Topological aspects of brane fields: solitons and higher-form symmetries

• Published in: arXiv.org 2024-01
• Main Authors: Pace, Salvatore D, Yu Leon Liu
• Format: Article
• Language: English
• Subjects: Anomalies; Branes; Broken symmetry; Classification; Gauge theory; Group theory; Homomorphisms; Homotopy theory; Order parameters; Solitary waves; Topology
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2311.09293

In this note, we classify topological solitons of \(n\)-brane fields, which are nonlocal fields that describe \(n\)-dimensional extended objects. We consider a class of \(n\)-brane fields that formally define a homomorphism from the \(n\)-fold loop space \(\Omega^n X_D\) of spacetime \(X_D\) to a space \(\mathcal{E}_n\). Examples of such \(n\)-brane fields are Wilson operators in \(n\)-form gauge theories. The solitons are singularities of the \(n\)-brane field, and we classify them using the homotopy theory of \({\mathbb{E}_n}\)-algebras. We find that the classification of codimension \({k+1}\) topological solitons with \({k\geq n}\) can be understood using homotopy groups of \(\mathcal{E}_n\). In particular, they are classified by \({\pi_{k-n}(\mathcal{E}_n)}\) when \({n>1}\) and by \({\pi_{k-n}(\mathcal{E}_n)}\) modulo a \({\pi_{1-n}(\mathcal{E}_n)}\) action when \({n=0}\) or \({1}\). However, for \({n>2}\), their classification goes beyond the homotopy groups of \(\mathcal{E}_n\) when \({k< n}\), which we explore through examples. We compare this classification to \(n\)-form \(\mathcal{E}_n\) gauge theory. We then apply this classification and consider an \({n}\)-form symmetry described by the abelian group \({G^{(n)}}\) that is spontaneously broken to \({H^{(n)}\subset G^{(n)}}\), for which the order parameter characterizing this symmetry breaking pattern is an \({n}\)-brane field with target space \({\mathcal{E}_n = G^{(n)}/H^{(n)}}\). We discuss this classification in the context of many examples, both with and without 't Hooft anomalies.