Subgroups Of SU Research Articles

Spinorial description for Lorentzian hs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{hs} $$\\end{document}-IKKT

We introduce a novel spinorial description for the higher-spin gauge theory induced by the IKKT matrix model on an FLRW spacetime with Lorentzian signature, called Lorentzian hs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{hs} $$\\end{document}-IKKT theory. The new description is based on Weyl spinors transforming under the space-like isometry subgroup SL(2, ℂ) of the structure group SO(2, 4) ≃ SU(2, 2). It allows us to exploit the full power of the spinor formalism in Lorentzian signature, in contrast to a previous formalism based on the compact subgroup SU(2)L × SU(2)R of SU(2, 2). Some cubic vertices of the Yang-Mills sector and the corresponding scattering amplitudes are computed. We observe that the n-point (for n ≥ 4) tree-level amplitudes are typically non-trivial on-shell, but exponentially suppressed in the late-time regime. While Lorentz invariance of the higher-spin amplitudes is not manifest, it is expected to be restored by higher-spin gauge invariance.

Estimates of Picard modular cusp forms

Abstract In this article, for n ≥ 2 {n\geq 2} , we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of SU ⁢ ( ( n , 1 ) , ℂ ) {\mathrm{SU}((n,1),\mathbb{C})} . The main result of the article is the following result. Let Γ ⊂ SU ⁢ ( ( 2 , 1 ) , 𝒪 K ) {\Gamma\subset\mathrm{SU}((2,1),\mathcal{O}_{K})} be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let ℬ Γ k {{{\mathcal{B}_{\Gamma}^{k}}}} denote the Bergman kernel associated to the 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} , complex vector space of weight-k cusp forms with respect to Γ. Let 𝔹 2 {\mathbb{B}^{2}} denote the 2-dimensional complex ball endowed with the hyperbolic metric, and let X Γ := Γ \ 𝔹 2 {X_{\Gamma}:=\Gamma\backslash\mathbb{B}^{2}} denote the quotient space, which is a noncompact complex manifold of dimension 2. Let | ⋅ | pet {|\cdot|_{\mathrm{pet}}} denote the point-wise Petersson norm on 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} . Then, for k ≫ 1 {k\gg 1} , we have the following estimate: sup z ∈ X Γ ⁡ | ℬ Γ k ⁢ ( z ) | pet = O Γ ⁢ ( k 5 2 ) , \sup_{z\in X_{\Gamma}}|{{\mathcal{B}_{\Gamma}^{k}}}(z)|_{\mathrm{pet}}=O_{% \Gamma}(k^{\frac{5}{2}}), where the implied constant depends only on Γ.

A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds

Let G G be a finite subgroup of S U ( 4 ) \mathrm {SU}(4) such that its elements have age at most one. In the first part of this paper, we define K K -theoretic stable pair invariants on a crepant resolution of the affine quotient C 4 / G {\mathbb {C}}^4/G , and conjecture a closed formula for their generating series in terms of the root system of G G . In the second part, we define degree zero Donaldson-Thomas invariants of Calabi-Yau 4-orbifolds, develop a vertex formalism that computes the invariants in the toric case, and conjecture closed formulae for their generating series for the quotient stacks [ C 4 / Z r ] [{\mathbb {C}}^4/{\mathbb {Z}}_r] , [ C 4 / Z 2 × Z 2 ] [{\mathbb {C}}^4/{\mathbb {Z}}_2\times {\mathbb {Z}}_2] . Combining these two parts, we formulate a crepant resolution correspondence which relates the above two theories.

Trinification from E6 symmetry breaking

In the context of E6 Grand Unified Theories (GUTs), an intriguing possibility for symmetry breaking to the Standard Model (SM) group involves an intermediate stage characterized by either SU(3) × SU(3) × SU(3) (trinification) or SU(6) × SU(2). The more common choices of SU(5) and SO(10) GUT symmetry groups do not offer such breaking chains. We argue that the presence of a real (rank 2 tensor) representation 650 of E6 in the scalar sector is the minimal and likely only reasonable possibility to obtain one of the novel intermediate stages. We analyze the renormalizable scalar potential of a single copy of the 650 and find vacuum solutions that support regularly embedded subgroups SU(3) × SU(3) × SU(3), SU(6) × SU(2), and SO(10) × U(1), as well as specially embedded subgroups F4 and SU(3) × G2 that do not contain the SM gauge symmetry. We show that for a suitable choice of parameters, each of the regular cases can be obtained as the lowest among the analyzed minima in the potential.

Composite topological structures in SO(10)

We explore a variety of composite topological structures that arise from the spontaneous breaking of SO(10) to SU(3)c× U(1)em via one of its maximal subgroups SU(5) × U(1)χ, SU(4)c× SU(2)L× SU(2)R, and SU(5) × U(1)X (also known as flipped SU(5)). They include i) a network of ℤ strings which develop monopoles and turn into necklaces with the structure of ℤ2 strings, ii) dumbbells connecting two different types of monopoles, or monopoles and antimonpoles, iii) starfish-like configurations, iv) polypole configurations, and v) walls bounded by a necklace. We display these structures both before and after the electroweak breaking. The appearance of these composite structures in the early universe and their astrophysical implications including gravitational wave emission would depend on the symmetry breaking patterns and scales, and the nature of the associated phase transitions.

More on fake GUT

It is remarkable that the matter fields in the Standard Model (SM) are apparently unified into the SU(5) representations. A straightforward explanation of this fact is to embed all the SM gauge groups into a simple group containing SU(5), i.e., the grand unified theory (GUT). Recently, however, a new framework “fake GUT” has been proposed. In this new framework, the apparent matter unification can be explained by a chiral gauge group G, G ⊃ SU(5). We emphasize that the SM matter fields are not necessarily embed- ded into the chiral representations to explain the apparent unification. In this paper, we discuss details of concrete realizations of the fake GUT model. We first study the model based on SU(5) × U(2)H, where SU(3)c in the SM is from SU(5) while SU(2)L× U(1)Y are from the diagonal subgroups of SU(5) × U(2)H. We also extend this model to the one based on a semi-simple group, SU(5) × SU(3)H, so that U(2)H is embedded in SU(3)H. We also show that this framework predicts rather different decay patterns of the proton, compared to the conventional GUT.

Higgs alignment of visible and dark gauge groups

We discuss a dark family of lepton-like particles with their own “private” gauge bosons वμ and धμ under a local SU’(2) × U’(1) symmetry. The product of dark and visible gauge groups SU’(2) × U’(1) × SUw(2) × UY(1) is broken dynamically to the diagonal (vector-like) subgroup SU(2) × U(1) through the coupling of two fields (math) to the Higgs field and the dark lepton-like particles. This defines a new Higgs portal, where the “dark leptons” can contribute to the dark matter and interact with Standard Model matter through Higgs exchange.

Exploring partial μ−τ reflection symmetry at DUNE and Hyper-Kamiokande

We study origin, consequences and testability of a hypothesis of `partial $\mu$-$\tau$' reflection symmetry. This symmetry predicts $ |U_{\mu i}|=|U_{\tau i}|~(i=1,2,3) $ for a single column of the leptonic mixing matrix $U$. Depending on whether this symmetry holds for the first or second column of $U$ different correlations between $\theta_{23}$ and $ \delta_{CP} $ can be obtained. This symmetry can be obtained using discrete flavour symmetries. In particular, all the subgroups of SU(3) with 3-dimensional irreducible representation which are classified as class C or D can lead to partial $\mu$-$\tau$ reflection symmetry. We show how the predictions of this symmetry compare with the allowed area in the $\sin^2\theta_{23} - \delta_{CP}$ plane as obtained from the global analysis of neutrino oscillation data. Furthermore, we study the possibility of testing these symmetries at the proposed DUNE and Hyper-Kamiokande (HK) experiments (T2HK, T2HKK), by incorporating the correlations between $\theta_{23}$ and $ \delta_{CP}$ predicted by the symmetries. We find that when simulated data of DUNE and HK is fitted with the symmetry predictions, the $\theta_{23}-\delta_{CP}$ parameter space gets largely restricted near the CP conserving values of $ \delta_{CP} $. Finally, we illustrate the capability of these experiments to distinguish between the two cases leading to partial $\mu-\tau$ symmetry namely $|U_{\mu1}| = |U_{\tau 1}|$ and $|U_{\mu 2}| = |U_{\tau 2}|$.

Automorphism groups that are subgroups of SU(n,1)

ABSTRACTWe study the question of when the automorphism group of a bounded domain in is a subgroup of the automorphism group of the unit ball in .

Alternative [SU(3)]4 model of leptonic color and dark matter

The alternative [SU(3)]4 model of leptonic color and dark matter is discussed. It unifies at MU∼1014 GeV and has the low-energy subgroup SU(3)q×SU(2)l×SU(2)L×SU(2)R×U(1)X with (u,h)R instead of (u,d)R as doublets under SU(2)R. It has the built-in global U(1) dark symmetry which is generalized B–L. In analogy to SU(3)q quark triplets, it has SU(2)l hemion doublets which have half-integral charges and are confined by SU(2)l gauge bosons (stickons). In analogy to quarkonia, their vector bound states (hemionia) are uniquely suited for exploration at a future e−e+ collider.

The effective supergravity of little string theory

In this work we present the minimal supersymmetric extension of the five-dimensional dilaton-gravity theory that captures the main properties of the holographic dual of little string theory. It is described by a particular gauging of mathcal {N}=2 supergravity coupled with one vector multiplet associated with the string dilaton, along the U(1) subgroup of SU(2) R-symmetry. The linear dilaton in the fifth coordinate solution of the equations of motion (with flat string frame metric) breaks half of the supersymmetries to mathcal {N}=1 in four dimensions. Interest in the linear dilaton model has lately been revived in the context of the clockwork mechanism, which has recently been proposed as a new source of exponential scale separation in field theory.

Hypothesis of friedmons as dark matter particles and hypothesis of initial cosmological constant decay

Hypothesis of friedmons as dark matter particles is proposed. Friedmons are stable particles with a mass of billion nucleon masses. These particles correspond to the not yet been discovered exact symmetry group dual to the SU(2) group: for the Standard model symmetries and dual symmetries, the roles of exact and broken symmetries and corresponding stable and unstable particles change places. The hypothesis of the decay of the primordial de Sitter vacuum of the Planck density to an asymptotic state of the expanding Universe with de Sitter vacuum of the observed critical density is proposed. The T -duality and S-duality hypotheses relating subgroups SU(3)×SU(2)×U(1) and dual subgroups S˜Ũ(3)× S˜Ũ(2) × ˜Ũ(1) with decay of the primordial symmetry group E(8) × ˜Ẽ(8) are proposed. In particular, these dualities relate the minimum Planck length 10−13 cm to the primordial curvature radius 10−13 cmof theMetagalaxy of the Planck density and its modern curvature radius of 1028 cm. That is, the probable relation of the Planck mass to the Metagalaxy mass of 1061 Planck masses is indicated.

Couplings in D(2, 1; \u03b1) superconformal mechanics from the SU(2) perspective

Dynamical realizations of the most general N = 4 superconformal group in one dimension D(2, 1; α) are reconsidered from the perspective of the R-symmetry subgroup SU(2). It is shown that any realization of the R-symmetry subalgebra in some phase space can be extended to a representation of the Lie superalgebra corresponding to D(2, 1; α). Novel couplings of arbitrary number of supermultiplets of the type (1, 4, 3) and (0, 4, 4) to a single supermultiplet of either the type (3, 4, 1), or (4, 4, 0) are constructed. D(2, 1; α) superconformal mechanics describing superparticles propagating near the horizon of the extreme Reissner-Nordström-AdS-dS black hole in four and five dimensions is considered. The parameter α is linked to the cosmological constant.

Quregisters, Symmetry Groups and Clifford Algebras

Natural one-to-one and two-to-one homomorphisms from SO(3) into SU(2) are built conventionally, and the collection of qubits, is identified with a subgroup of SU(2). This construction is suitable to be extended to corresponding tensor powers. The notions of qubits, quregisters and qugates are translated into the language of symmetry groups. The corresponding elements to entangled states in the tensor product of Hilbert spaces reflect entanglement properties as well, and in this way a notion of entanglement is realised in the tensor product of symmetry groups.

SU(2)L × SU(2)R minimal dark matter with 2 TeV W′

We construct the minimal dark matter models in the left–right symmetric extensions of the standard model (SM), where the gauge symmetry SU(3)C×SU(2)L×SU(2)R×U(1)B−L is broken into its subgroup SU(3)C×U(1)em by nonzero VEVs of a SU(2)R doublet HR and a SU(2)L×SU(2)R bidoublet H. A possible candidate of dark matter is explored in the framework of minimal dark matter considering SU(2)L,R multiplet scalar bosons and fermions. Then we focus on SU(2)R quintuplet fermions with B−L charges 0, 2 and 4 as the minimal dark matter candidates and investigate their phenomenology. We show that the dark matter in the model can provide observed relic density with 2 TeV W′ boson which is motivated by the ATLAS diboson excess and CMS eejj excess. The possible mass of dark matter is predicted for each B−L charge. We then estimate the scattering cross section of dark matter with nucleon and production cross section of charged components in the quintuplets at the LHC.

Δ(27) family symmetry and neutrino mixing

The observed neutrino mixing, having a near maximal atmospheric neutrino mixing angle and a large solar mixing angle, is close to tri-bi-maximal. This structure may be related to the existence of a discrete non-Abelian family symmetry. In this paper the family symmetry is the non-Abelian discrete group Δ(27), a subgroup of SU(3) with triplet and anti-triplet representations. Different frameworks are constructed in which the mixing follows from combining fermion mass terms with the vacuum structure enforced by the discrete symmetry. Mass terms for the fermions originate from familon triplets, anti-triplets or both. Vacuum alignment for the family symmetry breaking familons follows from simple invariants.

Testing the role of the Barbero-Immirzi parameter and the choice of connection in loop quantum gravity

We study the role of the Barbero-Immirzi parameter $\gamma$ and the choice of connection in the construction of (a symmetry-reduced version of) loop quantum gravity. We start with the four-dimensional Lorentzian Holst action that we reduce to three dimensions in a way that preserves the presence of $\gamma$. In the time gauge, the phase space of the resulting three-dimensional theory mimics exactly that of the four-dimensional one. Its quantization can be performed, and on the kinematical Hilbert space spanned by SU(2) spin network states the spectra of geometric operators are discrete and $\gamma$-dependent. However, because of the three-dimensional nature of the theory, its SU(2) Ashtekar-Barbero Hamiltonian constraint can be traded for the flatness constraint of an sl(2,C) connection, and we show that this latter has to satisfy a linear simplicity-like condition analogous to the one used in the construction of spin foam models. The physically relevant solution to this constraint singles out the non-compact subgroup SU(1,1), which in turn leads to the disappearance of the Barbero-Immirzi parameter and to a continuous length spectrum, in agreement with what is expected from Lorentzian three-dimensional gravity.

On the classification of 6D SCFTs and generalized ADE orbifolds

We study (1,0) and (2,0) 6D superconformal field theories (SCFTs) that can be constructed in F-theory. Quite surprisingly, all of them involve an orbifold singularity C^2 / G with G a discrete subgroup of U(2). When G is a subgroup of SU(2), all discrete subgroups are allowed, and this leads to the familiar ADE classification of (2,0) SCFTs. For more general U(2) subgroups, the allowed possibilities for G are not arbitrary and are given by certain generalizations of the A- and D-series. These theories should be viewed as the minimal 6D SCFTs. We obtain all other SCFTs by bringing in a number of E-string theories and/or decorating curves in the base by non-minimal gauge algebras. In this way we obtain a vast number of new 6D SCFTs, and we conjecture that our construction provides a full list.

Higher spin currents in Wolf space. Part I

For the N=4 superconformal coset theory described by SU(N+2)/SU(N) (that contains a Wolf space) with N=3, the N=2 WZW affine current algebra with constraints is obtained. The 16 generators of the large N=4 linear superconformal algebra are described by those WZW affine currents explicitly. By factoring out four spin-1/2 currents and the spin-1 current from these 16 generators, the remaining 11 generators (spin-2 current, four spin-3/2 currents, and six spin-1 currents) corresponding to the large N=4 nonlinear superconformal algebra are obtained. Based on the recent work by Gaberdiel and Gopakumar on the large N=4 holography, the extra 16 currents, with spin contents (1, 3/2, 3/2, 2), (3/2, 2, 2, 5/2), (3/2, 2, 2, 5/2), and (2, 5/2, 5/2, 3) described in terms of N=2 multiplets, are obtained and realized by the WZW affine currents. As a first step towards N=4 W algebra (which is NOT known so far), the operator product expansions (OPEs) between the above 11 currents and these extra 16 higher spin currents are found explicitly. It turns out that the composite fields with definite U(1) charges, made of above (11+16) currents (which commute with the Wolf space subgroup SU(N=3) x SU(2) x U(1) currents), occur in the right hand sides of these OPEs.

Nonabelian dark matter with resonant annihilation

We construct a model based on an extra gauge symmetry, SU(2)_X x U(1)_{B-L}, which can provide gauge bosons to serve as weakly-interacting massive particle dark matter. The stability of the dark matter is naturally guaranteed by a discrete Z_2 symmetry that is a subgroup of SU(2)_X. The dark matter interacts with standard model fermions by exchanging gauge bosons which are linear combinations of SU(2)_X x U(1)_{B-L} gauge bosons. With the appropriate choice of representation for the new scalar multiplet whose vacuum expectation value spontaneously breaks the SU(2)_X symmetry, the relation between the new gauge boson masses can naturally lead to resonant pair annihilation of the dark matter. After exploring the parameter space of the new gauge couplings subject to constraints from collider data and the observed relic density, we use the results to evaluate the cross section of the dark matter scattering off nucleons and compare it with data from the latest direct detection experiments. We find allowed parameter regions that can be probed by future direct searches for dark matter and LHC searches for new particles.