Flat Zipper-Unfolding Pairs for Platonic Solids

We show that four of the five Platonic solids' surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can "zipper-refold" to a flat doubly covered parallelogram, forming a rather compact representation of the surface. Thus these regular polyhedra have particular flat "zipper pairs." No such zipper pair exists for a dodecahedron, whose Hamiltonian unfoldings are "zip-rigid." This report is primarily an inventory of the possibilities, and raises more questions than it answers.Comment: 15 pages, 14 figures, 8 references. v2: Added one new figure. v3: Replaced Fig. 13 to remove a duplicate unfolding, reducing from 21 to 20 the distinct unfoldings. v4: Replaced Fig. 13 again, 18 distinct unfolding

A-D-E Classification of Conformal Field Theories

The ADE classification scheme is encountered in many areas of mathematics, most notably in the study of Lie algebras. Here such a scheme is shown to describe families of two-dimensional conformal field theories.Comment: 19 pages, 4 figures, 4 tables; review article to appear in Scholarpedia, http://www.scholarpedia.org

CFT, BCFT, ADE and all that

These pedagogical lectures present some material, classical or more recent, on (Rational) Conformal Field Theories and their general setting ``in the bulk'' or in the presence of a boundary. Two well posed problems are the classification of modular invariant partition functions and the determination of boundary conditions consistent with conformal invariance. It is shown why the two problems are intimately connected and how graphs -ADE Dynkin diagrams and their generalizations- appear in a natural way.Comment: Lectures at Bariloche, Argentina, January 2000. 36 pages, 4 figure

Tight-binding electronic spectra on graphs with spherical topology. II. The effect of spin-orbit interaction

This is the second of two papers devoted to tight-binding electronic spectra on graphs with the topology of the sphere. We investigate the problem of an electron subject to a spin-orbit interaction generated by the radial electric field of a static point charge sitting at the center of the sphere. The tight-binding Hamiltonian considered is a discretization on polyhedral graphs of the familiar form ${\bm L}\cdot{\bm S}$ of the spin-orbit Hamiltonian. It involves SU(2) hopping matrices of the form $\exp({\rm i}\mu{\bm n}\cdot{\bm\sigma})$ living on the oriented links of the graph. For a given structure, the dimensionless coupling constant $\mu$ is the only parameter of the model. An analysis of the energy spectrum is carried out for the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron and icosahedron) and the C$_{60}$ fullerene. Except for the latter, the $\mu$-dependence of all the energy levels is obtained analytically in closed form. Rather unexpectedly, the spectra are symmetric under the exchange $\mu\leftrightarrow\Theta-\mu$, where $\Theta$ is the common arc length of the links. For the symmetric point $\mu=\Theta/2$, the problem can be exactly mapped onto a tight-binding model in the presence of the magnetic field generated by a Dirac monopole, studied recently. The dependence of the total energy at half filling on $\mu$ is investigated in all examples.Comment: 25 pages, 15 figures, 12 tables. Various kinds of minor improvement