Representations Of SO Research Articles

Partial parameterization of orthogonal wavelet matrix filters

In this paper we propose a procedure which allows the construction of a large family of FIR d×d matrix wavelet filters by exploiting the one-to-one correspondence between QMF systems and orthogonal operators which commute with the shifts by two. A characterization of the class of filters of full rank type that can be obtained with such procedure is given. In particular, we restrict our attention to a special construction based on the representation of SO(2d) in terms of the elements of its Lie algebra. Explicit expressions for the filters in the case d=2 are given, as a result of a local analysis of the parameterization obtained from perturbing the Haar system.

Higgs low-energy theorem (and its corrections) in composite models

Abstract The Higgs low-energy theorem gives a simple and elegant way to estimate the couplings of the Higgs boson to massless gluons and photons induced by loops of heavy particles. We extend this theorem to take into account possible nonlinear Higgs interactions as well as new states resulting from a strong dynamics at the origin of the breaking of the electroweak symmetry. We show that, while it approximates with an accuracy of order a few percents single Higgs production, it receives corrections of order 50% for double Higgs production. A full one-loop computation of the gg → hh cross section is explicitly performed in MCHM5, the minimal composite Higgs model based on the SO(5)/SO(4) coset with the Standard Model fermions embedded into the fundamental representation of SO(5). In particular we take into account the contributions of all fermionic resonances, which give sizeable (negative) corrections to the result obtained considering only the Higgs nonlinearities. Constraints from electroweak precision and flavor data on the top partners are analyzed in detail, as well as direct searches at the LHC for these new fermions called to play a crucial role in the electroweak symmetry breaking dynamics.

Half-maximal supergravity in three dimensions: supergeometry, differential forms and algebraic structure

The half-maximal supergravity theories in three dimensions, which have local $SO(8)\xz SO(n)$ and rigid SO(8,n) symmetries, are discussed in a superspace setting starting from the superconformal theory. The on-shell theory is obtained by imposing further constraints; it is essentially a non-linear sigma model that induces a Poincar\'e supergeometry. The deformations of the geometry due to gauging are briefly discussed. The possible $p$-form field strengths are studied using supersymmetry and SO(8,n) symmetry. The set of such forms obeying consistent Bianchi identities constitutes a Lie super co-algebra while the demand that these identities admit solutions places a further constraint on the possible representations of SO(8,n) that the forms transform under which can be easily understood using superspace cohomology. The dual Lie superalgebra can then be identified as the positive sector of a Borcherds superalgebra that extends the Lie algebra of the duality group. In addition to the known $p=2,3,4$ forms, which we construct explicitly, there are five-forms that can be non-zero in supergravity, while all forms with $p>5$ vanish. It is shown that some six-forms can have non-trivial contributions at order $\a'$.

Charging the superconformal index

The superconformal index is an important invariant of superconformal field theories. In this note we refine the superconformal index by inserting the charge conjugation operator C. We construct a matrix integral for this charged index for N=4 SYM with SU(N) gauge group. The key ingredient for the construction is a "charged character," which reduces to Tr(C) for singlet representations of the gauge group. For each irreducible real SU(N) representation, we conjecture that this charged character is equal to the standard character for a corresponding representation of SO(N+1) or SP(N-1), for N even or odd respectively. The matrix integral for the charged index passes tests for small N and for N -> infinity. Like the ordinary superconformal index, for N=4 SYM the charged index is independent of N in the large-N limit.

The Chow ring of the classifying space [formula omitted

We compute the Chow ring of the classifying space BSO ( 2 n , C ) in the sense of Totaro using the fibration Gl ( 2 n ) / SO ( 2 n ) → BSO ( 2 n ) → BGl ( 2 n ) and a computation of the Chow ring of Gl ( 2 n ) / SO ( 2 n ) in a previous paper. We find this Chow ring is generated by Chern classes and a characteristic class defined by Edidin and Graham which maps to 2 n − 1 times the Euler class under the usual class map from the Chow ring to ordinary cohomology. Moreover, we show this class represents 1 / 2 n − 1 ( n − 1 ) ! times the nth Chern class of the representation of SO ( 2 n ) whose highest weight vector is twice that of the half-spin representation.

On the effect of resonances in composite Higgs phenomenology

We consider a generic composite Higgs model based on the coset SO(5)/SO(4) and study its phenomenology beyond the leading low-energy effective lagrangian approximation. Our basic goal is to introduce in a controllable and simple way the lowest-lying, possibly narrow, resonances that may exist is such models. We do so by proposing a criterion that we call partial UV completion. We characterize the simplest cases, corresponding respectively to a scalar in either singlet or tensor representation of SO(4) and to vectors in the adjoint of SO(4). We study the impact of these resonances on the signals associated to high-energy vector boson scattering, pointing out for each resonance the characteristic patterns of depletion and enhancement with respect to the leading-order chiral lagrangian. En route we derive the O(p 4) general chiral lagrangian and discuss its peculiar accidental and approximate symmetries.

Symmetry and mass degeneration in multi-Higgs-doublet models

We investigate possible symmetry properties of the scalar sector of Multi-Higgs-Doublet Models, and, to some extent, the generalization of such models to gauge groups other than SU(2) L × U(1) Y . In models with C (charge conjugation) invariance, and where certain quartic terms are not present, the scalar potential is invariant under a group larger than the gauge group, O(4) when the Higgs fields are doublets. If the Higgs fields develop aligned vacuum expectation values, this symmetry will break to an O(3) subgroup, which in general is further broken by loop corrections involving the gauge bosons. Assuming such corrections are small, the physical properties of the Higgs sector will approximately organize into representations of SO(3). If the vacuum expectation values of the Higgs fields are aligned in the direction of the C even fields, the mass spectra of the charged and C odd sectors will be degenerate. Moreover, if the Higgs fields develop a pair of non-aligned vacuum expectation values, so that the charge conjugation symmetry is spontaneously broken (but not the U(1) electromagnetic gauge invariance), a pair of light charged pseudo-Goldstone bosons will appear.

Spectral Theory of the Atiyah–Patodi–Singer Operator on Compact Flat Manifolds

We study the spectral theory of the Dirac-type boundary operator \(\mathcal{D}\) defined by Atiyah, Patodi, and Singer, acting on smooth even forms of a compact flat Riemannian manifold M. We give an explicit formula for the multiplicities of the eigenvalues of \(\mathcal{D}\) in terms of values of characters of exterior representations of SO(n), where n=dim M. As a consequence, we give large families of \(\mathcal{D}\)-isospectral flat manifolds that are non-homeomorphic to each other. Furthermore, we derive expressions for the eta series in terms of special values of Hurwitz zeta functions and, as a result, we obtain a simple explicit expression of the eta invariant.

Nonrelativistic counterparts of twistors and the realizations of Galilean conformal algebra

Using the notion of Galilean conformal algebra (GCA) in arbitrary space dimension d, we introduce for d=3 quantized nonrelativistic counterpart of twistors as the spinorial representation of SO(2,1)⊕SO(3) which is the maximal semisimple subalgebra of three-dimensional GCA. The GC-covariant quantization of such nonrelativistic spinors, which shall be called also Galilean twistors, is presented. We consider for d=3 the general spinorial matrix realizations of GCA, which are further promoted to quantum-mechanical operator representations, expressed as bilinears in quantized Galilean twistors components. For arbitrary Hermitian quantum-mechanical Galilean twistor realizations we obtain the result that the representations of GCA with positive-definite Hamiltonian do not exist. For non-positive H we construct for N⩾2 the Hermitian Galilean N-twistor realizations of GCA; for N=2 such realization is provided explicitly.

On the topology and the geometry of SO(3)-manifolds

Consider the non-standard embedding of SO(3) into SO(5) given by the five-dimensional irreducible representation of SO(3), henceforth called SO(3)ir. In this note, we study the topology and the differential geometry of five-dimensional Riemannian manifolds carrying such an SO(3)ir structure, i.e., with a reduction of the frame bundle to SO(3)ir.

Green's functions and non-singlet glueballs on deformed conifolds

We study the Laplacian on Stenzel spaces (generalized deformed conifolds), which are tangent bundles of spheres endowed with Ricci flat metrics. The (2d − 2)-dimensional Stenzel space has SO(d) symmetry and can be embedded in through the equation ∑di = 1zi2 = ϵ2. We discuss Green's function with a source at a point on the Sd − 1 zero section of TSd − 1. Its calculation is complicated by mixing between different harmonics with the same SO(d) quantum numbers due to the explicit breaking by the ϵ-deformation of the U(1) symmetry that rotates zi by a phase. A similar mixing affects the spectrum of normal modes of warped deformed conifolds that appear in gauge/gravity duality. We solve the mixing problem numerically to determine certain bound state spectra in various representations of SO(d) for the d = 4 and d = 5 examples.

Discrete components of some complementary series

We show that Complementary Series representations of SO(n,1), which are sufficiently close to a cohomological repre- sentation contain discretely, Complementary Series of SO(m,1) also sufficiently close to cohomological representations, provided that the degree of the cohomological representation does not ex- ceed m/2. We prove, as a consequence, that the cohomological represen- tation of degree i of the group SO(n,1) contains discretely, the cohomological representation of degree i of the subgroup SO(m,1) if i ≤ m/2. As a global application, we show that if G/Q is a semisimple algebraic group such that G(R) = SO(n,1) up to compact fac- tors, and if we assume that for all n, the tempered cohomologi- cal representations are not limits of complementary series in the automorphic dual of SO(n,1), then for all n, non-tempered coho- mological representations are isolated in the automorphic dual of G. This reduces conjectures of Bergeron to the case of tempered cohomological representations.

Formula omitted], generalised space–time and IIA string theory

As advocated in hep-th/0307098 we construct the non-linear realisation of the semi-direct product of E11 and its first fundamental representation at lowest level from the IIA viewpoint. We find a theory that is SO(10,10)⊗GL(1) invariant and contains the fields of gravity, a two form and a dilaton but which depend on coordinates which belong to the vector representation of SO(10,10). The resulting Lagrangian agrees that of recent work on the so-called doubled field theory. However, the construction given in this Letter is straightforward and systematic. It also reveals the relevant underlying symmetries and opens the way to include the Ramond–Ramond, and higher level, fields together with additional coordinates of the generalised space–time.

Plasma transport coefficients: An application of SO(3) group

The appearance of anisotropy in certain properties such as conductivity and diffusion coefficients could be induced by the application of external magnetic fields. Using the matricial representation of SO(3) group it was possible to demonstrate that this anisotropy is characterized by a rotation of plasma current. This work presented in a didatical way the relation between the external magnetic field and some plasma parameters with the rotation angle.

Formula omitted] deformations of the minimal unitary representation of [formula omitted] as massless 6 D conformal supermultiplets

Minimal unitary representation of SO ⁎ ( 8 ) ≃ SO ( 6 , 2 ) realized over the Hilbert space of functions of five variables and its deformations labeled by the spin t of an SU ( 2 ) subgroup correspond to massless conformal fields in six dimensions as was shown in [S. Fernando, M. Gunaydin, arXiv:1005.3580]. In this paper we study the minimal unitary supermultiplet of OSp ( 8 ⁎ | 2 N ) with the even subgroup SO ⁎ ( 8 ) × USp ( 2 N ) and its deformations using quasiconformal methods. We show that the minimal unitary supermultiplet of OSp ( 8 ⁎ | 2 N ) admits deformations labeled uniquely by the spin t of an SU ( 2 ) subgroup of the little group SO ( 4 ) of lightlike vectors in six dimensions. We construct the deformed minimal unitary representations and show that they correspond to massless 6 D conformal supermultiplets. The minimal unitary supermultiplet of OSp ( 8 ⁎ | 4 ) is the massless supermultiplet of ( 2 , 0 ) conformal field theory that is believed to be dual to M-theory on AdS 7 × S 4 . We study its deformations in further detail and show that they are isomorphic to the doubleton supermultiplets constructed by using twistorial oscillators.

Alephnull Anti-de Sitter supergravity with Lorentz Chern–Simons term in 3D

We formulate supergravity in three-dimensional Anti-de Sitter (AdS) space–time with an arbitrary number of supersymmetries with a Lorentz Chern–Simons term. Our field content is (eμm,ψμA,AμAB,ωμmn,λmnA), where the gravitino ψμA and the gaugino λmnA are in the vectorial representation of SO(N) for N∀, whose gauge field is AμAB. The ωμmn is a spin connection regarded as an independent field. Both ωμmn and AμAB have their Chern–Simons (CS) terms. Local N∀ (ℵ0) supersymmetry requires the coefficients of these CS terms be proportional to the gravitino mass. Differently from most conventional works, our supersymmetry transformation for ωμmn is proportional to the gaugino λmnA. The solution for the scalar curvature is a negative constant, and our space–time is AdS. Despite the Lorentz CS term, ωμmn can be algebraically solved. The Lorentz and SO(N) CS terms serve as the gravitational and SO(N) gauge anomalies for the two-dimensional boundary superconformal field theory. We also compute the charges for ℵ0 supersymmetry, SO(N) and Lorentz symmetries, and also show that Witten–Nester charges are positive definite.

Minimal unitary representation of [formula omitted] and its [formula omitted] deformations as massless 6 D conformal fields and their supersymmetric extensions

We study the minimal unitary representation (minrep) of SO ( 6 , 2 ) over an Hilbert space of functions of five variables, obtained by quantizing its quasiconformal realization. The minrep of SO ( 6 , 2 ) , which coincides with the minrep of SO ∗ ( 8 ) similarly constructed, corresponds to a massless conformal scalar field in six spacetime dimensions. There exists a family of “deformations” of the minrep of SO ∗ ( 8 ) labeled by the spin t of an SU ( 2 ) T subgroup of the little group SO ( 4 ) of lightlike vectors. These deformations labeled by t are positive energy unitary irreducible representations of SO ∗ ( 8 ) that describe massless conformal fields in six dimensions. The SU ( 2 ) T spin t is the six-dimensional counterpart of U ( 1 ) deformations of the minrep of 4D conformal group SU ( 2 , 2 ) labeled by helicity. We also construct the supersymmetric extensions of the minimal unitary representation of SO ∗ ( 8 ) to minimal unitary representations of OSp ( 8 ∗ | 2 N ) that describe massless six-dimensional conformal supermultiplets. The minimal unitary supermultiplet of OSp ( 8 ∗ | 4 ) is the massless supermultiplet of ( 2 , 0 ) conformal field theory that is believed to be dual to M-theory on AdS 7 × S 4 .

Composition series of the induced representations of SO(5) using intertwining operators

Composition series of the induced representations of SO(5) using intertwining operators

Lifting of characters on p-adic orthogonal and metaplectic groups

AbstractLet F be a p-adic field. Consider a dual pair $({\rm SO}(2n+1)_+, \widetilde {{\rm Sp}}(2n)),$ where SO(2n+1)+ is the split orthogonal group and $\widetilde {{\rm Sp}}(2n)$ is the metaplectic cover of the symplectic group Sp(2n) over F. We study lifting of characters between orthogonal and metaplectic groups. We say that a representation of SO(2n+1)+ lifts to a representation of $\widetilde {{\rm Sp}}(2n)$ if their characters on corresponding conjugacy classes are equal up to a transfer factor. We study properties of this transfer factor, which is essentially the character of the difference of the two halves of the oscillator representation. We show that the lifting commutes with parabolic induction. These results were motivated by the paper ‘Lifting of characters on orthogonal and metaplectic groups’ by Adams who considered the case F=ℝ.

Discrete components of some complementary series representations

We show that the restriction of the complementary series representations of SO(n, 1) to SO(m, 1) (m < n) contains complementary series representations of SO(m, 1) discretely, provided that the continuous parameter is sufficiently close to the first point of reducibility and the representation of M — the compact part of the Levi- is a sufficiently small fundamental representation. We prove, as a consequence, that the cohomological representation of degree i of the group SO(n, 1) contains discretely, for i ≤ m/2, the cohomological representation of degree i of the subgroup SO(m, 1) if i ≤ m/2. As a global application, we show that if G/ℚ is a semisimple algebraic group such that G(ℝ) = SO(n, 1) up to compact factors, and if we assume that for all n, the tempered cohomological representations are not limits of complementary series in the automorphic dual of SO(n, 1), then for all n, non-tempered cohomological representations are isolated in the automorphic dual of G. This reduces conjectures of Bergeron to the case of tempered cohomological representations.