Symmetric tangled Platonic polyhedra

Conventional embeddings of the edge-graphs of Platonic polyhedra, {f, z}, where f, z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway’s two-dimensional (2D) orbifold notation (equivalent to Schönflies symbols Ih, Oh, and Td). Tangled Platonic {f, z} polyhedra—which cannot lie on the sphere without edge-crossings—are constructed as windings of helices with three, five, seven,… strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the “θz” polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo.

Quarks Mixing in Chiral Symmetries

We discuss a subject of the quarks mixing in SU4∗SU4 and SU6∗SU6 symmetries trying to calculate the quarks mixing angles and the complex phase responsible for the CP non-conservation on the basis of the Gell-Mann Oakes Renner model. Assuming symmetry breaking in a limit of exact sub-symmetries for simultaneous quarks rotations in both electric charge sub-spaces we can estimate all mention above parameters. A perfect agreement of the experimental value of the Cabibbo angles with a sum of simultaneous quarks mixing angles in doublets (u,c) and (d,s) in the SU4∗SU4 symmetry suggests that a quarks mixing is realized in a maximal allowed range. The same assumption used for the SU6∗SU6 and a simultaneous maximal allowed quarks mixing in both electric charge triplets (u,c,t) and (d,s,b) gives a perfect agreement with the experimental value of the Cabibbo angle and estimation on the angles Θ2 and Θ3 as well as a bond for the complex phase δ.

Boundary states for chiral symmetries in two dimensions

We study boundary states for Dirac fermions in d = 1 + 1 dimensions that preserve Abelian chiral symmetries, meaning that the left- and right-moving fermions carry different charges. We derive simple expressions, in terms of the fermion charge assignments, for the boundary central charge and for the ground state degeneracy of the system when two different boundary conditions are imposed at either end of an interval. We show that all such boundary states fall into one of two classes, related to SPT phases supported by (−1)F , which are characterised by the existence of an unpaired Majorana zero mode.

Chiral symmetries in nuclear physics

In this report the theoretical concepts of a chirally symmetric meson field theory are reviewed and an overview of the most relevant applications in nuclear physics is given. This includes a unified description of the vacuum properties of hadrons, finite nuclei and hot, dense and strange nuclear matter in an extended chiral SU(3)_L x SU(3)_R σ-ω model.

Instantaneous Bethe–Salpeter view of Goldstone-type pseudoscalar mesons

Describing the lightest pseudoscalar mesons as bound states of quark and antiquark within the framework of an instantaneous Bethe–Salpeter formalism constructed such as to retain (in contrast to Salpeter’s equation) as much information on the relativistic effects provided by the full quark propagator as conceivable allows for a surprisingly simple implementation of their near masslessness mandatory for their interpretability as pseudo-Goldstone bosons related to the spontaneous breaking of the chiral symmetries of quantum chromodynamics.