International Journal of Modern Physics A

Several ideas for solving the problem of fermion mass hierarchy and mixing and specific supersymmetric models that realize it are reviewed. In particular, we discuss many models based on SO(10) in four dimensions combined with a family symmetry to accommodate fermion mass hierarchy and mixing, including the case of neutrinos. These models are compared and various tests that can be used to distinguish these models are suggested. We also include a discussion of a few SO(10) models in higher space–time dimensions.

We describe an expedient way to derive the CIV-DV prepotential in power series expansion in S_i. This is based on integrations of equations for its derivatives ∂ℱ/∂T_m with respect to additional (Whitham) moduli T_m. For illustrative purposes, we calculate explicitly the leading terms of the expansion and explicitly check some components of the WDVV equations to the leading order. Extension to any higher order is simple and algorithmic.

We explicitly construct the creation operators for the quantum field configurations associated to quantum membranes (two-branes) in BF and generalized Chern–Simons theories in a space–time of dimension D=5. The creation operators for quantum excitations carrying topological charge are also obtained in the same theories. For the case of D=5 generalized Chern–Simons theory, we show that this operator actually creates an open string with a topological charge at its tip. It is shown that a duality structure exists in general, relating the membrane and topological excitation operators and the corresponding dual algebra is derived. Composite topologically charged membranes are shown to possess generalized statistics that may, in particular, be fermionic. This is the first step for the bosonization procedure in these theories. Potential applications in the full quantization of two-branes is also briefly discussed.

We study the perturbations on both "radial" and "angular" components of the quintessence with an internal Abelian and non-Abelian symmetry. The properties of the perturbation on the "radial" component depend on the specific potential of the model and is similiar for both Abelian and non-Abelian case. We show that the cosine-type potential is very interesting for the O(N) quintessence model and also give a critical condition of instability for the potential. While the properties of perturbations on "angular" components depend on whether the internal symmetry is Abelian or non-Abelian, which we have discussed respectively. In the non-Abelian case, the fluctuation of the "angular" component will increase rapidly with time while in the Abelian case it will not.

In space–time dimensions larger than 2, whenever a global symmetry G is spontaneously broken to a subgroup H, and G and H are Lie groups, there are Nambu–Goldstone modes described by fields with values in G/H. In two-dimensional space–times as well, models where fields take values in G/H are of considerable interest even though in that case there is no spontaneous breaking of continuous symmetries. We consider such models when the world sheet is a two-sphere and describe their fuzzy analogs for G = SU(N+1), H = S(U(N-1) ⊗ U(1)) ≃ U(N) and . More generally our methods give fuzzy versions of continuum models on S² when the target spaces are Grassmannians and flag manifolds described by (N+1) × (N+1) projectors of rank ≤ (N+1)/2. These fuzzy models are finite-dimensional matrix models which nevertheless retain all the essential continuum topological features like solitonic sectors. They seem well suited for numerical work.

Using Einstein, Landau–Lifshitz, Papapetrou and Weinberg energy–momentum complexes, we explicitly evaluate the energy and momentum distributions associated with a nonstatic and circularly symmetric three-dimensional space–time. The gravitational background under study is an exact solution of Einstein's equations in the presence of a cosmological constant and a null fluid. It can be regarded as the three-dimensional analogue of the Vaidya metric and represents a nonstatic spinless (2+1)-dimensional black hole with an outflux of null radiation. All four above-mentioned prescriptions give exactly the same energy and momentum distributions for the specific black hole background. Therefore, the results obtained here provide evidence in support of the claim that for a given gravitational background, different energy–momentum complexes can give identical results in three dimensions. Furthermore, in the limit of zero cosmological constant, the results presented here reproduce those obtained by Virbhadra. He utilized the Landau–Lifshitz energy–momentum complex for the same (2+1)-dimensional black hole background in the absence of a cosmological constant.