Irreducible Representations Of SO Research Articles

Particles of a de Sitter Universe

The de Sitter spacetime is a maximally symmetric spacetime. It is one of the vacuum solutions to Einstein equations with a cosmological constant. It is the solution with a positive cosmological constant and describes a universe undergoing accelerated expansion. Among the possible signs for a cosmological constant, this solution is relevant for primordial and late-time cosmology. In the case of a zero cosmological constant, studies on the representations of its isometry group have led to a broader understanding of particle physics. The isometry group of d+1-dimensional de Sitter is the group SO(d+1,1), whose representations are well known. Given this insight, what can we learn about the elementary degrees of freedom in a four dimensional de Sitter universe by exploring how the unitary irreducible representations of SO(4,1) present themselves in cosmological setups? This article aims to summarize recent advances along this line that benefit towards a broader understanding of quantum field theory and holography at different signs of the cosmological constant. Particular focus is given to the manifestation of SO(4,1) representations at the late-time boundary of de Sitter. The discussion is concluded by pointing towards future questions at the late-time boundary and the static patch with a focus on the representations.

Anatomy of scalar mediated proton decays in SO(10) models

Realistic models based on the renormalizable grand unified theories have varieties of scalars, many of which are capable of mediating baryon (B) and lepton (L) number non-conserving processes. We identify all such scalar fields residing in 10, overline{mathbf{126}} and 120 dimensional irreducible representations of SO(10) which can induce baryon and lepton number violating interactions through the leading order d = 6 and d = 7 operators. Explicitly computing their couplings with the standard model fermions, we derive the effective operators including the possibility of mixing between the scalars stemming from a given representation. We find that such interactions at d = 6 are mediated by only three sets of scalars: T(3, 1, −1/3), mathcal{T} (3, 1, −4/3) and mathbbm{T} (3, 3, −1/3) and their conjugates. In the models with 10 and overline{mathbf{126}} , only the first has appropriate couplings to mediate the proton decay. While mathcal{T} and mathbbm{T} can induce baryon number violating interactions when 120 is present, mathcal{T} does not contribute to the proton decay at tree level because of its flavour antisymmetric coupling. Three additional colour triplets and their conjugates can mediate nucleon decay via d = 7 operators which violate also the B − L. We give general expressions for partial widths of proton in terms of the fundamental Yukawa couplings and use these results to explicitly compute the proton lifetime and branching ratios for the minimal non-supersymmetric SO(10) model based on 10 and overline{mathbf{126}} Higgs. We find that the proton preferably decays into overline{nu} K+ or μ+K0 and list several distinct features of scalar mediated proton decay. If the latter dominates over the gauge mediated contributions, the proton decay spectrum provides a direct probe to the flavour structure of the underlying grand unified theory.

AdS (super)projectors in three dimensions and partial masslessness

We derive the transverse projection operators for fields with arbitrary integer and half-integer spin on three-dimensional anti-de Sitter space, AdS3. The projectors are constructed in terms of the quadratic Casimir operators of the isometry group SO(2, 2) of AdS3. Their poles are demonstrated to correspond to (partially) massless fields. As an application, we make use of the projectors to recast the conformal and topologically massive higher-spin actions in AdS3 into a manifestly gauge-invariant and factorised form. We also propose operators which isolate the component of a field that is transverse and carries a definite helicity. Such fields correspond to irreducible representations of SO(2, 2). Our results are then extended to the case of mathcal{N} = 1 AdS3 supersymmetry.

Rotational symmetry of the multiple-relaxation-time collision model

We point out that the minimal components of the tensorial moments of the distribution that can be independently relaxed in collision without violating rotational symmetry are its irreducible representation (irrep) of SO(3), and a generic multiple-relaxation-time collision model can be constructed by independently relaxing these components. As the simplest example the irreps of the second moment are the traceless deviatoric stress and an isotropic tensor which is conserved in monatomic gases. Applying the decomposition to the thermal lattice Boltzmann model for polyatomic gases [Phys. Rev. E 77, 035701(R) (2008)PRESCM1539-375510.1103/PhysRevE.77.035701], the shear and bulk viscosities are decoupled by two independent relaxation times. The hydrodynamic equation of the model is obtained via Chapman-Enskog calculation and verified by numerical simulation.

Remarks on a melonic field theory with cubic interaction

We revisit the Amit-Roginsky (AR) model in the light of recent studies on Sachdev-Ye-Kitaev (SYK) and tensor models, with which it shares some important features. It is a model of N scalar fields transforming in an N-dimensional irreducible representation of SO(3). The most relevant (in renormalization group sense) invariant interaction is cubic in the fields and mediated by a Wigner 3jm symbol. The latter can be viewed as a particular rank-3 tensor coupling, thus highlighting the similarity to the SYK model, in which the tensor coupling is however random and of even rank. As in the SYK and tensor models, in the large-N limit the perturbative expansion is dominated by melonic diagrams. The lack of randomness, and the rapidly growing number of invariants that can be built with n fields, makes the AR model somewhat closer to tensor models. We review the results from the old work of Amit and Roginsky with the hindsight of recent developments, correcting and completing some of their statements, in particular concerning the spectrum of the operator product expansion of two fundamental fields. For 5.74 < d < 6 the fixed-point theory defines a real CFT, while for smaller d complex dimensions appear, after a merging of the lowest dimension with its shadow. We also introduce and study a long-range version of the model, for which the cubic interaction is exactly marginal at large N , and we find a real and unitary CFT for any d < 6, both for real and imaginary coupling constant, up to some critical coupling.

A multiple-relaxation-time collision model for nonequilibrium flows

Despite yielding correct hydrodynamics in the continuum limit, the Bhatnagar–Gross–Krook collision model is too simplistic to model the full details of the collision, which becomes increasingly important as the quasi-equilibrium assumption breaks down. In a recent phenomenological collision model, independent relaxation rates are assigned to the components of the tensorial Hermite expansion of the distribution corresponding to the irreducible representations of SO(3), yielding arguably the most general form of multirelaxation without violating rotation symmetry. Here we show that by using the relaxation rates obtained analytically from Boltzmann collision term with Maxwell molecular model, lattice Boltzmann method yields results in good agreement with the accurate fast spectral method in simulation of the spontaneous Rayleigh–Brillouin scattering problem. The hydrodynamically insignificant relaxation rates of the higher moments are found to be significant as the Knudsen number increases. These results suggest that with properly tuned relaxation rates, the collision model could potentially mimic the behavior of arbitrary collision kernels.

Exactly Solvable Magnet of Conformal Spins in Four Dimensions.

We provide the eigenfunctions for a quantum chain of N conformal spins with nearest-neighbor interaction and open boundary conditions in the irreducible representation of SO(1,5) of scaling dimension Δ=2-iλ and spin numbers ℓ=ℓ[over ˙]=0. The spectrum of the model is separated into N equal contributions, each dependent on a quantum number Y_{a}=[ν_{a},n_{a}] which labels a representation of the principal series. The eigenfunctions are orthogonal and we computed the spectral measure by means of a new star-triangle identity. Any portion of a conformal Feynmann diagram with square lattice topology can be represented in terms of separated variables, and we reproduce the all-loop "fishnet" integrals computed by B. Basso and L. Dixon via bootstrap techniques. We conjecture that the proposed eigenfunctions form a complete set and provide a tool for the direct computation of conformal data in the fishnet limit of the supersymmetric N=4 Yang-Mills theory at finite order in the coupling, by means of a cutting-and-gluing procedure on the square lattice.

On the SO(n + 3) to SO(n) branching multiplicity space

We study the decomposition as an SO(3)-module of the multiplicity space corresponding to the branching from SO(n+3) to SO(n). Here, SO(n) (resp. SO(3)) is considered embedded in SO(n+3) in the upper left-hand block (resp. lower right-hand block). We show that when the highest weight of the irreducible representation of SO(n) interlaces the highest weight of the irreducible representation of SO(n+3), then the multiplicity space decomposes as a tensor product of ⌊(n+2)/2⌋ reducible representations of SO(3).

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.

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 .

Some evidence about the dynamical group SO (4, 2) symmetries of the periodic table of elements

It is a great pleasure to see that this symposium is held in honor of Professor E. U. Condon to whom I owe so much, whom I admire so much, and whom I respect so much. I am Professor Condon's fourth and last student. I do not know of any rule which states that the last should be the least. I am afraid in my case this is inevitable, since the first three are F. Seitz, E. M. McMillan, and G. H. Shortley. The subject of the order of electron shells in neutral atoms (Madelung's rule) and in highly ionized ions (hydrogenic order) is reviewed. The results and the predictions of the central field approximations are surveyed. Since the symmetries of Madelung's rule can be explained by the chain and since there are two irreducible representations of SO (3, 2), one with (n + l) = even and the other with (n + l) = odd, into which the SO (4, 2) representation, whose multiplicity pattern coincides with the structure of periodic table, reduces, when considered along periods the physical properties of normal configurations of atoms or lightly ionized ions are expected to fall on separate smooth curves according to whether (n + l) is even or odd. By using observed and theoretical data it is shown that this is indeed the case. In particular it is also demonstrated that if along the periods three points of the same kind are known, corresponding to the ionization potentials for normal states of atoms, the fourth point can be extrapolated very accurately. For example, by this method the ionization potential for the normal state of francium is determined to be 87(IP)7s = 32060 ± 30 cm-1.

Branching laws for minimal holomorphic representations

In this paper we study the branching law for the restriction from SU ( n , m ) to SO ( n , m ) of the minimal representation in the analytic continuation of the scalar holomorphic discrete series. We identify the group decomposition with the spectral decomposition of the action of the Casimir operator on the subspace of S ( O ( n ) × O ( m ) ) -invariants. The Plancherel measure of the decomposition defines an L 2 -space of functions, for which certain continuous dual Hahn polynomials furnish an orthonormal basis. It turns out that the measure has point masses precisely when n − m > 2 . Under these conditions we construct an irreducible representation of SO ( n , m ) , identify it with a parabolically induced representation, and construct a unitary embedding into the representation space for the minimal representation of SU ( n , m ) .

Linearizing N=3 nonlinear supersymmetry in two dimensions

Abstract We investigate for N = 3 supersymmetry (SUSY) in D = 2 the algebraic relation between the Volkov–Akulov (VA) model of nonlinear (NL) SUSY and a (renormalizable) SO(3) vector supermultiplet of linear (L) SUSY. We derive SUSY and SO(3) invariant relations between component fields of the vector supermultiplet and Nambu–Goldstone fermions of the VA model at leading orders by using three arbitrary dimensionless parameters which can be recasted as the vacuum expectation values of auxiliary fields in the vector supermultiplet. Two different irreducible representations of SO(3) super-Poincare symmetry which appear in the same massless state are compatible with each other in the linearization of NL SUSY. The equivalence of a NL SUSY VA action to a free L SUSY action containing the Fayet–Iliopoulos D term which indicates a spontaneously SUSY breaking is also discussed explicitly according to the SUSY invariant relations.

Geometric interpretations of two branching theorems of D.E. Littlewood

D.E. Littlewood proved two branching theorems for decomposing the restriction of an irreducible finite-dimensional representation of a unitary group to a symmetric subgroup. One is for restriction of a representation of U(n) to the rotation group SO(n) when the given representation τλ of U(n) has nonnegative highest weight λ of depth ⩽n/2. It says that the multiplicity in τλ|SO(n) of an irreducible representation of SO(n) of highest weight ν is the sum over μ of the multiplicities of τλ in the U(n) tensor product τμ⊗τν, the allowable μ’s being all even nonnegative highest weights for U(n). Littlewood’s proof is character-theoretic. The present paper gives a geometric interpretation of this theorem involving the tensor products τμ⊗τν explicitly. The geometric interpretation has an application to the construction of small infinite-dimensional unitary representations of indefinite orthogonal groups and, for each of these representations, to the determination of its restriction to a maximal compact subgroup. The other Littlewood branching theorem is for restriction from U(2r) to the rank-r quaternion unitary group Sp(r). It concerns nonnegative highest weights for U(2r) of depth ⩽r, and its statement is of the same general kind. The present paper finds an analogous geometric interpretation for this theorem also.

Nuclear, Algebraic Surfaces and SO(10)

In this paper nuclear representations of the Dirac ring, developed over many years, are shown to be a particular case of a theorem in algebraic geometry which at the same time associates them with a Hodge decomposition of a Kaehler manifold. This yields a shape illustrated by Figs. 1,2,3 which in some cases is independent of any appeal to a symmetry group. However because the odd-A nuclear representations are in the infinitesimal ring of SO(4) and the internal space of each representation is in a Kaehler (even Calabi Yau) manifold K; the group SO(10) = SO(4) × K can give additional information. This paper develops the very fruitful symbiosis between algebra and irreducible representations of SO(10) and shows that the odd-A nucleus is bound by a self-generating string field, whose quanta are the mesons, which therefore dispenses with nuclear forces and reduces nuclear structure to geometry. An appendix gives and example of how the theory may be applied to laser cooling.

Nuclear Algebraic Geometry and SO(10)

In this contribution nuclear representations of the Dirac ring, developed over many years, are shown to be a particular case of a theorem in algebraic geometry which at the same time associates them with a Hodge decomposition of a Kaehler manifold. This yields a shape that in some cases is independent of any appeal to a symmetry group. However, because the nuclear representations are in the infinitesimal ring of SO(4) and the internal space of each representation is in a Kaehler (even Calabi-Yau) manifold K; the group SO(10) = SO(4) × K can give additional information. This paper develops the very fruitful symbiosis between algebra and irreducible representations of SO(10) and covers some aspects of string theory.

Abelian decomposition of SO(2N) Yang-Mills theory

Faddeev and Niemi have proposed a decomposition of SU(N) Yang-Mills theory in terms of new variables, appropriate for describing the theory in the infrared limit. We extend this method to SO(2N) Yang-Mills theory. We find that the SO(2N) connection decomposes according to irreducible representations of SO(N). The low energy limit of the decomposed theory is expected to describe soliton-like configurations with nontrivial topological numbers. How the method of decomposition generalizes for $SO(2N+1)$ Yang-Mills theory is also discussed.

Supersymmetric structure of space–time and matter: superon–graviton model

A unified description of space–time and matter is proposed by using a single irreducible representation of SO(10) super-Poincaré algebra ( SO(10) SPA). All (observed) elementary particles except the graviton are the (massless) eigenstates of SO(10) SPA composed of fundamental Nambu–Goldstone fermions with spin 1/2, superons associating with the spontaneous breakdown of the supertranslation of the space–time. The systematic investigations of the Standard Model (SM) and SU(5) GUT by using superon diagrams may reveal the generation structure, the stability of the proton, K 0– K ̄ 0 , B 0– B ̄ 0 and D 0– D ̄ 0 mixings, CP-violation, the atmospheric and solar neutrino deficits and the absence of the electroweak lepton–flavor mixing. The fundamental action of the superon–graviton model (SGM) for space–time and matter is proposed, which is invariant under a new supersymmetry.

Abelian decomposition of Sp(2 N) Yang–Mills theory

In the previous paper, we generalized the method of Abelian decomposition to the case of SO( N) Yang–Mills theory. This method that was proposed by Faddeev and Niemi introduces a set of variables for describing the infrared limit of a Yang–Mills theory. Here, we extend the decomposition method further to the general case of four-dimensional Sp(2 N) Yang–Mills theory. We find that the Sp(2 N) connection decomposes according to irreducible representations of SO( N).

Analogies between finite-dimensional irreducible representations of SO(2n) and infinite-dimensional irreducible representations of Sp(2n,R). II. Plethysms

The basic spin difference character Δ″ of SO(2n) is a useful device in dealing with characters of irreducible spinor representations of SO(2n). It is shown here that its kth-fold symmetrized powers, or plethysms, associated with partitions κ of k factorize in such a way that Δ″⊗{κ}=(Δ″)r(κ)Πκ, where r(κ) is the Frobenius rank of κ. The analogy between SO(2n) and Sp(2n,R) is shown to be such that the plethysms of the basic harmonic or metaplectic character Δ̃ of Sp(2n,R) factorize in the same way to give Δ̃⊗{κ}=(Δ̃)r(κ)Π̃κ. Moreover, the analogy is shown to extend to the explicit decompositions into characters of irreducible representations of SO(2n) and Sp(2n,R) not only for the plethysms themselves, but also for their factors Πκ and Π̃κ. Explicit formulas are derived for each of these decompositions, expressed in terms of various group–subgroup branching rule multiplicities, particularly those defined by the restriction from O(k) to the symmetric group Sk. Illustrative examples are included, as well as an extension to the symmetrized powers of certain basic tensor difference characters of both SO(2n) and Sp(2n,R).