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

Abstract

In this note, we classify topological solitons of nn-brane fields, which are nonlocal fields that describe nn-dimensional extended objects. We consider a class of nn-brane fields that formally define a homomorphism from the nn-fold loop space \Omega^n X_DΩnXD of spacetime X_DXD to a space \mathcal{E}_nℰn. Examples of such nn-brane fields are Wilson operators in nn-form gauge theories. The solitons are singularities of the nn-brane field, and we classify them using the homotopy theory of {\mathbb{E}_n}𝔼n-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}_nℰn. In particular, they are classified by {\pi_{k-n}(\mathcal{E}_n)}πk−n(ℰn) when {n>1}n>1 and by {\pi_{k-n}(\mathcal{E}_n)}πk−n(ℰn) modulo a {\pi_{1-n}(\mathcal{E}_n)}π1−n(ℰn) 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}_nℰn when {k< n}k<n, which we explore through examples. We compare this classification to nn-form \mathcal{E}_nℰn 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)}}ℰn=G(n)/H(n). We discuss this classification in the context of many examples, both with and without ’t Hooft anomalies.