Discrete Subgroups Of SU Research Articles

Black holes and (0,4) SCFTs from type IIB on K3

We study the central charges and levels of 2d mathcal{N} = (0, 4) superconformal field theories that are dual to four- and five-dimensional BPS black holes in compactifications of type IIB string theory on a K3 surface. They arise from wrapping a D3-brane on a curve C inside K3 and have transverse space either an ALE or ALF space. These D3-branes have an AdS3 × S3/Γ near horizon geometry where Γ is a discrete subgroup of SU(2). We compute the central charges and levels of the 2d SCFTs both in the microscopic picture and from six-dimensional mathcal{N} = (2, 0) supergravity. These quantities determine the black hole entropy via Cardy’s formula. We find agreement between the microscopic and macroscopic computations. The contributions from one-loop quantum corrections to the macroscopic result are crucial for this matching.

Combining symmetry breaking and restoration with configuration interaction: Extension to z -signature symmetry in the case of the Lipkin model

Background: Ab initio many-body methods whose numerical cost scales polynomially with the number of particles have been developed over the past fifteen years to tackle closed-shell mid-mass nuclei. Open-shell nuclei have been further addressed by implementing variants based on the concept of spontaneous symmetry breaking (and restoration). Purpose: In order to access the spectroscopy of open-shell nuclei more systematically while controlling the numerical cost, we design a novel many-body method that combines the merit of breaking and restoring symmetries with those brought about by low-rank individual excitations. Methods: The recently proposed truncated configuration-interaction method based on optimized symmetry-broken and -restored states is extended to the z-signature symmetry associated with a discrete subgroup of SU(2). The highly-truncated N-body Hilbert subspace within which the Hamiltonian is diagonalized is spanned by a z-signature broken and restored Slater determinant vacuum and associated low-rank excitations. Results: The proposed method provides an excellent reproduction of the ground-state energy and of low-lying excitation energies of various z-signatures and total angular momenta. In doing so, the successive benefits of (i) breaking the symmetry, (ii) restoring the symmetry, (iii) including low-rank particle-hole excitations and (iv) optimizing the amount by which the underlying vacuum breaks the symmetry are illustrated. Conclusions: The numerical cost of the newly designed variational method is polynomial with respect to the system size. The present study confirms the results obtained previously for the attractive pairing Hamiltonian in connection with the breaking and restoration of U(1) global gauge symmetry. These two studies constitute a strong motivation to apply this method to realistic nuclear Hamiltonians.

Residual Z2 symmetries and leptonic mixing patterns from finite discrete subgroups of U(3)

We study embedding of non-commuting Z2 and Zm, m ≥ 3 symmetries in discrete subgroups (DSG) of U(3) and analytically work out the mixing patterns implied by the assumption that Z2 and Zm describe the residual symmetries of the neutrino and the charged lepton mass matrices respectively. Both Z2 and Zm are assumed to be subgroups of a larger discrete symmetry group Gf possessing three dimensional faithful irreducible representation. The residual symmetries predict the magnitude of a column of the leptonic mixing matrix UPMNS which are studied here assuming Gf as the DSG of SU(3) designated as type C and D and large number of DSG of U(3) which are not in SU(3). These include the known group series Σ(3n3), Tn(m), Δ(3n2, m), Δ(6n2, m) and Δ′(6n2, j, k). It is shown that the predictions for a column of |UPMNS| in these group series and the C and D types of groups are all contained in the predictions of the Δ(6N2) groups for some integer N. The Δ(6N2) groups therefore represent a sufficient set of Gf to obtain predictions of the residual symmetries Z2 and Zm.

CKM and PMNS mixing matrices from discrete subgroups of SU(2)

Remaining within the realm of the Standard Model(SM) local gauge group, this first principles derivation of both the PMNS and CKM matrices utilizes quaternion generators of the three discrete (i.e., finite) binary rotational subgroups of SU(2) called [3,3,2], [4,3,2], and [5,3,2] for three lepton families in R3 and four related discrete binary rotational subgroups [3,3,3], [4,3,3], [3,4,3], and [5,3,3] represented by four quark families in R4. The traditional 3x3 CKM matrix is extracted as a submatrix of the 4x4 CKM4 matrix. If these two additional quarks b' and t' of a 4th quark family exist, there is the possibility that the SM lagrangian may apply all the way down to the Planck scale. There are then numerous other important consequences. The Weinberg angle is derived using these same quaternion generators, and the triangle anomaly cancellation is satisfied even though there is an obvious mismatch of three lepton families to four quark families. In a discrete space, one can also use these generators to derive a unique connection from the electroweak local gauge group SU(2)L x U(1)Y acting in R4 to the discrete group Weyl E8 in R8. By considering Lorentz transformations in discrete (3,1)-D spacetime, one obtains another Weyl E8 discrete symmetry group in R8, so that the combined symmetry is Weyl E8 x Weyl E8 = "discrete" SO(9,1) in 10-D spacetime. This unique connection is in direct contrast to the 10500 possible connections for superstring theory!

Discrete flavor symmetries for degenerate solar neutrino pair and their predictions

Flavour symmetries appropriate for describing a neutrino spectrum with degenerate solar pair and a third massive or massless neutrino are discussed. We demand that the required residual symmetries of the leptonic mass matrices be subgroups of some discrete symmetry group $G_f$. $G_f$ can be a subgroup of SU(3) if the third neutrino is massive and we derive general results on the mixing angle predictions for various discrete subgroups of SU(3). The main results are: (a) All the SU(3) subgroups of type C fail in simultaneously giving correct $\theta_{13}$ and $\theta_{23}$. (b) All the groups of type D can predict a relation $\cos^2\theta_{13} \sin^2\theta_{23}=\frac{1}{3}$ among the mixing angles which appears to be a good zeroth order approximation. Among these, various $\Delta(6n^2)$ groups with $n\geq 8$ can simultaneously lead also to $\sin^2 \theta_{13}$ in agreement with global fit at 3$\sigma$. (c) The group $\Sigma(168)\cong PSL(2,7)$ predicts near to the best fit value for $\theta_{13}$ and $\theta_{23}$ within the 1$\sigma$ range. All discrete subgroups of U(3) with order $<512$ and having three dimensional irreducible representation are considered as possible $G_f$ when the third neutrino is massless. Only seven of them are shown to be viable and three of these can correctly predict $\theta_{13}$ and/or $\theta_{23}$. The solar angle remains undetermined at the leading order in all the cases due to degeneracy in the masses. A class of general perturbations which can correctly reproduce all the observables are discussed in the context of several groups which offer good leading order predictions.

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.

Maximal neutrino mixing from a minimal flavor symmetry

We study a number of models, based on a non-Abelian discrete group, that successfully reproduce the simple and predictive Yukawa textures usually associated with U(2) theories of flavor. These models allow for solutions to the solar and atmospheric neutrino problems that do not require altering successful predictions for the charged fermions or introducing sterile neutrinos. Although Yukawa matrices are hierarchical in the models we consider, the mixing between second- and third-generation neutrinos is naturally large. We first present a quantitative analysis of a minimal model proposed in earlier work, consisting of a global fit to fermion masses and mixing angles, including the most important renormalization group effects. We then propose two new variant models: The first reproduces all important features of the $\mathrm{SU}(5)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(2)$ unified theory with neither SU(5) nor U(2). The second demonstrates that discrete subgroups of SU(2) can be used in constructing viable supersymmetric theories of flavor without scalar universality even though SU(2) by itself cannot.

Non-abelian finite gauge theories

We study orbifolds of ${\cal N} = 4$ U(n) super-Yang-Mills theory given by discrete subgroups of SU(2) and SU(3). We have reached many interesting observations that have graph-theoretic interpretations. For the subgroups of SU(2), we have shown how the matter content agrees with current quiver theories and have offered a possible explanation. In the case of SU(3) we have constructed a catalogue of candidates for finite (chiral) ${\cal N}=1$ theories, giving the gauge group and matter content. Finally, we conjecture a McKay-type correspondence for Gorenstein singularities in dimension 3 with modular invariants of WZW conformal models. This implies a connection between a class of finite ${\cal N}=1$ supersymmetric gauge theories in four dimensions and the classification of affine SU(3) modular invariant partition functions in two dimensions.

Geometry, topology, and physics of non-Abelian lattices

After reviewing in some detail the notion of non-Euclidean lattices, whose domain of physical realization lies mostly in the novel carbon structures of the family offullerenes, we have discussed a number of physical problems denned over such lattices. We have shown that the group-theoretical definition of these lattices leads to “designing” new tubular regular structures, endowed with symmetries unheard of in the frame of customary crystallography, which combine features of extreme complexity and, at the same time, of great regularity. We have compared the role of the non-Abelian symmetries which these super-lattices are characterized by, with that of (discrete) harmonic (Fourier) lattice symmetry typical of customary crystallographic lattices. Many novel features enter into play, due to thenon-flatness of the related lattice geometry, which led us to a novel—sometimes unexpected—insight into the dynamical and/or thermodynamical properties of various physical systems which have these lattices as ambient space. We have analyzed how lattice topology bears on the complex combinatorics (related to loop-counting) of the classical Ising model. These lattices, even though finite, are, of course, much closer to being three-dimensional than regular 2D lattices simply equipped with periodic boundary conditions. We have shown, on the other hand, how the relation between the lattice symmetry (for example, in the case of fullerene, the discrete subgroup ofSU(2) that we have denotedg60 and the symmetry proper to the Hamiltonian of quantum systems of many itinerant interacting electrons (Hubbard-like models) allows us to reduce the calculation of the system spectral properties to a “size” that can be dealt with numerically with present-day numerical exact diagonalization techniques much more easily than a regular 3D cluster with a quite smaller number of sites.

Branes and fundamental groups

We consider bound states of D-branes wrapped around cycles with non-trivial fundamental groups of finite order. We find a new mechanism for binding D-branes by turning on flat discrete abelian and non-abelian gauge fields on their worldvolume. As a concrete application we study type IIB in the background where an S^3/G shrinks, where G is a discrete subgroup of SU(2) acting freely on S^3.

Do cosmic strings determine the number of generations?

What distinguishes the generations above the electroweak scale? We propose a model where three families of leptons and the electroweak bosons are assigned to the irreducible representations of the local binary tetrahedral group of order 24, a discrete subgroup of SU (2), and the local discrete symmetry Z 3. The scattering cross sections of leptons off the cosmic strings associated with the elements of the local discrete symmetry distinguish the members of the lepton families. The possible relevance of the cosmic strings to E 8 unification and the families of quarks are discussed.

SU(2) current algebra orbifolds of the Gaussian model

We examine a set of new c = 1 two-dimensional conformal field theories on the torus created via an orbifold construction on the SU(2) × SU(2) currents of the gaussian model. The partition functions of these models are calculated; a study of the discrete symmetries of these theories exhibits some of the remarkable structural similarities between the discrete subgroups of SU(2) and the simply laced Lie algebras.

Monte Carlo integration for lattice gauge theories with fermions

A Monte Carlo procedure is constructed for lattice gauge theories with fermions by replacing integration over fermion degrees of freedom in the path integral by conventional integration over effective boson degrees of freedom. The method is applied to gauge theories over two discrete subgroups of SU(2).

Monte Ćarlo calculations and a model of the phase structure for gauge theories on discrete subgroups of SU(2)

A Monte Carlo technique is used to evaluate path integrals for lattice gauge theories with gauge groups given by the SU(2) images of the rotational symmetries of a tetrahedron, a cube, and an icosahedron, called $\overline{T}$, $\overline{O}$, and $\overline{I}$, respectively. The coupling-constant dependence of the mean action per plaquette provides evidence for two phases, in each of these theories, separated by a first-order phase transition. The critical gauge coupling constant moves toward zero as the order of the group is increased. The mean plaquette action, the expectations of square gauge loops, and an estimate of the string tension for the largest group $\overline{I}$ agree with Creutz's results for SU(2) over a wide range of coupling constants. A model for the phase transitions in $\overline{T}$, $\overline{O}$, $\overline{I}$, and ${Z}_{N}$ is discussed which predicts critical coupling constants close to the observed values and suggests, as expected, that these transitions are a special property of gauge theories over discrete groups.

Phase structure of non-Abelian lattice gauge theories

The phase structure of four-dimensional lattice gauge theories based on finite non-Abelian groups is studied by Monte Carlo computations. All models examined exhibit a two-phase structure with a first-order phase transition. In three systems where the gauge group is a discrete subgroup of SU(2) the critical temperature moves toward zero as the order of the group increases and the high-temperature phase has confining properties.