International Journal of Modern Physics A

This paper is a rudimentary introduction, geared at nonspecialists, to how noncommutative field theories arise in physics and their applications to string theory, particle physics and condensed matter systems.

The generic supersymmetric version of the Standard Model would have the minimal list of superfields incorporating the Standard Model particles, and a Lagrangian dictated by the Standard Model gauge symmetries. To be phenomenologically viable, soft supersymmetry breaking terms have to be included. In the most popular version of the supersymmetric Standard Model, an ad hoc discrete symmetry, called R parity, is added in by hand. While there has been a lot of various kinds of R-parity violation studies in the literature, the complete version of supersymmetry without R parity is not popularly appreciated. In this article, we present a pedagogical review of the formulation of this generic supersymmetric Standard Model and give a detailed discussion on the basic conceptual issues involved. Unfortunately, there are quite some confusing, or even plainly wrong, statements on the issues within the literature of R-parity violations. We aim at clarifying these issues here. We will first discuss our formulation, about which readers are urged to read without bias from previous acquired perspectives on the topic. Based on the formulation, we will then address the various issues. In relation to phenomenology, our review here will not go beyond tree-level mass matrices. But we will give a careful discussion of mass matrices of all the matter fields involved. Useful expressions for perturbative diagonalizations of the mass matrices at the phenomenologically interesting limit of corresponds to small neutrino masses are derived. All these expressions are given in the fully generic setting, with information on complex phases of parameters retained. Such expressions have been shown to be useful in the analyses of various phenomenological features.

We find that the IIA matrix models defined on the non-compact C³/Z₆, C²/Z₂ and C²/Z₄ orbifolds preserve supersymmetry where the fermions are on-mass-shell Majorana–Weyl fermions. In these examples supersymmetry is preserved both in the orbifolded space and in the non-orbifolded space at the same time. The matrix model on C³/Z₆ orbifold has the same supersymmetry as the case of C³/Z₃ orbifold which was pointed out previously. On the other hand the matrix models on C²/Z₂ and C²/Z₄ orbifold have a half of the supersymmetry. We further find that the matrix model on C²/Z₂ orbifold with a parity-like identification preserves supersymmetry.

The radiative transfer equations for multiple inverse Compton scattering of the Cosmic Microwave Background Radiation (CMBR) by the hot intra-cluster electrons are solved numerically. The spherical isothermal and inhomogeneous β model has been considered for the electron distribution. The anisotropy of the CMBR caused by scattering, known as thermal Sunyaev–Zel'dovich effect, along the radial axis of the medium is compared with the analytical solution of Kompaneets equation. The X-ray data of several clusters of galaxies at low redshifts provide an estimation of the central electron density n₀ to be of the order 10^-3. It is found that for this value of n₀ the effect of multiple scattering is negligible. The numerically calculated anisotropy along the radial axis matches well with the analytical solution that describes single scattering. The result incorporating multiple scattering is fitted with the recent observation of Sunyaev–Zel'dovich effect in the cluster Abell 2163. It is shown that if n₀ is greater by an order of magnitude, which could be possible for cluster of galaxies at comparatively higher redshift, multiple scattering would play a significant role at the Wien region of the anisotropy spectrum. A fitting formula for the correction to the Sunyaev–Zel'dovich effect due to multiple scattering is provided.

We investigate the quantization of the bosonic string model which has a local U(1)_V×U(1)_A gauge invariance as well as the general coordinate and Weyl invariance on the world-sheet. The model is quantized by Lagrangian and Hamiltonian BRST formulations á la Batalin, Fradkin and Vilkovisky and noncovariant light-cone gauge formulation. Upon the quantization the model turns out to be formulated consistently in (26+2)-dimensional background space–time involving two time-like coordinates.

We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S₃, the dihedral group D₄ and the quaternion group Q. Poincaré duality holds in every case, and under some assumptions (essentially the existence of a top form) we find that it must hold in general.

A short review of the bicovariant (noncommutative) differential calculus on finite G is given for selfconsistency. Exterior derivative, exterior product, metric, Hodge dual, connections, torsion, curvature, and biinvariant integration can be defined algebraically. A projector decomposition of the braiding operator is found, and used in constructing the projector on the space of two-forms. By means of the braiding operator and the metric a knot invariant is defined for any finite group.

We show that it is possible to construct supersymmetric three-generation models with the Standard Model gauge group in the framework of non-simply-connected elliptically fibered Calabi–Yau threefolds, without section but with a bi-section. The fibrations on a cover Calabi–Yau threefold, where the model has six generations of SU(5) and the bundle is given via the spectral cover description, use a different description of the elliptic fiber which leads to more than one global section. We present two examples of a possible cover Calabi–Yau threefold with a free involution: one is a fiber product of rational elliptic surfaces dP₉; another example is an elliptic fibration over a Hirzebruch surface. We compute the necessary amount of chiral matter by "turning on" a further parameter which is related to singularities of the fibration and the branching of the spectral cover.