Abelian Discrete Group Research Articles

Noncompact lattice Higgs model with Abelian discrete gauge groups: Phase diagram and gauge symmetry enlargement

We study the phase diagram and phase transitions of the three dimensional multicomponent lattice Higgs model with noncompact Abelian discrete groups. The model with noncompact U(1) gauge group is known to undergo, for a sufficiently large number of scalar fields $N$, a continuous transition associated to the charged fixed point of the continuous Abelian Higgs field theory. We show that in the model with gauge group ${\mathbb{Z}}_{q}^{(nc)}\ensuremath{\equiv}2\ensuremath{\pi}\mathbb{Z}/q$ only critical transitions in the orthogonal universality classes are present for small values of $N$, while a symmetry enlargement to the continuous Abelian Higgs universality class happens when $q\ensuremath{\ge}5$ and $N$ is large enough.

Evolution of the gauge couplings and Weinberg angle in 5-dimensions for a G2 gauge group

We explicitly test, in a simplified 5-dimensional model with G2 gauge symmetry, the evolution of the gauge couplings. We assume that all the matter fields are propagating in the bulk, and consider orbifolds based on Abelian discrete groups which lead to 5-dimensional gauge theories compactified on an S1/Z2. The gauge couplings evolution is derived at one-loop level and used to test the impact on lower energy observables, in particular the Weinberg angle.

Evolution of the gauge couplings and Weinberg angle in 5-dimensions for an SU(5) and flipped SU(5) gauge group

We explicitly test, in a simplified 5-dimensional model with SU(5) and SU(5) × U(1)′ gauge symmetries, the evolution of the gauge couplings. We assume that all the matter fields are propagating in the bulk, and consider orbifolds based on Abelian discrete groups which lead to 5-dimensional gauge theories compactified on an S1/Z2. The gauge couplings evolution is derived at one-loop level and used to test the impact on lower energy observables, in particular the Weinberg angle.

Growth functions for some uniformly amenable groups

Abstract We present a simple constructive proof of the fact that every abelian discrete group is uniformly amenable. We improve the growth function obtained earlier and find the optimal growth function in a particular case. We also compute a growth function for some non-abelian uniformly amenable group.

On reduced amalgamated free products of C*-algebras and the MF property

We establish the MF property of the reduced group $ C^* $-algebra of an amalgamated free product of countable Abelian discrete groups. This result is then used to give a characterization of the amalgamated free products of Abelian groups for which the BDF semigroup of the reduced group $ C^* $-algebra is a group. Along the way we get a tensor product factorization of the corresponding group von Neumann algebra. We end the exposition by applying the ideas from the first part to give a few more examples of groups with a reduced group $ C^* $-algebra which is MF.

Discrete flavor symmetries and models of neutrino mixing

We review the application of non abelian discrete groups to the theory of neutrino masses and mixing, which is strongly suggested by the agreement of the Tri-Bimaximal mixing pattern with experiment. After summarizing the motivation and the formalism, we discuss specific models, based on A4, S4 and other finite groups, and their phenomenological implications, including lepton flavor violating processes, leptogenesis and the extension to quarks. In alternative to Tri-Bimaximal mixing the application of discrete flavor symmetries to quark-lepton complementarity and Bimaximal Mixing is also considered.

Invariants of the half-liberated orthogonal group

Le groupe orthogonal semi-libéré O n * est un groupe quantique intermédiaire entre le groupe orthogonal O n et sa version libre O n + . Nous discutons ici ses propriétés algébriques de base, et nous classifions ses représentations irréductibles. Cette classification est établie grâce à une mise en relation avec le groupe U n et des méthodes inspirées de la théorie des algèbres de Lie. Un groupe discret non abélien joue le rôle de réseau des poids. Nous utilisons ces résultats pour montrer que le groupe quantique discret dual est à croissance polynomiale.

Standard model Higgs boson from higher dimensional gauge fields

We consider the possibility that the standard model Higgs fields may originate from extra components of higher dimensional gauge fields. Theories of this type considered before have had problems accommodating the standard model fermion content and Yukawa couplings different from the gauge coupling. Considering orbifolds based on Abelian discrete groups we are led to a 6 dimensional ${G}_{2}$ gauge theory compactified on ${T}^{2}{/Z}_{4}.$ This theory can naturally produce the SM Higgs fields with the right quantum numbers while predicting the value of the weak mixing angle ${\mathrm{sin}}^{2}{\ensuremath{\theta}}_{W}=0.25$ at the tree level, close to the experimentally observed one. The quartic scalar coupling for the Higgs boson is generated by the higher dimensional gauge interaction and predicts the existence of a light Higgs boson. We point out that one can write a quadratically divergent counterterm for Higgs boson mass localized to the orbifold fixed point. However, we calculate these operators and show that higher dimensional gauge interactions do not generate them at least at one loop. Fermions are introduced at orbifold fixed points, making it easy to accommodate the standard model fermion content. Yukawa interactions are generated by Wilson lines. They may be generated by the exchange of massive bulk fermions, and the fermion mass hierarchy can be obtained. Around a TeV, the first KK modes would appear as well as additional fermion modes localized at the fixed point needed to cancel the quadratic divergences from the Yukawa interactions. The cutoff scale of the theory could be a few times 10 TeV.

On riesz subsets of abelian discrete groups

We study the class of the Riesz subsets of abelian discrete groups, that is, the sets for which the F. and M. Riesz theorem extends. We show that the “classical” tools of the theory — Riesz projections, localization in the Bohr sense, products — are leading to Riesz sets which are satisfying nice additional properties, e.g., the Mooney-Havin result extends to this class. We give an alternative proof of a result of A. B. Alexandrov, and we improve a construction of H. P. Rosenthal. The connection is made between this class and theM-structure theory. We show a result of convergence at the boundary for holomorphic functions on the polydisc. The Bourgain-Davis result on convergence of analytic martingales is improved.

Decomposition theorems for certain C*-crossed products

Let G be an abelian discrete group, A a unital C*-algebra and an action of G on A, i.e. (A, G,) is a C*-dynamical system. Let K denote the kernel ker of and put R = G/K. The main purpose of this article is to determine the roles of K and R in the crossed product G A. This goal is achieved in Section 2, where we prove that G A is *-isomorphic to a twisted crossed product of R with C*(K) A with respect to the action 1 and a 2-cocycle related to the 2-cocycle determined by the extension G of R by K. Here is the obvious action of R on A.