A noncommutative geometric approach to left-right symmetric weak interactions

By combining the generalized exterior algebra of forms over a noncommutative algebra with the gauging of discrete directions and the associated Higgs fields, we consider the construction of the bosonic sector of left-right symmetric models of the form SU(2) _L SU(2) _R U(1). We see that within this formalism maximal use can be made of the gauge connection associated with the noncommutative graded algebra.     

SU(l|m|n)阶化代数及其规范模型

本文将阶化李代数SU(m|n)扩充到SU(l|m|n),并讨论了它的局部规范理论。找到了规范不变的拉格郎日量,并将SU(2|1)弱电模型推广到统一强、弱、电相互作用模型。 关键词：     

Regular Basis and R-Matrices for the ^(su)(n)k Knizhnik–Zamolodchikov Equation

Dynamical R-matrix relations are derived for the group-valued chiral vertex operators in the SU(n) WZNW model from the KZ equation for a general four-point function including two step operators. They fit the exchange relations of the U _q (sl _n ) covariant quantum matrix algebra derived previously by solving the dynamical Yang–Baxter equation. As a byproduct, we extend the regular basis introduced earlier for SU(2) chiral fields to SU(n) step operators and display the corresponding triangular matrix representation of the braid group.     

Can the JPC = 1++ mesons form an ideal nonet?

In the framework of asymptotic SU(3) and the chiral SU(3) ? SU(3) charge algebra, it is argued that the 1⁺⁺ mesons can form an (almost) ideal nonet along with the 1^-- mesons. Experimental implications are also discussed.     

Higgs fields and the determination of the Weinberg angle

SU(2) × U(1) gauge theories, in which the Higgs fields transform as doublets under SU(2) are interpreted as pure Yang-Mills theories in six dimensions, the components of the gauge potentials in the extra dimensions playing the role of the Higgs' fields. Two consistent theories are discovered: one in which SU(2) × U(1) is embedded in SU(3) and the vector bosons remain massless - and another where SU(2) × U(1) is embedded in the graded Lie algebra SU(2|1), the symmetry is spontaneously broken in a natural fashion and the theory is equivalent to that of Weinberg and Salam, with a specific value 30° for the Weinberg angle and a prediction of the Higgs' mass.     

Departures from chiral symmetry

We review the physical concepts supporting the notion of an approximate hadron symmetry with special emphasis on the Nambu-Goldstone realizations of chiral SU (2) × SU (2) and SU (3) × SU (3). We stress the role of perturbation theory in the symmetry breaking as the technical instrument to connect broken symmetries with experiment. This is an alternate to the treatments that stress PCAC and current algebra. We find that chiral SU (2) × SU (2) is a good hadron symmetry to within 7% making it the best hadron symmetry after isotopic symmetry. The nonrenormalization theorem, Σ-terms, K_l3 decay, η→3π decay, the Goldberger-Treiman relation and many other specific processes and their relation to approximate chiral symmetry are discussed.     

SU(2) ⊗ U(1) gauge theories of the generalized σ model for π0, η, η' → γγ decays

By embedding the chiral current-mixing gauge theories in the SU(2)_L ? SU(2)_R generalized σ model, it is shown that the correct sign and magnitude for π⁰ → γγ decay, as well as the SU(3) relation of π⁰, η, η' → γγ decays can be obtained within the framework of SU(2) ? U(1) gauge theories of weak and electromagnetic interactions.     

Light Quark Hadron Mass Scale

With a symmetry procedure based on Noether's theorem, the field equation of motion is obtained from the Dirac Hamiltonian H(D_μ) of a massless quark interacting with a gluon. The equation of motion is the Yang-Mills equation with external current which is spin-dependent and follows from the group algebra. In addition to the pure gauge solution we find a gauge covariant solution which follows from current conservation and sets the mass scale m₀/M = g². This gluon field is due to the density of dipole moments squared and represents four harmonic oscillators with quadratic constraints; the gluon can be written as a string potential or as a 1/x potential with a sharp cutoff. The chiral symmetry group G^spin × G^D gives the light quark hadron degenerate multiplet mass spectrum in terms of m₀[SU(2) × SU(2)] with the spinorial decomposition and the multipole breaks into dipoles. Scaling from atomic lengths it is found that g = em₀/nM for light quarks is the quark charge e/3 renormalized by m₀/M and g is magnetic. Thus quarks occur at the ends of spinning magnetic strings with dipole lengths ∼m₀⁻¹. The mass scale is that of a degenerate magnetic multipole with charge n = 3, 4… .     

Fusion rings and geometry

The algebraic structure of fusion rings in rational conformal field theories is analyzed in detail in this paper. A formalism which closely parallels classical tools in the study of the cohomology of homogeneous spaces is developed for fusion rings, in general, and for current algebra theories, in particular. It is shown that fusion rings lead to a natural orthogonal polynomial structure. The rings are expressed through generators and relations. The relations are then derived from some potentials leading to an identification of the fusion rings with deformations of affine varieties. In general, the fusion algebras are mapped to affine varieties which are the locus of the relations. The connection with modular transformations is investigated in this picture. It is explained how chiral algebras, arising inN=2 superconformal field theory, can be derived from fusion rings. In particular, it is argued that theories of the typeSU(N) _k /SU(n–1) are theN=2 counterparts of Grassmann manifolds and that there is a natural identification of the chiral fields with Schubert varieties, which is a graded algebra isomorphism.Supported in part by NSF grant PHY 89-04035 supplemented by funds from NASA     

A symmetry reduction scheme of the Dirac algebra without dimensional defects

In relating the Dirac algebra to homogeneous coordinates of a projective geometry, we present a simple geometric scheme which allows to identify various Lie algebras and Lie groups well-known from classical physics as well as from quantum field theory. We introduce a 1 -point-compactification and quaternionic Möbius transformations, and we use SU* (4) and a symmetry reduction scheme without dimensional defects to identify transformations and particle representations thoroughly. As such, two subsequent nonlinear σ models SU*(4)/U Sp(4) and U Sp(4)/SU(2) × U(1) emerge as well as a possible double coset decomposition of SU*(4) with respect to SU(2) × U(1). Whereas the first model leads to equivalence classes of hyperbolic manifolds and naturally introduces coordinates and velocities, the second coset model leads to a Hermitian symmetric (vector) space (Kählerian space) of real dimension 6, i.e., to a 3-dimensional complex space with a global symplectic and a local SU(2) × U(1) symmetry which allows to identify the (local) gauge group of electroweak interactions as well as under certain assumptions it admits compact SU(3) transformations as automorphisms of this 3-dimensional (hyper)complex vector space. In the limit of low energies, this geometric SU*(4) scheme naturally yields the (compact) group SU(4) to describe “chiral symmetry” and conserved isospin of hadrons as well as the low-dimensional hadron representations. Last not least, with respect to some of the SU*(4) generators we find a multiplication table which (up to signs) is identical with the octonions represented in the Fano plane.     

Derivation of the Bosonic Part of the Electroweak Chiral Lagrangian from a General Underlying Technicolor Theory

Bosonic part of SU(2)L U(1)Y effective chiral Lagrangian for electroweak symmetry breaking is derived from an underlying technicolor theory with no approximation. The underlying theory is assumed to be the most general gauge theory without fundamental scalars. A condensate is required to exist in the theory which breaks SU(2)L U(1)Y dynamically to U(1)em and the anomaly of the theory caused by gauge interaction must be cancelled. The formulation offers general definitions in terms of underlying theory for the low energy constants in effective chiral Lagrangian.``     

Chiral anomalies and constraints on the gauge group in higher-dimensional supersymmetric Yang-Mills theories

Chiral anomalies for gauge theories in any even dimension are computed and the results applied to supersymmetric theories in D = 6, 8 and 10. For D = 8 there is an anomalous chiral U(1) invariance, just as in D = 4, except for certain special groups. For D = 6 and D = 10 there is no anomalous chiral U(1) symmetry, but the gauge current is anomalous except for certain “anomaly-free” groups. For D = 6 the group is thereby constrained to be one of {SU(2), SU(3), exceptional}, while for D = 10 it is constrained to be one of {SU(n) n ≤ 5, USp(4), E₈}.     

How chiral are type-II superstrings?

We study the type-II superstrings in four dimensions by studying vacua where massless chiral multiplets transform as complex representations of the non-abelian gauge group. We show that the gauge group can only be SU(3) and that such fields transform as 3 of SU(3). However, attempts to obtain the theory with N=1 supergravity fail. It turns out that the “different” constructions via asymmetric orbifolds give the same massless spectrum with necessarily N=2 supergravity.     

Geometric Phase in Quaternionic Quantum Mechanics

Quaternion quantum mechanics is examined at the level of unbroken SU(2) gauge symmetry. A general quaternionic phase expression is derived from formal properties of the quaternion algebra.     

QCD Dirac spectrum and components of the gauge field

We analyze the relation between the Dirac spectrum and the gauge field in SU(3) lattice QCD. We focus on how a certain component of the gauge field is related to the Dirac spectrum. First, we consider momentum components of the gauge field. It turns out that the broad momentum region is relevant for the low-lying Dirac spectrum and topological charges. The connection with chiral random matrix theory is also discussed. Second, we consider an SU(2) subgroup component of the SU(3) gauge field. The SU(2) subgroup component behaves like the SU(2) gauge field in the low-lying Dirac spectrum.     

Chiral symmetry breaking and vacuum structure in QCD

The solution of the axial U(1) problem, the role of the topology of the gauge group in forcing the breaking of axial symmetry in any irreducible representation of the observable algebra and the θ vacua structure are revisited in the temporal gauge with attention to the mathematical consistency of the derivations. Both realizations with strong and weak Gauss law are discussed; the control of the general mechanisms and structures is obtained on the basis of the localization of the (large) gauge transformations and the local generation of the chiral symmetry. The Schwinger model in the temporal gauge exactly reproduces the general results.     

A supersymmetric left-right confining model of the weak interactions

We extend the supersymmetric, confining theory of weak interactions to a left-right symmetric model. This model is based on the gauge group SU(M)_SC×SU(2)_R×SU(2)_L×SU(3)_c×U(1) and is more natural as far as supersymmetry breaking is concerned. Supersymmetry protects chiral symmetries from spontaneous breakdown and allows a solution to the strong CP problem. This model can accommodate at most three generations of quarks and leptons.     

The soliton supermultiplet in 4 + 1 dimensions

We consider the SO(4) = SU(2) ? USp(2) Clifford algebra, obtained by the supersymmetry algebra for the N = 2 supersymmetric Yang-Mills theory in 4+1 dimensions, which, in the phase of unbroken gauge symmetry, has a topological charge as central charge. We find that, even if the Higgs mechanism is absent, the massive soliton supermultiplet contains the same number of states as the massless supermultiplet of elementary particles.     

Gauge Field Theories on Clifford Algebras

We present a straightforward model of the U(1) gauge equations of Dirac and Maxwell, as well as the U(n) Yang–Mills equations where all fields and gauge transformations take values in a Clifford algebra. When expressed in terms of the Clifford components of the fields, the equations display various gauge symmetries which we intestigate for all Clifford algebras. In particular, for the Pauli algebra, the Dirace CA equations possess the SU(2) × U(1)-symmetry.     

Locally gauge-invariant formulation of parastatistics

A locally gauge-invariant formulation of parastatistics, which is equivalent to a Yang-Mills gauge theory, is given, using a complex Clifford algebra (case of SU(N)) or a real Clifford algebra (case of SO(N)). In particular, for the SU(3) case, the gauged theory of para-Fermi quarks is equivalent to quantum chromodynamics.