Irreducible SO Research Articles

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.

A note on the topology of irreducible SO(3)-manifolds

A smooth, connected, oriented 5-manifold M is an irreducibleSO(3)-manifold if there exists a rank 3 vector bundle η over M such that the tangent bundle is isomorphic to the bundle of symmetric trace-free endomorphisms of η. In this article we give necessary and sufficient conditions for the existence of irreducible SO(3)-manifolds. We also provide some new examples of irreducible SO(3)-manifolds.

Projectors and seed conformal blocks for traceless mixed-symmetry tensors

In this paper we derive the projectors to all irreducible SO(d) representations (traceless mixed-symmetry tensors) that appear in the partial wave decomposition of a conformal correlator of four stress-tensors in d dimensions. These projectors are given in a closed form for arbitrary length $l_1$ of the first row of the Young diagram. The appearance of Gegenbauer polynomials leads directly to recursion relations in $l_1$ for seed conformal blocks. Further results include a differential operator that generates the projectors to traceless mixed-symmetry tensors and the general normalization constant of the shadow operator.

Normality of the Twistor Space of a 5-Manifold with an Irreducible SO(3)-Structure

A manifold with an irreducible SO(3)-structure is a 5-manifold M whose structure group can be reduced to the group SO(3), non-standardly imbedded in SO(5). The study of such manifolds has been initiated by Bobienski and Nurowski who, in particular, have shown that one can define four CR-structures on a twistor-like 7-dimensional space associated with M. In the present paper, it is observed that these CR-structures are induced by almost contact metric structures. The purpose of the paper is to study the problem of normality of these structures. The main result gives necessary and sufficient conditions for normality in geometric terms of the base manifold M. Examples illustrating this result are presented at the end of the paper.

Interbasis expansions for the isotropic 3D harmonic oscillator and bivariate Krawtchouk polynomials

An explicit expression for the general bivariate Krawtchouk polynomials is obtained in terms of the standard Krawtchouk and dual Hahn polynomials. The bivariate Krawtchouk polynomials occur as matrix elements of the unitary reducible representations of SO(3) on the energy eigenspaces of the three-dimensional isotropic harmonic oscillator and the explicit formula is obtained from the decomposition of these representations into their irreducible components. The decomposition entails expanding the Cartesian basis states in the spherical bases that span irreducible SO(3) representations. The overlap coefficients are obtained from the Clebsch–Gordan problem for the Lie algebra.

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

In the present paper, we consider a five-dimensional Riemannian manifold with an irreducible SO(3)-structure as an example of an abstract statistical manifold. We prove that if a five-dimensional Riemannian manifold with an irreducible SO(3)-structure is a statistical manifold of constant curvature, then the metric of the Riemannian manifold is an Einstein metric. In addition, we show that a five-dimensional Euclidean sphere with an irreducible SO(3)-structure cannot be a conjugate symmetric statistical manifold. Finally, we show some results for a five-dimensional Riemannian manifold with a nearly integrable SO(3)-structure. For example, we prove that the structure tensor of a nearly integrable SO(3)-structure on a five-dimensional Riemannian manifold is a harmonic symmetric tensor and it defines the first integral of third order of the equations of geodesics. Moreover, we consider some topological properties of five-dimensional compact and conformally flat Riemannian manifolds with irreducible SO(3)-structure.

Cosmological perturbations from a group theoretical point of view

We present a new approach to cosmological perturbations based on the theory of Lie groups and their representations. After rederiving the standard covariant formalism from SO(3) considerations, we provide a new expansion of the perturbed Friedmann–Lemaître–Robertson–Walker (FLRW) metric in terms of the irreducible representations of the Lorentz group. The resulting decomposition splits into (scalar, scalar), (scalar, vector) and (vector, vector) terms. These equations directly correspond to the standard Lifshitz classification of cosmological perturbations using scalar, vector and tensor modes which arise from the irreducible SO(3) representation of the spatial part of the metric. While the Lorentz group basis matches the underlying local symmetries of the FLRW spacetime better than SO(3), the new equations do not provide further simplification compared to the standard cosmological perturbation theory. We conjecture that this is due to the fact that the so(3, 1) ∼ su(2) × su(2) Lorentz algebra has no pair of commuting generators commuting with any of the translation group generators.

Irreducible SO(3) geometry in dimension five

We consider the nonstandard inclusion of SO(3) in SO(5) associated with a 5-dimensional irreducible representation. The tensor ϒ representing this reduction is found to be given by a ternary symmetric form with special properties. A 5-dimensional manifold ( M , g , ϒ) with Riemannian metric g and ternary form generated by such a tensor has a corresponding SO(3) structure, whose Gray-Hervella type classification is established using 𝔰𝔬(3)-valued connections with torsion. Structures with antisymmetric torsions, we call them the nearly integrable SO(3) structures, are studied in detail. In particular, it is shown that the integrable models (those with vanishing torsion) are isometric to the symmetric spaces M + = SU(3)/SO(3), M − = SL(3, R )/SO(3), M 0 = ℝ 5 . We also find all nearly integrable SO(3) structures with transitive symmetry groups of dimension d > 5 and some examples for which d = 5. Given an SO(3) structure ( M , g , ϒ), we define its “twistor space” 𝕋 to be the 𝕊 2 -bundle of those unit 2-forms on M which span ℝ 3 = 𝔰𝔬(3). The 7-dimensional twistor manifold 𝕋 is then naturally equipped with several CR and G 2 structures. The ensuing integrability conditions are discussed and interpreted in terms of the Gray-Hervella type classification.

Scaling of the Irreducible SO(3)-Invariants of Velocity Correlations in Turbulence

The scaling behavior of the SO(3) irreducible amplitudes d'n(r) of velocity structure tensors is numerically examined for Navier-Stokes turbulence. Here, l characterizes the irreducible representation by the index of the corresponding Legendre polynomial, and n denotes the tensorial rank, i.e., the order of the moment. For moments of different order n but with the same representation index l extended self-similarity (ESS) towards large scales is found. Intermittency seems to increase with l. We estimate that a crossover behavior between different inertial subrange scaling regimes in the longitudinal and transversal structure functions will hardly be detectable for achievable Reynolds numbers.

Top quark mass in supersymmetric SO(10) unification.

The successful prediction of $\sin^2\theta_W$ suggests that the effective theory beneath the GUT scale is the two-Higgs MSSM. If we further assume that the unified gauge group contains SO(10), that the two light Higgs doublets lie mostly in a single irreducible SO(10) representation, and that the $t$, $b$ and $\tau$ masses originate in renormalizable Yukawa interactions of the form $16_3 O 16_3$, then also the top quark mass can be predicted in terms of the MSSM parameters. To compute $m_t$ we present a precise analytic approximation to the solution of the 2-loop renormalization group equations, and study supersymmetric and GUT threshold corrections and the input value of the $b$ quark mass. The large ratio of top to bottom quark masses derives from a large ratio, $\tan\beta$, of Higgs vacuum expectation values. We point out that when $\tan\beta$ is large, so are certain corrections to the $b$ quark mass prediction, unless a particular hierarchy exists in the parameters of the model. With such a hierarchy, which may result from approximate symmetries, the top mass prediction depends only weakly on the spectrum. Our results may be applied to any supersymmetric SO(10)-like model as long as $\lambda_t\simeq \lambda_b\simeq\lambda_\tau$ at the GUT scale and there are no intermediate mass scales in the desert.

Symmetric spaces and strongly isotropy irreducible spaces

A strongly isotropy irreducible homogeneous space is a connected effective homogeneous space L/H where H is a compact subgroup of a Lie group L such that the identity component H 0 of H acts irreducibly on T~H(L/H). If we assume instead that H, but not necessarily H 0, acts irreducibly on Tetc(L/H), then L/H is called an isotropy irreducible homogeneous space. These spaces are interesting because by Schur's Lemma they all admit an L-invariant Einstein metric, and because they include the irreducible Riemannian symmetric spaces as a sub-family. The nonsymmetric strongly isotropy irreducible homogeneous spaces were classified independently by Manturov [M1,2,3], Wolf [Wol], and later by Kr~mer [Kr], with some minor omissions in the first two references. In addition, in [Wol] a detailed study of the geometric properties of strongly isotropy irreducible spaces was carried out. The isotropy irreducible spaces which are not strongly isotropy irreducible have recently been classified in [WZ3]. In a different direction, we present in this paper a direct proof of a beautiful observation of Wall concerning a correspondence between the compact simply connected irreducible symmetric spaces on the one hand and the compact strongly isotropy irreducible quotients of the classical groups on the other hand (cf. the remarks added in proof in [Wol, pp. 147-148]). This correspondence has a number of exceptions: certain Grassmannians and the isotropy irreducible space SO(7)/G 2, which is diffeomorphic to ~pT. These exceptions appear at first sight to be even more mysterious than the correspondence. But our main result is that there is an explanation, using general principles, of this correspondence as well as all its exceptions. In particular, this means that the classification of the non-symmetric strongly isotropy irreducible quotients of the classical groups may be read off from that of the irreducible symmetric spaces. Consider a compact simply connected irreducible symmetric space G/K, where G is the identity component of the isometry group, and K is the isotropy group,

Irreducible SO(2) extended superfields

All the irreducible SO(2) extended scalar superfields are obtained in SO(2)-invariant form. These superfields can be labelled by the eigenvalues of three Casimir operators of the enlarged algebra containing chiral transformation. They may be still more simply classified by introducing two invariant simple superspaces. Lagrangians and supersymmetry transformation rules are obtained from the explicit form of the irreducible superfields. The highest superspin multiplet is identified as the multiplet for the linearized SO(2) Weyl supergravity. The linearized SO(2) Poincaré supergravity is obtained by coupling this multiplet with the chiral superfields.