On the classification of topological defects and textures

TL;DR

This paper introduces a systematic method using higher category theory to classify topological sectors in sigma models, addressing longstanding classification challenges in various dimensions.

Contribution

It develops a novel framework based on higher categorical generalizations of the fundamental group for classifying topological defects in sigma models.

Findings

Recovered known topological classifications

Derived new classifications in higher dimensions

Provided a qualitative description of the mathematical framework

Abstract

Ordered phases resulting from spontaneously broken continuous symmetries are effectively described by sigma models of maps to the coset space of Goldstone modes. A classic problem is to classify the topological sectors of the sigma model. In simple cases, this can be expressed in terms of homotopy groups of the target space; but in general, it is a complicated affair -- even in two dimensions -- and a general method is lacking. In this letter, we introduce a technique to systematically classify topological sectors of sigma models in various dimensions using a framework based on higher categorical generalizations of the fundamental group. As a demonstration, we recover some known results and obtain new ones. The technique and relevant mathematical structures are described only qualitatively in this letter; details can be found in our companion paper.