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We provide an explicit description of the Poincare duals of each generator of the rational cohomology ring of the S U ( 2 ) character variety for a genus g surface with central extension — equivalently, that of the moduli space of stable holomorphic bundles of rank 2 and odd degree.

New analysis regarding the structure of center vortices is presented: Using data from gluonic SU(2) lattice simulation with Wilson action, a correlation of fluctuations in color space to the curvature of vortex fluxes was found. Finite size effects of the S2-homogeneity hint at color homogeneous regions on the vortex surface.

We classify $SU(2)$-cyclic and $SU(2)$-abelian 3-manifolds, for which every representation of the fundamental group into $SU(2)$ has cyclic or abelian image respectively, among geometric 3-manifolds which are not hyperbolic. As an application, we give examples of hyperbolic 3-manifolds which do not admit degree-1 maps to any Seifert fibered manifold other than $S^3$ or a lens space. We also produce infinitely many one-cusped hyperbolic manifolds with at least four $SU(2)$-cyclic Dehn fillings, one more than the number of cyclic fillings allowed by the cyclic surgery theorem., Comment: 36 pages, 4 figures. v2: accepted version; added Lemma 2.4 to streamline parts of section 2

We study quantum properties of SU(2|1) supersymmetric (deformed $$ \mathcal{N} $$ N = 4, d = 1 supersymmetric) extension of the superintegrable Smorodinsky-Winternitz system on a complex Euclidian space ℂN. The full set of wave functions is constructed and the energy spectrum is calculated. It is shown that SU(2|1) supersymmetry implies the bosonic and fermionic states to belong to separate energy levels, thus exhibiting the “even-odd” splitting of the spectra. The superextended hidden symmetry operators are also defined and their action on SU(2|1) multiplets of the wave functions is given. An equivalent description of the same system in terms of superconformal SU(2|1, 1) quantum mechanics is considered and a new representation of the hidden symmetry generators in terms of the SU(2|1, 1) ones is found.

We describe the connected components of the space $\text {Hom}(\Gamma ,SU(2))$ of homomorphisms for a discrete nilpotent group $\Gamma$. The connected components arising from homomorphisms with non-abelian image turn out to be homeomorphic to $\mathbb {RP}^{3}$. We give explicit calculations when $\Gamma$ is a finitely generated free nilpotent group. In the second part of the paper, we study the filtration $B_{\text {com}} SU(2)=B(2,SU(2))\subset \cdots \subset B(q,SU(2))\subset \cdots$ of the classifying space $BSU(2)$ (introduced by Adem, Cohen and Torres-Giese), showing that for every $q\geq 2$, the inclusions induce a homology isomorphism with coefficients over a ring in which 2 is invertible. Most of the computations are done for $SO(3)$ and $U(2)$ as well.

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an [Formula: see text]-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at [Formula: see text]. Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for all representations of dimension at most [Formula: see text]. Along the way, we compute the Hilbert series of the module of covariants of an arbitrary [Formula: see text]- or [Formula: see text]-module as well as its first three Laurent coefficients.

We explicitly show that the groups of $2 \times 2$ unitary matrices with determinant equal to 1 whose entries are double or dual numbers are homomorphic to ${\rm SO}(2,1)$ or to the group of rigid motions of the Euclidean plane, respectively, and we introduce the corresponding two-component spinors. We show that with the aid of the double numbers we can find generating functions for separable solutions of the Laplace equation in the $(2 + 1)$ Minkowski space, which contain special functions that also appear in the solution of the Laplace equation in the three-dimensional Euclidean space, in spheroidal and toroidal coordinates.

We show that, under a natural scaling, the small-time behavior of the logarithmic derivatives of the subelliptic heat kernel on $SU(2)$ converges to their analogues on the Heisenberg group at time 1. Realizing $SU(2)$ as $\mathbb{S}^3$, we then generalize these results to higher-order odd-dimensional spheres equipped with their natural subRiemannian structure, where the limiting spaces are now the higher-dimensional Heisenberg groups., 32 pages, corrected errors in preliminary estimates in Section 2.1

We study supersymmetry breaking deformations of the $\mathcal{N}=1$ 5d fixed point known as $E_1$, the UV completion of $SU(2)$ super-Yang-Mills. The phases of the non-supersymmetric theory can be characterized by Chern-Simons terms involving background $U(1)$ gauge fields, allowing us to identify a phase transition at strong coupling. We propose that this may signify the emergence of a non-trivial, non-supersymmetric CFT in $d=4+1$ dimensions., 18 pages, 2 figures. v2: minor typos fixed, references added

In search for an effective model of quark confinement we study the vacuum of SU(2) quantum chromodynamic with lattice simulations using Wilson action. Assuming that center vortices are the relevant excitations causing confinement, we analyzed their physical size and their color structure. We present confirmations for a vanishing thickness of center vortices in the continuum limit and hints at their color structure. This is the first time that algorithms for the detection of thick center vortices based on non-trivial center regions has been used.

Starting with the braided quantum group SU q ( 2 ) for a complex deformation parameter q we perform the construction of the quotient SU q ( 2 ) / T which serves as a model of a quantum sphere. Then we follow the reasoning of Podleś who for real q classified quantum spaces with the action of SU q ( 2 ) with appropriate spectral properties. These properties can also be expressed in the context of the braided quantum SU q ( 2 ) (with complex q) and we find that they lead to precisely the same family of quantum spaces as found by Podleś for the real parameter | q | .

In this paper we first provide the proof of $SU(2)$ Y-Map convergence. Then, by using $SU(1,1)$ LQG simplicity constraints we define $SU(1,1)$ Y-Map from infinitely differentiable with a compact support functions on $SU(1,1)$ to the functions (not necessarily square integrable) on $SL(2,C)$, and prove its convergence as well., arXiv admin note: text overlap with arXiv:1509.01312

We study the semiclassical limit of a class of invariant tensors for infinite-dimensional unitary representations of $\mathrm{SL}(2,\mathbb{C})$ of the principal series, corresponding to generalized Clebsch-Gordan coefficients with $n\geq3$ legs. We find critical configurations of the quantum labels with a power-law decay of the invariants. They describe 3d polygons that can be deformed into one another via a Lorentz transformation. This is defined viewing the edge vectors of the polygons are the electric part of bivectors satisfying a (frame-dependent) relation between their electric and magnetic parts known as $\gamma$-simplicity in the loop quantum gravity literature. The frame depends on the SU(2) spin labelling the basis elements of the invariants. We compute a saddle point approximation using the critical points and provide a leading-order approximation of the invariants. The power-law is universal if the SU(2) spins have their lowest value, and $n$-dependent otherwise. As a side result, we provide a compact formula for $\gamma$-simplicity in arbitrary frames. The results have applications to the current EPRL model, but also to future research aiming at going beyond the use of fixed time gauge in spin foam models., Comment: 29 pages and 3 figures. v2:minor corrections and a subsection added to match published version

The SU(2)-Lie algebraic approach based on the dynamical group U1(2)×U2(2)×⋯×Un(2) to describe a set of n local oscillators is analysed from the point of view of its connection with configuration sp...

We introduce a formalism for describing four-dimensional scattering amplitudes for particles of any mass and spin. This naturally extends the familiar spinor-helicity formalism for massless particles to one where these variables carry an extra SU(2) little group index for massive particles, with the amplitudes for spin S particles transforming as symmetric rank 2S tensors. We systematically characterise all possible three particle amplitudes compatible with Poincare symmetry. Unitarity, in the form of consistent factorization, imposes algebraic conditions that can be used to construct all possible four-particle tree amplitudes. This also gives us a convenient basis in which to expand all possible four-particle amplitudes in terms of what can be called "spinning polynomials". Many general results of quantum field theory follow the analysis of four-particle scattering, ranging from the set of all possible consistent theories for massless particles, to spin-statistics, and the Weinberg-Witten theorem. We also find a transparent understanding for why massive particles of sufficiently high spin can not be "elementary". The Higgs and Super-Higgs mechanisms are naturally discovered as an infrared unification of many disparate helicity amplitudes into a smaller number of massive amplitudes, with a simple understanding for why this can't be extended to Higgsing for gravitons. We illustrate a number of applications of the formalism at one-loop, giving few-line computations of the electron (g-2) as well as the beta function and rational terms in QCD. "Off-shell" observables like correlation functions and form-factors can be thought of as scattering amplitudes with external "probe" particles of general mass and spin, so all these objects--amplitudes, form factors and correlators, can be studied from a common on-shell perspective., Comment: 79 pages, published version, multiple typos corrected and references updated

We present a major update on our investigations of SU(2) gauge theory with one Dirac flavor in the adjoint representation on the lattice. In particular we consider larger volumes, as well as four different values of the gauge coupling. We provide results for the spectrum including gluonic, fermionic, and hybrid observables, Polyakov loops, and the anomalous dimension of the fermionic condensate from the Dirac mode number. These data confirm that the theory is close to the lower boundary of the conformal window for adjoint fermions. Our investigations provide important insights regarding the realization of different infrared scenarios that have been conjectured for this theory., 19 pages, 10 figures. Version accepted for publication in PRD

We investigate the phenomenology of an extension of the Standard Model (SM) by a non-abelian gauge group $SU(2)_{HS}$ where all SM particles are singlets under this gauge group, and a new scalar representation $\phi$ that is singlet under SM gauge group and doublet under $SU(2)_{HS}$. In this model, the dark matter (DM) candidates are the three mass degenerate dark photons $A_{i}$ $(i=1,2,3)$ of $SU(2)_{HS}$; and the hidden sector interacts with the (SM) particles through the Higgs portal interactions. Consequently, there will be a new CP-even scalar $\eta$ that could be either heavier or lighter than the SM-like Higgs. By taking into account all theoretical and experimental constraints such as perturbativity, unitarity, vacuum stability, non-SM Higgs decays, DM direct detection, DM relic density, we found viable DM is possible in the range from GeV to TeV. Within the viable parameters space, the both of the triple Higgs coupling and the di-Higgs production at LHC14 could be enhanced or reduced depending on the scalar mixing and the mass of the scalar particle $\eta$., Comment: 22 pages & 11 figures

We introduce an axion-inflation model embedded in the left-right symmetric extension of the Standard Model in which ${W}_{R}$ is coupled to the axion. This model merges three milestones of modern cosmology, i.e., inflation, cold dark matter, and baryon asymmetry. Thus, it can naturally explain the observed coincidences among cosmological parameters, i.e., ${\ensuremath{\eta}}_{\mathsf{B}}\ensuremath{\approx}{P}_{\ensuremath{\zeta}}$ and ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{DM}}\ensuremath{\simeq}5{\mathrm{\ensuremath{\Omega}}}_{\mathsf{B}}$. The source of asymmetry is spontaneous $CP$ violation in the physics of inflation, and the lightest right-handed neutrino is the cold dark matter candidate with mass ${m}_{{N}_{1}}\ensuremath{\sim}1\text{ }\text{ }\mathrm{GeV}$. The introduced mechanism does not rely on the largeness of the unconstrained $CP$-violating phases in the neutrino sector or fine-tuned masses for the heaviest right-handed neutrinos. It has two unknown fundamental scales, i.e., scale of inflation ${\mathrm{\ensuremath{\Lambda}}}_{\mathrm{inf}}=\sqrt{H{M}_{\mathrm{Pl}}}$ and left-right symmetry breaking ${\mathrm{\ensuremath{\Lambda}}}_{F}$. Sufficient matter asymmetry demands that ${\mathrm{\ensuremath{\Lambda}}}_{\mathrm{inf}}\ensuremath{\approx}{\mathrm{\ensuremath{\Lambda}}}_{F}$. Baryon asymmetry and dark matter today are remnants of a pure quantum effect (chiral anomaly) in inflation, which, thanks to flavor effects, has been memorized by cosmic evolution.

The generalized SU(2) Proca theory (GSU2P) is a variant of the well known generalized Proca theory where the vector field belongs to the Lie algebra of the SU(2) group of global transformations under which the action is made invariant. New interesting possibilities arise in this framework because of the existence of new interactions of purely non-Abelian character and new configurations of the vector field resulting in spatial spherical symmetry and the cosmological dynamics being driven by the propagating degrees of freedom. We study the two-dimensional phase space of the system that results when the cosmic triad configuration is employed in the Friedmann-Lemaitre-Robertson-Walker background and find an attractor curve whose attraction basin both covers almost all the allowed region and does not include a Big-Bang singularity. Such an attractor curve corresponds to a primordial inflationary solution that has the following characteristic properties: 1.) it is a de Sitter solution whose Hubble parameter is regulated by a generalized version of the SU(2) group coupling constant, 2.) it is constant-roll including, as a limiting case, the slow-roll variety, 3.) a number of e-folds $N > 60$ is easily reached, 4.) it has a graceful exit into a radiation dominated period powered by the canonical kinetic term of the vector field and the Einstein-Hilbert term. The free parameters of the action are chosen such that the tensor sector of the theory is the same as that of general relativity at least up to second-order perturbations, thereby avoiding the presence of ghost and Laplacian instabilities in the tensor sector as well as making the gravity waves propagate at light speed. This is a proof of concept of the interesting properties we could find in this scenario when the coupling constants be replaced by general coupling functions and more terms be discovered in the GSU2P., LaTeX file in WileYMSP-template style, 28 pages, 20 figures. Version published in Annalen der Physik. Dedicated to Carlos J. Quimbay Herrera, Marta A. Losada Falk, and David H. Lyth

At the classical level, the SU(2/1) superalgebra offers a natural description of the elementary particles: leptons and quarks massless states, graded by their chirality, fit the smallest irreducible representations of SU(2/1). Our new proposition is to pair the left/right space-time chirality with the superalgebra chirality and to study the model at the one-loop quantum level. If, despite the fact that they are non-Hermitian, we use the odd matrices of SU(2/1) to minimally couple an oriented complex Higgs scalar field to the chiral Fermions, novel anomalies occur. They affect the scalar propagators and vertices. However, these undesired new terms cancel out, together with the Adler-Bell-Jackiw vector anomalies, because the quarks compensate the leptons. The unexpected and striking consequence is that the scalar propagator must be normalized using the antisymmetric super-Killing metric and the scalar-vector vertex must use the symmetric d_aij structure constants of the superalgebra. Despite this extraordinary structure, the resulting Lagrangian is actually Hermitian., A new anomaly free SU(2/1) superalgebra QFT model with a surprising super covariant scalar Lagrangian is presented. 12 pages, 30 references, plus 9 annexes giving in extenso all the matrices of SU(2/1) for leptons, quarks and the three families indecomposable representations. In v2, some citations were modified. In v3, relative to the version published in JHEP, we fixed a sign in equation H.3

We study the effects of addition of the Chern-Simons (CS) term in the minimal Yang-Mills (YM) matrix model composed of two $2\ifmmode\times\else\texttimes\fi{}2$ matrices with $SU(2)$ gauge and $SO(2)$ global symmetry. We obtain the Hamiltonian of this system in appropriate coordinates and demonstrate that its dynamics is sensitive to the values of both the CS coupling, $\ensuremath{\kappa}$, and the conserved conjugate momentum, ${p}_{\ensuremath{\phi}}$, associated to the $SO(2)$ symmetry. We examine the behavior of the emerging chaotic dynamics by computing the Lyapunov exponents and plotting the Poincar\'e sections as these two parameters are varied and, in particular, find that the largest Lyapunov exponents evaluated within a range of values of $\ensuremath{\kappa}$ are above what is computed at $\ensuremath{\kappa}=0$, for $\ensuremath{\kappa}{p}_{\ensuremath{\phi}}l0$. We also give estimates of the critical exponents for the Lyapunov exponent as the system transits from the chaotic to nonchaotic phase with ${p}_{\ensuremath{\phi}}$ approaching to a critical value.

We study string breaking in the three dimensional SU(2) Higgs model, using values of the gauge coupling for which the confinement-like and Higgs-like regions of the phase diagram are separated just by a smooth crossover. We show that even in the presence of string breaking, the confining part of the interquark potential is well described by the Effective String Theory and that also the fine details of the effective string, like the higher order terms of the Nambu-Goto action or the boundary correction, can be precisely extracted from the fits and agree with the effective string predictions. We comment on the implications of these results for QCD simulations with dynamical quarks., 9 pages, 6 eps figures, minor changes and some references added. To be published on Phys. Rev. D

This paper considers homomorphism of the Lie group SU(2) to the Lie group SO(3) of all rotations of 3- dimensional Euclidean space from Observers Mathematics point of view. In our work, we proved that in Observers Mathematics the probability of spin-j transformation is a homomorphism (representation) of SU(2 ) to the set of matrix transformations of a linear space of polynomial functions is less than 1, and got corresponding results for elementary fermions and bosons. As a continuation of these results we proved here the following theorems: Theorem 1. In Observers Mathematics the probability of two-to-one transformation of SU(2) to SO(3) is Lie groups homomorphism (representation) is less than 1. Theorem 2. The probability of two-to-one transformation and spin-j transformation (j = 1) are equivalent in Observers Mathematics is less than 1.

We propose a vector dark matter model with an exotic dark SU(2) gauge group. Two Higgs triplets are introduced to spontaneously break the symmetry. All of the dark gauge bosons become massive, and the lightest one is a viable vector DM candidate. Its stability is guaranteed by a remaining Z_2 symmetry. We study the parameter space constrained by the Higgs measurement data, the dark matter relic density, and direct and indirect detection experiments. We find numerous parameter points satisfying all the constraints, and they could be further tested in future experiments. Similar methodology can be used to construct vector dark matter models from an arbitrary SO(N) gauge group., 25 pages, 5 figures

We compute the large N limit of the partition function of the Euclidean Yang--Mills measure with structure group SU(N) or U(N) on all closed compact surfaces, orientable or not, excepted for the sphere and the projective plane. This limit is finite as opposed to the case of the sphere and presumably the projective plane. We expect that the results we present might give an insight towards the master field on these surfaces., 25 pages, 4 figures

Non-Abelian gauge field plays an important role in high-energy physics through Yang-Mill theory, in condensed-matter physics, for instance, with anomalous quantum Hall effect, and in quantum information. In its SU(2) symmetry version, Hamiltonians with non-Abelian gauge field reduce to spin-orbit coupling-like systems, taking the general form: H = p 2 /2m+ A.p /m , where m is the mass of the particle, p the momentum and A the non-Abelian gauge field with 2x2 matrix components, decomposed on SU(2) Lie group generators. These class of Hamiltonians can be simulate with ultracold gas [1] using various methods [2] . The topological properties of spin-orbit-coupled system have been explore analysing the energy bands in 2D Fermionic system [3] . Here, the eigenenergies have momentum dependence, leading to Rabi flopping with periodic change of quasi-momentum, even if the non-Abelian gauge field is homogenous. This oscillation is similar to relativistic Zitterbewegung predicted in Dirac equation and simulated in ultracold system [4] .

Previously we have shown that the algebra $$\begin{aligned} J_3 [{\mathbb {C}}\otimes {\mathbb {O}}] \otimes C \ell (4,{\mathbb {C}}), \end{aligned}$$ given by the tensor product of the complex exceptional Jordan $$J_3 [{\mathbb {C}}\otimes {\mathbb {O}}]$$ and the complex Clifford algebra $$C \ell (4,{\mathbb {C}})$$ , can describe all of the spinorial degrees of freedom of three generations of fermions in four-space-time dimensions. We extend our construction to show that it also includes the degrees of freedom of three sets of pairs of complex scalar Higgs-doublets $$\{{\mathbf {H}}^{(m)}_L, {\mathbf {H}}^{(m)}_R\}; m = 1,2,3$$ , and their $$\mathrm {CPT}$$ conjugates. Furthermore, a close inspection of the fermion structure of each generation reveals that it fits naturally with the sixteen complex-dimensional representation of the internal left/right symmetric gauge group $$G_{LR} = \mathrm {SU}(3)_C \times \mathrm {SU}(2)_L \times \mathrm {SU}(2)_R \times \mathrm {U}(1)$$ . It is reviewed how the latter group emerges from the intersection of $$\mathrm {SO}(10)$$ and $$\mathrm {SU}(3) \times \mathrm {SU}(3) \times \mathrm {SU}(3)$$ in $$E_6$$ . In the concluding remarks we briefly discuss the role that the extra Higgs fields may have as dark matter candidates; the construction of Chern–Simons-like matrix cubic actions; hexaquarks; supersymmetry and Clifford bundles over the complex-octonionic projective plane $$({\mathbb {C}}\otimes {\mathbb {O}}) {\mathbb {P}}^2$$ whose isometry group is $$E_6$$ .

Tree-level dynamical stability of scalar field potentials in renormalizable theories can in principle be expressed in terms of positivity conditions on quartic polynomial structures. However, these conditions cannot always be cast in a fully analytical resolved form, involving only the couplings and being valid for all field directions. In this paper we consider such forms in three physically motivated models involving $SU(2)$ triplet scalar fields: the Type-II seesaw model, the Georgi-Machacek model, and a generalized two-triplet model. A detailed analysis of the latter model allows one to establish the full set of necessary and sufficient boundedness-from-below conditions. These can serve as a guide, together with unitarity and vacuum structure constraints, for consistent phenomenological (tree-level) studies. They also provide a seed for improved loop-level conditions and encompass in particular the leading ones for the more specific Georgi-Machacek case. Incidentally, we present complete proofs of various properties and also derive general positivity conditions on quartic polynomials that are equivalent to but much simpler than the ones used in the literature.