On the classification of topological defects and textures

2018/10/30 by J. P. Ang, Ang, J. P., Abhishodh Prakash +1
Computer Science · Mathematics · Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Strongly Correlated Electrons (cond-mat.str-el) #Theoretical and Computational Physics #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1810.12964

openalex publication_date 2018/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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