Global anomalies in Yang-Mills theories in higher dimensions

A general method is described for finding a class of fermion representations that can lead to Z₂ anomalies. These anomalies are shown to exist in arbitrary D = 4k dimensions. Examples are given.     

Global gravitational anomalies from perturbative chiral anomalies

The relation between (4k ? 2)-dimensional global gravitational anomalies and perturbative chiral anomalies in 4k dimensions is clarified using an open space generalization of the Atiyah-Patodi-Singer index theorem. These anomalies are then reduced to a chiral anomaly in a two-dimensional Schwinger model. It is argued that “all” anomalies can be similarly reduced.     

Non perturbative effective potentials of quantum oscillators

Non perturbative analogues of the Gaussian effective potential (GEP) are defined for quantum oscillators obeyingq—or (q,p)—deformed commutation relations. These are called the non perturbativeq-effective potential (NP _q EP) and the non perturbativeqp effective potential (NP _qp EP), in the respective cases. A system-specific effective potential (SSEP) is also introduced by means of an additional minimization with respect to theq orq andp parameters. The method is applied toq and (q,p) oscillators of the quartic and sextic types. The SSEP in the case of ground states of theq-oscillators corresponds toq=1, which is the ordinary bosonic limit. A potential shape transition that involves the conversion of a double well to a single well or vice versa, is seen to exist in the case of quantum oscillators sitting in a double well potential.     

Estimating higher order perturbative coefficients using Borel transform

A new method of estimating higher order perturbative coefficients is discussed. It exploits the rapid, asymptotic growth of perturbative coefficients and the information on the singularities in the complex Borel plane. A comparison with other methods is made in several Quantum Chromodynamics (QCD) expansions.     

Non perturbative calculation of the imaginary part of vacuum polarization

We derive a formula for the unitarity relation of the vacuum polarization which includes arbitrary number of soft photons in the vertex functions intermediate states. This quasi-elastic approximation has a spectral form, the soft photon contribution appearing as a broadened delta function. An interpolating formula for the vertices is given which incorporates the asymptotic behavior found by many authors.     

Discrete symmetries in higher dimensions

We consider the constraints imposed by causality on the transformations of time reversal ?, charge conjugationC, and parityP in higher dimensional space-time.     

Grand unification in higher dimensions

We have recently proposed an alternative picture for the physics at the scale of gauge coupling unification, where the unified symmetry is realized in higher dimensions but is broken locally by a symmetry breaking defect. Gauge coupling unification, the quantum numbers of quarks and leptons and the longevity of the proton arise as phenomena of the symmetrical bulk, while the lightness of the Higgs doublets and the masses of the light quarks and leptons probe the symmetry breaking defect. Moreover, the framework is extremely predictive if the effective higher dimensional theory is valid over a large energy interval up to the scale of strong coupling. Precise agreement with experiments is obtained in the simplest theory—SU(5) in five dimensions with two Higgs multiplets propagating in the bulk. The weak mixing angle is predicted to be sin²θ_w=0.2313±0.0004, which fits the data with extraordinary accuracy. The compactification scale and the strong coupling scale are determined to be and , respectively. Proton decay with a lifetime of order is expected with a variety of final states such as e⁺π⁰, and several aspects of flavor, including large neutrino mixing angles, are understood by the geometrical locations of the matter fields. When combined with a particular supersymmetry breaking mechanism, the theory predicts large lepton flavor violating μ→e and τ→μ transitions, with all superpartner masses determined by only two free parameters. The predicted value of the bottom quark mass from Yukawa unification agrees well with the data. This paper is mainly a review of the work presented in hep-ph/0103125, hep-ph/0111068, and hep-ph/0205067 [1], [2] and [3].     

Confined quantum electrodynamics in 1+1 dimensions: A perturbative analysis

The spectrum of confined QED in 1+1 dimensions is analysed using perturbation theory. The mass spectra of systems made up of massless fermions are calculated toO(e ²) and compared to the mass spectra obtained using nonperturbative methods. Systems containing heavy fermions are also studied and an analogy with the 3+1 dimensional Bag model is pointed out.     

Chiral anomalies in even and odd dimensions

Odd dimensional Yang-Mills theories with an extra topological mass term, defined by the Chern-Simons secondary characteristic, are discussed. It is shown in detail how the topological mass affects the equal time charge commutation relations and how the modified commutation relations are related to non-abelian chiral anomalies in even dimensions. We also study the SU(3) chiral model (Wess-Zumino model) in four dimensions and we show how a gauge invariant interaction with an external SU(3) vector potential can be defined with the help of the Chern-Simons characteristic in five dimensions.     

The Vaidya solution in higher dimensions

The Vaidya metric representing the gravitational field of a radiating star is generalized to spacetimes of dimensions greater than four.     

Quantum Hall effect in higher dimensions

Following recent work on the quantum Hall effect on S⁴, we solve the Landau problem on the complex projective spaces $\text{C}\text{P}{}^{\mathrm{k}}$ and discuss quantum Hall states for such spaces. Unlike the case of S⁴, a finite spatial density can be obtained with a finite number of internal states for each particle. We treat the case of $\text{C}\text{P}{}^2$ in some detail considering both Abelian and nonAbelian background fields. The wavefunctions are obtained and incompressibility of the Hall states is shown. The case of $\text{C}\text{P}{}^3$ is related to the case of S⁴.     

Explicit counteraction algorithms in higher dimensions

Using the background field method we construct algorithms for the one-loop counterterms of a field theory in a space-time of dimension 2, 4, 6, 8 or 10. From the d = 6 algorithm we demonstrate the one-loop finiteness of N = 2 supersymmetric Yang-Mills theories and also N = 1 Yang-Mills theory. All other N = 1 Yang-Mills theories + N = 1 matter theories in d = 6 are shown to have a divergent one-loop S-matrix.

We also present partial results for two- and three-loop algorithms in d = 6 and d = 4 respectively.     

Integrable classical systems in higher dimensions

A general method for the construction of the second constant of motion (up to second order) for higher-dimensional classical systems is carried out. Correspondingly, the first- and the second-order potential equations are obtained whose solutions can directly provide the integrable systems.     

Calogero-Sutherland type models in higher dimensions

We construct two different Calogero-Sutherland type models with only two-body interactions in arbitrary dimensions. We obtain some exact wave functions, including the ground states, of these two models for an arbitrary number of spinless nonrelativistic particles.     

The superconformal algebra in higher dimensions

The superconformal algebra for 4/4N-dimensional super-Minkowski space (d=4) can be identified with the simple superalgebra su (2,2/N). For even-dimension d=5,6 the superconformal algebra can be identified with a real form of the simple superalgebras F(4), D(4,1) respectively in Kac's classification. For even-dimension d>-7 it is impossible to define a superconformal algebra satisfying three natural conditions: (1) it acts as infinitesimal automorphisms on super-Minkowski space; (2) this action extends the natural action of the super-Poincaré algebra; (3) when the action of the even part of the superconformal algebra is reduced to an infinitesimal action on ordinary Minkowski space, it extends the natural action of the conformal algebra so (2, d).     

Shape invariant potentials in higher dimensions

In this paper we investigate the shape invariance property of a potential in one dimension. We show that a simple ansatz allows us to reconstruct all the known shape invariant potentials in one dimension. This ansatz can be easily extended to arrive at a large class of new shape invariant potentials in arbitrary dimensions. A reformulation of the shape invariance property and possible generalizations are proposed. These may lead to an important extension of the shape invariance property to Hamiltonians that are related to standard potential problems via space time transformations, which are found useful in path integral formulation of quantum mechanics.