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

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