Topological signatures in CMB temperature anisotropy maps

We propose an alternative formalism to simulate CMB temperature maps in $\Lambda$CDM universes with nontrivial spatial topologies. This formalism avoids the need to explicitly compute the eigenmodes of the Laplacian operator in the spatial sections. Instead, the covariance matrix of the coefficients of the spherical harmonic decomposition of the temperature anisotropies is expressed in terms of the elements of the covering group of the space. We obtain a decomposition of the correlation matrix that isolates the topological contribution to the CMB temperature anisotropies out of the simply connected contribution. A further decomposition of the topological signature of the correlation matrix for an arbitrary topology allows us to compute it in terms of correlation matrices corresponding to simpler topologies, for which closed quadrature formulae might be derived. We also use this decomposition to show that CMB temperature maps of (not too large) multiply connected universes must show ``patterns of alignment'', and propose a method to look for these patterns, thus opening the door to the development of new methods for detecting the topology of our Universe even when the injectivity radius of space is slightly larger than the radius of the last scattering surface. We illustrate all these features with the simplest examples, those of flat homogeneous manifolds, i.e., tori, with special attention given to the cylinder, i.e., $T^1$ topology.Comment: 25 pages, 7 eps figures, revtex4, submitted to PR

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.Comment: 17 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1011.427

Defect Perturbations in Landau-Ginzburg Models

Perturbations of B-type defects in Landau-Ginzburg models are considered. In particular, the effect of perturbations of defects on their fusion is analyzed in the framework of matrix factorizations. As an application, it is discussed how fusion with perturbed defects induces perturbations on boundary conditions. It is shown that in some classes of models all boundary perturbations can be obtained in this way. Moreover, a universal class of perturbed defects is constructed, whose fusion under certain conditions obey braid relations. The functors obtained by fusing these defects with boundary conditions are twist functors as introduced in the work of Seidel and Thomas.Comment: 46 page

Problems on Polytopes, Their Groups, and Realizations

The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete Geometry, to appear

Electro and magneto statics of topological insulators as modeled by planar, spherical and cylindrical θ\theta boundaries: Green function approach

The Green function (GF) method is used to analyze the boundary effects produced by a Chern Simons (CS) extension to electrodynamics. We consider the electromagnetic field coupled to a $\theta$ term that is piecewise constant in different regions of space, separated by a common interface $\Sigma$, the $\theta$ boundary, model which we will refer to as $\theta$ electrodynamics ($\theta$ ED). This model provides a correct low energy effective action for describing topological insulators (TI). In this work we construct the static GF in $\theta$ ED for different geometrical configurations of the $\theta$ boundary, namely: planar, spherical and cylindrical $\theta$ interfaces. Also we adapt the standard Green theorem to include the effects of the $\theta$ boundary. These are the most important results of our work, since they allow to obtain the corresponding static electric and magnetic fields for arbitrary sources and arbitrary boundary conditions in the given geometries. Also, the method provides a well defined starting point for either analytical or numerical approximations in the cases where the exact analytical calculations are not possible. Explicit solutions for simple cases in each of the aforementioned geometries for $\theta$ boundaries are provided. The adapted Green theorem is illustrated by studying the problem of a point like electric charge interacting with a planar TI with prescribed boundary conditions. Our generalization, when particularized to specific cases, is successfully compared with previously reported results, most of which have been obtained by using the methods of images.Comment: 24 pages, 4 figures, accepted for publication in PRD. arXiv admin note: text overlap with arXiv:1511.0117