Spherical Orbifolds Research Articles

Spherical orbifold tutte embeddings

This work presents an algorithm for injectively parameterizing surfaces into spherical target domains called spherical orbifolds. Spherical orbifolds are cone surfaces that are generated from symmetry groups of the sphere. The surface is mapped the spherical orbifold via an extension of Tutte's embedding. This embedding is proven to be bijective under mild additional assumptions, which hold in all experiments performed. This work also completes the adaptation of Tutte's embedding to orbifolds of the three classic geometries - Euclidean, hyperbolic and spherical - where the first two were recently addressed. The spherical orbifold embeddings approximate conformal maps and require relatively low computational times. The constant positive curvature of the spherical orbifolds, along with the flexibility of their cone angles, enables producing embeddings with lower isometric distortion compared to their Euclidean counterparts, a fact that makes spherical orbifolds a natural candidate for surface parameterization.

Dark matter and localised fermions from spherical orbifolds?

We study a class of six-dimensional models based on positive curvature surfaces (spherical 2-orbifolds) as extra-spaces. Using the Newman-Penrose formalism, we discuss the particle spectrum in this class of models. The fermion spectrum problem, which has been addressed with flux compactifications in the past, can be avoided using localised fermions. In this framework, we find that there are four types of geometry compatible with the existence of a stable dark matter candidate and we study the simplest case in detail. Using the complementarity between collider resonance searches and relic density constraints, we show that this class of models is under tension, unless the model lies in a funnel region characterised by a resonant Higgs s-channel in the dark matter annihilation.

Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis

We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology, the three-sphere is tiled into copies of a fundamental domain under the corresponding deck group. We employ the point symmetry of each Platonic manifold to construct its fundamental domain as a spherical orbifold. While the three-sphere supports an orthonormal complete basis for harmonic analysis formed by Wigner polynomials, a given spherical orbifold leads to a selection of a specific subbasis. The resulting selection rules find applications in cosmic topology, probed by the cosmic microwave background.

Spherical Orbifolds for Cosmic Topology

Harmonic analysis is a tool to infer cosmic topology from the measured astrophysical cosmic microwave background CMB radiation. For overall positive curvature, Platonic spherical manifolds are candidates for this analysis. We combine the specific point symmetry of the Platonic manifolds with their deck transformations. This analysis in topology leads from manifolds to orbifolds. We discuss the deck transformations of the orbifolds and give eigenmodes for the harmonic analysis as linear combinations of Wigner polynomials on the 3-sphere. These provide new tools for detecting cosmic topology from the CMB radiation.

Block regularization of the Kepler problem on surfaces of revolution with positive constant curvature

We consider the Kepler problem on surfaces of revolution that are homeomorphic to S 2 and have constant Gaussian curvature. We show that the system is maximally superintegrable, finding constants of motion that generalize the Runge–Lenz vector. Then, using such first integrals, we determine the class of surfaces that lead to block-regularizable collision singularities. In particular we show that the singularities are always regularizable if the surfaces are spherical orbifolds of revolution with constant curvature.

Branes, waves and AdS orbifolds

The embedding of a Taub-NUT space in the directions transverse to the world volume of branes describes branes at (spherical) orbifold singularities. Similarly, the embedding of a pp-wave in the brane world volume yields an AdS orbifold. In case of the D1-D5--brane system, the AdS3 orbifolds yields a BTZ black hole; as we will show, the same holds for D3-branes corresponding to AdS5. In addition we will show that the AdS orbifolds and the spherical orbifolds are U-dual to each other. However in contrast to spherical orbifolds the AdS orbifolds lead to a running coupling, which is in the IR inverse to the coupling of the spherical orbifold. A discussion of the general pp-wave solution in AdS space is added.