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

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 ${\mathrm{\Omega }}^n{X}_D$ 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 $k\ge n$ 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 ${\mathcal{E}}_n=G^{(n)}\mathrm{/}{H}^{(n)}$. We discuss this classification in the context of many examples, both with and without 't Hooft anomalies.