Topological excitations and Postnikov data

Upon spontaneously breaking a continuous symmetry, the classical topological defects and textures of the resulting system are characterized by the homotopy groups of the resulting order parameter manifold. For example, in 2D magnetic systems with SO(3) symmetry spontaneously broken down to O(2), the order parameter manifold is two dimensional real projective space. The homotopy theory of this space implies that there is one species of vortex which is its own anti-vortex and that the textures of this system are skyrmions. Given the richness of mathematical theory of homotopy, one may wonder if it can tell us more than a basic zoology of excitations. In our work, we show that homotopy theory not only predicts the types of classical topological excitations, it also predicts how these excitations interact with one another. This information is contained in the so-called "Postnikov data." In this talk, we will show how "homotopy types" can be used to understand subtle information about classical topological excitations and its implications symmetry broken quantum topological order.