Subgroups Of SU Research Articles

Three-dimensional symmetry-protected topological phase close to antiferromagnetic Néel order

It is well known that the Haldane phase of a one-dimensional spin-1 chain is a symmetry-protected topological (SPT) phase, which is described by a nonlinear sigma model (NLSM) with a $\ensuremath{\Theta}$ term at $\ensuremath{\Theta}=2\ensuremath{\pi}$. In this work we study a three-dimensional (3d) SPT phase of a SU(2$N$) antiferromagnetic spin system with a self-conjugate representation on every site. The spin-ordered N\'eel phase of this system has a ground state manifold $\mathcal{M}=\frac{\mathrm{U}(2N)}{\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)}$, and this system is described by a NLSM defined with manifold $\mathcal{M}$. Since the homotopy group ${\ensuremath{\pi}}_{4}[\mathcal{M}]=\mathbb{Z}$ for $N>1$, this NLSM can naturally have a $\ensuremath{\Theta}$ term. We will argue that when $\ensuremath{\Theta}=2\ensuremath{\pi}$ this NLSM describes a SPT phase. This SPT phase is protected by the SU(2$N$) spin symmetry, or its subgroup SU($N$)$\ifmmode\times\else\texttimes\fi{}$SU$(N)\ensuremath{\rtimes}{Z}_{2}$, without assuming any other discrete symmetry. We will also construct a trial SU(2$N$) spin state on a 3d lattice; we argue that the long-wavelength physics of this state is precisely described by the aforementioned NLSM with $\ensuremath{\Theta}=2\ensuremath{\pi}$.

On 6d N = (2, 0) theory compactified on a Riemann surface with finite area

We study 6d N=(2,0) theory of type SU(N) compactified on Riemann surfaces with finite area, including spheres with fewer than three punctures. The Higgs branch, whose metric is inversely proportional to the total area of the Riemann surface, is discussed in detail. We show that the zero-area limit, which gives us a genuine 4d theory, can involve a Wigner-Inonu contraction of global symmetries of the six-dimensional theory. We show how this explains why subgroups of SU(N) can appear as the gauge group in the 4d limit. As a by-product we suggest that half-BPS codimension-two defects in the six-dimensional (2,0) theory have an operator product expansion whose operator product coefficients are four-dimensional field theories.

The Complex Quantum-State of Black-Hole and Thermostatistics.

In this paper the quantum aspects are described in-detail with the help of a new type of energy source called latent energy group SU(6) of the super unified theory of SU(11). The thermodynamics of general self gravitating systems created by the energy group SU(6) and some related topics such as complex space-time (i.e. Pseudo-Space-Time) are also briefly discussed. The thermodynamic connection is based on Hawking celebrated application of quantum theory to Black-Hole. We know that macroscopic systems have definite and precise energies, subject to definite conservation principle. In this paper the generalized theory of black-hole is examined in-detail. We study the quantum state of black-hole. We consider our universe begins through the symmetry breaking of the super unified theory with gauss group SU(11) (  SU(6)  SU(5)  U(1)) from 7-dimensionsl space-time. The subgroup SU(6) has been interpreted as a new type of energy source other than SU(5) (SU(5)  SU (3)  SU(2)  U(1), where SU(3), the strong energy group, SU(2) the weak-energy group and U(1), the electro-dynamics). The energy group SU(6) are responsible for the situation of the so-called black-hole and U(1), also for the creation of Schwarzschild mass 'M' and angular momentum 'J', as U(1) are responsible for the creation of electric charge 'e' ( where M & J- depends on 'e' as shown in this paper ). We consider the line element proposed by the quantum-vacuum Kantowski-Sachs universe without cosmological constant. We also consider the scale factor b(= iR) of the metric to be imaginary and the other a( = R) is real, of the Kantowski-Sachs quantum-state with complex space- time. Solving the metric tensors we get the energy tensors in the 10(= 4+6) dimensional spacetime under the exchange RI ↔ R. We think from 10-dimensional to 7-dimensional stage, the situation of primordial black-hole formed by the energy group SU(11). The fact is obviously defined within 10- dimensional to7-dimensional flat universe and then the black-hole formed within 7-dimensional to 4- dimensional closed universe through the symmetry breaking of the energy group SU(11), the negative pressure by is define the matter energy density and is define the latent energy density. At the early stage The universe then begins its journey through the phase transition system.

Dyons and Magnetic Monopoles Revisited

We construct dyon solutions in SU(N) with topological electric and magnetic charge. Assuming a |Φ| 4 —like potential for the Higgs field, we show that the mass of the dyons is relatively insensitive to the coupling parameter λ characterizing the potential. We then apply the methodology of constructing dyon solutions in SU(N) to G2. In order to define the electromagnetic field consistently in the manner that we propose, we find that dyon solutions exist only when G2 is considered under the action of its maximal and regular subgroup SU(3). In this case we find two different types of dyons, one of which has properties identical to dyons in SU(3). The other dyon has some properties which are seemingly atypical, eg, the magnetic charge g m = 4π 3/e, which differs from the 't Hooft/Polyakov monopole where g m = 4π 1/e.

Quaternion Octonion Reformulation of Grand Unified Theories

In this paper, Grand Unified theories are discussed in terms of quaternions and octonions by using the relation between quaternion basis elements with Pauli matrices and Octonions with Gell Mann \lambda matrices. Connection between the unitary groups of GUTs and the normed division algebra has been established to re-describe the SU(5)gauge group. We have thus described the SU(5)gauge group and its subgroup SU(3)_{C}\times SU(2)_{L}\times U(1) by using quaternion and octonion basis elements. As such the connection between U(1) gauge group and complex number, SU(2) gauge group and quaternions and SU(3) and octonions is established. It is concluded that the division algebra approach to the the theory of unification of fundamental interactions as the case of GUTs leads to the consequences towards the new understanding of these theories which incorporate the existence of magnetic monopole and dyon.

Interaction of orientable object fields with gauge fields

We consider a scalar field f(g) on the Poincaré group M(3, 1). This scalar field describes objects that are characterized by a position x and an orientation z, g=(x,z). The field f(x, z) admits two kinds of transformations, corresponding to a change of the space-fixed reference frame, as well as to a change of the body-fixed reference frame, which form the group M(3, 1)ext×M(3, 1)int, and also phase transformations U(1)ch of orientational variables z. Elementary particles considered as elementary orientable objects are described by the scalar functions transforming according to irreps of the group M(3, 1)ext×M(3, 1)int×U(1)ch. Correspondingly, their continuous symmetries can be divided into external, which form the Poincaré group M(3, 1)ext, and internal M(3, 1)int×U(1)ch. The assumption that the internal symmetries in the theory of orientable objects are gauge ones allows one to obtain important features of the known fundamental interactions—the electroweak and the gravitational. Localization of the group of the right translations T(4)int leads to the teleparallel theory of gravity, which is equivalent to general relativity. Localization of the compact subgroup SU(2)int×U(1)ch leads to the theory of electroweak interactions. Thus, the suggested approach can be considered as a possible way to gravitational–electroweak unification.

Non-Abelian monopole-vortex complex

In the context of softly broken N=2 supersymmetric quantum chromodynamics (SQCD), with a hierarchical gauge symmetry breaking SU(N+1) -> U(N) -> 1, at scales v1 and v2, respectively, where v1 >> v2, we construct monopole-vortex complex soliton-like solutions and examine their properties. They represent the minimum of the static energy under the constraint that the monopole and antimonopole positions sitting at the extremes of the vortex are kept fixed. They interpolate the 't Hooft-Polyakov-like regular monopole solution near the monopole centers and a vortex solution far from them and in between. The main result, obtained in the theory with Nf=N equal-mass flavors, is concerned with the existence of exact orientational CP(N-1) zero modes, arising from the exact color-flavor diagonal SU(N)_{C+F} global symmetry. The "unbroken" subgroup SU(N) \subset SU(N+1) with which the na\"ive notion of non-Abelian monopoles and the related difficulties were associated, is explicitly broken at low energies. The monopole transforms nevertheless according to the fundamental representation of a new exact, unbroken SU(N) symmetry group, as does the vortex attached to it. We argue that this explains the origin of the dual non-Abelian gauge symmetry.

The BQP-hardness of approximating the Jones polynomial

A celebrated important result due to Freedman et al (2002 Commun. Math. Phys.227 605–22) states that providing additive approximations of the Jones polynomial at the kth root of unity, for constant k=5 and k⩾7, is BQP-hard. Together with the algorithmic results of Aharonov et al (2005) and Freedman et al (2002 Commun. Math. Phys.227 587–603), this gives perhaps the most natural BQP-complete problem known today and motivates further study of the topic. In this paper, we focus on the universality proof; we extend the result of Freedman et al (2002) to ks that grow polynomially with the number of strands and crossings in the link, thus extending the BQP-hardness of Jones polynomial approximations to all values to which the AJL algorithm applies (Aharonov et al 2005), proving that for all those values, the problems are BQP-complete. As a side benefit, we derive a fairly elementary proof of the Freedman et al density result, without referring to advanced results from Lie algebra representation theory, making this important result accessible to a wider audience in the computer science research community. We make use of two general lemmas we prove, the bridge lemma and the decoupling lemma, which provide tools for establishing the density of subgroups in SU(n). Those tools seem to be of independent interest in more general contexts of proving the quantum universality. Our result also implies a completely classical statement, that the multiplicative approximations of the Jones polynomial, at exactly the same values, are #P-hard, via a recent result due to Kuperberg (2009 arXiv:0908.0512). Since the first publication of those results in their preliminary form (Aharonov and Arad 2006 arXiv:quant-ph/0605181), the methods we present here have been used in several other contexts (Aharonov and Arad 2007 arXiv:quant-ph/0702008; Peter and Stephen 2008 Quantum Inf. Comput.8 681). The present paper is an improved and extended version of the results presented by Aharonov and Arad (2006) and includes discussions of the developments since then.

Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: SU(3)

We complete the computation of spectral measures for SU(3) nimrep graphs arising in subfactor theory, namely the SU(3) ADE graphs associated with SU(3) modular invariants and the McKay graphs of finite subgroups of SU(3). For the SU(2) graphs the spectral measures distill onto very special subsets of the semicircle/circle, whilst for the SU(3) graphs the spectral measures distill onto very special subsets of the discoid/torus. The theory of nimreps allows us to compute these measures precisely. We have previously determined spectral measures for some nimrep graphs arising in subfactor theory, particularly those associated with all SU(2) modular invariants, all subgroups of SU(2), the torus, SU(3), and some SU(3) graphs.

The Bernstein–Gelfand–Gelfand complex and Kasparov theory for [formula omitted

For G = SL ( 3 , C ) , we construct an element of G -equivariant analytic K-homology from the Bernstein–Gelfand–Gelfand complex for G . This furnishes an explicit splitting of the restriction map from the Kasparov representation ring R ( G ) to the representation ring R ( K ) of its maximal compact subgroup SU ( 3 ) , and the splitting factors through the equivariant K-homology of the flag variety X of G . In particular, we obtain a new model for the γ-element of G . The construction is made using SU ( 3 ) -harmonic analysis associated to the canonical fibrations of X . On this matter, we prove results which demonstrate the compatibility of both the G -action and the order zero longitudinal pseudodifferential operators with the SU ( 3 ) -harmonic analysis.

Dynamical symmetry reduction and discrete tomography of a Ξ atom

When implemented using a reasonable Hamiltonian, the tomography of a three-level Ξ atom is complicated by the equidistant energy levels of the atom. This restricts the possible transformations to those in the SO(3) subgroup of SU(3). Although complete reconstruction is possible for a single Ξ atom using a continuous set of tomograms, the discrete optimal set of tomograms, related to mutually unbiased bases in dimension 3, are not accessible by time evolution. We discuss here the search for an optimal set of discrete basis states compatible with the reduced SO(3) symmetry of the system.

Generalized twisted quantum doubles and the McKay Correspondence

We consider a class of quasiHopf algebras which we call generalized twisted quantum doubles. They are abelian extensions H = C [ G ¯ ] ∗ ⋈ C [ G ] ( G is a finite group, G ¯ a homomorphic image, and * denotes the dual algebra), possibly twisted by a 3-cocycle, and are a natural generalization of the twisted quantum double construction of Dijkgraaf, Pasquier and Roche. We show that if G is a subgroup of SU 2 ( C ) then H exhibits an orbifold McKay Correspondence: certain fusion rules of H define a graph with connected components indexed by conjugacy classes of G ¯ , each connected component being an extended affine Diagram of type ADE whose McKay correspondent is the subgroup of G stabilizing an element in the conjugacy class. This reduces to the original McKay Correspondence when G ¯ = 1 .

Holonomy Groups Coming from F-Theory Compactification

We study holonomy groups coming from F-theory compactifications. We focus mainly on SO(8) as 12−4=8 and subgroups SU(4), Spin(7), G 2 and SU(3) suitable for descent from F-theory, M-theory and Superstring theories. We consider the relation of these groups with the octonions, which is striking and reinforces their role in higher dimensions and dualities. These holonomy groups are related in various mathematical forms, which we exhibit.

Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory

We determine spectral measures for some nimrep graphs arising in subfactor theory, particularly those associated with SU(3) modular invariants. Our methods also give an alternative approach to deriving the results of Banica and Bisch for ADE graphs and subgroups of SU(2) and explain the connection between their results for affine ADE graphs and the Kostant polynomials. We also look at the Hilbert generating series of associated pre-projective algebras.

Kähler flat manifolds

Using a criterion of Johnson-Rees [9] we give a list of all four and six dimensional flat Kähler manifolds. We calculate their R–cohomology, including the Hodge numbers. As a corollary, we classify all flat complex manifolds of dimension 3 whose holonomy groups are subgroups of SU(3). Moreover, we define a family of flat Kähler manifolds which are generalizations of the oriented Hantzsche-Wendt Riemannian manifolds [14].

Explicit connections with SU(2)-monodromy

The pure braid group Γ of a quadruply-punctured Riemann sphere acts on the SL(2, ℂ)-moduli ℳ of the representation variety of such sphere. The points in ℳ are classified into Γ-orbits. We show that, in this case, the monodromy groups of many explicit solutions to the Riemann-Hilbert problem are subgroups of SU(2). Most of these solutions are examples of representations that have dense images in SU(2), but with finite Γ-orbits in ℳ. These examples relate to explicit immersions of constant mean curvature surfaces.

A-D-E QUIVERS AND BARYONIC OPERATORS

We study baryonic operators of the gauge theory on multiple D3-branes at the tip of the conifold orbifolded by a discrete subgroup of SU(2). The string theory analysis predicts that the number and the order of the fixed points of this discrete subgroup acting on S2 are directly reflected in the spectrum of baryonic operators on the corresponding quiver gauge theory constructed from two Dynkin diagrams of the corresponding type. We confirm the prediction by utilizing techniques to enumerate baryonic operators of the quiver gauge theory which includes the gauge groups with different ranks.

On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices

Parametrization of 4 ◊ 4-matrices G of the complex linear group GL(4,C) in terms of four complex 4-vector parameters (k,m,n,l) is investigated. Additional restrictions separating some subgroups of GL(4,C) are given explicitly. In the given parametrization, the problem of inverting any 4◊4 matrix G is solved. Expression for determinant of any matrix G is found: detG = F(k,m,n,l). Unitarity conditions G + = G 1 have been formulated in the form of non-linear cubic algebraic equations including complex conjugation. Several simplest solutions of these unitarity equations have been found: three 2-parametric subgroups G1, G2, G3 - each of subgroups consists of two commuting Abelian unitary groups; 4-parametric unitary subgroup consisting of a product of a 3-parametric group isomorphic SU(2) and 1- parametric Abelian group. The Dirac basis of generators k, being of Gell-Mann type, substantially differs from the basis i used in the literature on SU(4) group, formulas relating them are found - they permit to separate SU(3) subgroup in SU(4). Special way to list 15 Dirac generators of GL(4,C) can be used { k} = { i j ( iV j = K L M)}, which permit to factorize SU(4) transformations according to S = e i~a~ e i ~b~ e ikK e ilL e imM , where two first factors commute with each other and are isomorphic to SU(2) group, the three last ones are 3-parametric groups, each of them consisting of three Abelian commuting unitary subgroups. Besides, the structure of fifteen Dirac matrices k permits to separate twenty 3-parametric subgroups in SU(4) isomorphic to SU(2); those subgroups might be used as bigger elementary blocks in constructing of a general transformation SU(4). It is shown how one can specify the present approach for the pseudounitary group SU(2,2) and SU(3,1).

Simple finite non-Abelian flavor groups

The recently measured unexpected neutrino mixing patterns have caused a resurgence of interest in the study of finite flavor groups with two- and three-dimensional irreducible representations. This paper details the mathematics of the two finite simple groups with such representations, the icosahedral group A5, a subgroup of SO(3), and PSL2(7), a subgroup of SU(3).

Geometric phase forN-level systems through unitary integration

Geometric phases are important in quantum physics and are now central to fault-tolerant quantum computation. For spin $1∕2$, the Bloch sphere ${S}^{2}$, together with a U(1) phase, provides a complete SU(2) description. We generalize to $N$-level systems and $\mathrm{SU}(N)$ in terms of a $2(N\ensuremath{-}1)$-dimensional base space and reduction to a $(N\ensuremath{-}1)$-level problem, paralleling closely the two-dimensional case. This iteratively solves the time evolution of an $N$-level system and gives $(N\ensuremath{-}1)$ geometric phases explicitly. A complete analytical construction of an ${S}^{4}$ Bloch-like sphere for two qubits is given for the Spin(5) or SO(5) subgroup of SU(4).