonions into a copy of the complex plane plus the rest. This description, however, proceeded via the exceptional Jordan algebras J3(O),J2(O) and, in this sense, remained indirect. One of the goals of this paper is to provide as explicit a description as possible and also to clarify the underlying geometry. The other goal is to emphasize the role played by different complex structures in the spaces O and O2. We provide a new characterization of GSM: The group GSM is the subgroup of Spin(9) that commutes with a certain complex structure JR in the space O2 of Spin(9) spinors. The complex structure JR is parameterized by a choice of a unit imaginary octonion. This characterization of GSM is essentially octonionic in the sense that JR is restrictive because octonions are non-associative. The quaternionic analog of JR is the complex structure in the space H2 of Spin(5) spinors that commutes with all Spin(5) transformations.