International Journal of Modern Physics A

In this paper, we discuss off-shell representations of N-extended supersymmetry in one dimension, i.e. N-extended supersymmetric quantum mechanics, and following earlier work on the subject, we codify them in terms of graphs called Adinkras. This framework provides a method of generating all Adinkras with the same topology, and so also all the corresponding irreducible supersymmetric multiplets. We develop some graph theoretic techniques to understand these diagrams in terms of a relatively small amount of information, namely, at what heights various vertices of the graph should be "hung."

We then show how Adinkras that are the graphs of N-dimensional cubes can be obtained for superfields satisfying constraints that involve superderivatives. This dramatically widens the range of supermultiplets that can be described using the superspace formalism and also organizes them. Other topologies for Adinkras are possible, and we show that it is reasonable that these are also the result of constraining superfields using superderivatives.

We arrange the family of Adinkras with an N-cubical topology, and so also the sequence of corresponding irreducible supersymmetric multiplets, in a cyclic sequence, which we call the main sequence. We produce the N = 1 and N = 2 main sequences in detail, and indicate some aspects of the situation for higher N.

A simple algorithm to calculate the group theory factor entering in anomalies at four and six dimensions for SU(N) and SO(N) groups in terms of the Casimir invariants of their subgroups is presented. Explicit examples of some of the lower dimensional representations of SU(n), n ≤ 5 and SO(10) groups are presented, which could be used for model building in four and six dimensions.

We study a statistical model of random plane partitions. The statistical model has interpretations as five-dimensional supersymmetric SU(N) Yang–Mills on ℝ⁴ × S¹ and as Kähler gravity on local SU(N) geometry. At the thermodynamic limit a typical plane partition called the limit shape dominates in the statistical model. The limit shape is linked with a hyperelliptic curve, which is a five-dimensional version of the SU(N) Seiberg–Witten curve. Amoebas and the Ronkin functions play intermediary roles between the limit shape and the hyperelliptic curve. In particular, the Ronkin function realizes an integration of thermodynamical density of the main diagonal partitions, along one-dimensional slice of it and thereby is interpreted as the counting function of gauge instantons. The radius of S¹ can be identified with the inverse temperature of the statistical model. The large radius limit of the five-dimensional Yang–Mills is the low temperature limit of the statistical model, where the statistical model is frozen to a ground state that is associated with the local SU(N) geometry. We also show that the low temperature limit corresponds to a certain degeneration of amoebas and the Ronkin functions known as tropical geometry.

We use (4+1) split to derive the 4D induced energy density ρ and pressure p of the 5D Ricci-flat cosmological solutions which are characterized by having a bounce instead of a bang. The solutions contain two arbitrary functions of time t and, therefore, are mathematically rich in giving various cosmological models. By using four known energy conditions (null, weak, strong, and dominant) to pick out and study physically meaningful solutions, we find that the 4D part of the 5D solutions asymptotically approaches to the standard 4D FRW models and the expansion of the universe is decelerating for normal induced matter for which all the four energy conditions are satisfied. We also find that quintessence might be normal or abnormal, depending on the parameter w of the equation of state. If -1 ≤ w < -1/3, the expansion of the universe is accelerating and the quintessence is abnormal because the strong energy condition is violated while other three are satisfied. For phantom, all the four energy conditions are violated. Before the bounce, all the four energy conditions are violated, implying that the cosmic matter before the bounce could be explained as a phantom that has a large negative pressure and makes the universe bouncing. In the early times after the bounce, the dominant energy condition is violated, while the other three are satisfied, and so the cosmic matter could be explained as a super-luminal acoustic matter.

Superbubbles are shells in the interstellar medium produced by the simultaneous explosions of many supernova remnants. The solutions of the mathematical diffusion and of the Fourier expansion in 1D, 2D and 3D were deduced in order to describe the diffusion of nucleons from such structures. The mean number of visits in the the case of the Levy flights in 1D was computed with a Monte Carlo simulation. The diffusion of cosmic rays has its physical explanation in the relativistic Larmor gyro-radius which is energy dependent. The mathematical solution of the diffusion equation in 1D with variable diffusion coefficient was computed. Variable diffusion coefficient means magnetic field variable with the altitude from the Galactic plane. The analytical solutions allow us to calibrate the code that describes the Monte Carlo diffusion. The maximum energy that can be extracted from the superbubbles is deduced. The concentration of cosmic rays is a function of the distance from the nearest superbubble and the selected energy. The interaction of the cosmic rays on the target material allows us to trace the theoretical map of the diffuse Galactic continuum gamma-rays. The streaming of the cosmic rays from the Gould Belts that contains the sun at its internal was described by a Monte Carlo simulation. Ten new formulas are derived.

The angular distribution of fission fragments has been measured for proton-induced fission of ¹⁹⁷Au and ²⁰⁹Bi nuclei at several energies between 25 and 30 MeV using surface-barrier detectors. The measured anisotropies are found to be in agreement with the predictions of the standard saddle-point statistical model. The measured anisotropies for neither of the nuclei show any anomalous behavior as a function of both energy and entrance channel mass asymmetry. The fission cross sections of ¹⁹⁷Au and ²⁰⁹Bi nuclei were also measured and compared with that of the previous works.

Classical equations of motion for three-dimensional σ-models in curved background are solved by a transformation that follows from the Poisson–Lie T-plurality and transform them into the equations in the flat background. Transformations of coordinates that make the metric constant are found and used for solving the flat model. The Poisson–Lie transformation is explicitly performed by solving the PDE's for auxiliary functions and finding the relevant transformation of coordinates in the Drinfel'd double. String conditions for the solutions are preserved by the Poisson–Lie transformations. Therefore we are able to specify the type of σ-model solutions that solve also equations of motion of three-dimensional relativistic strings in the curved backgrounds. Some simple examples are given.

We discuss the symmetry properties of the reparametrization invariant model of an interacting relativistic particle where the electromagnetic field is taken as the constant background field. The direct coupling between the relativistic particle and the electromagnetic gauge potential is a special case of the above with a specific set of subtleties involved in it. For the above model, we demonstrate the existence of a time–space noncommutativity (NC) in the space–time structure from the symmetry considerations alone. We further show that the NC and commutativity properties of this model are different aspects of a unique continuous gauge symmetry that is derived from the nonstandard gauge-type symmetry transformations by requiring their consistency with (i) the equations of motion, and (ii) the expressions for the canonical momenta, derived from the Lagrangians. We provide a detailed discussion on the noncommutative deformation of the Poincaré algebra.