TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.16.5.128
TI  - Topological aspects of brane fields: Solitons and higher-form symmetries
PY  - 2024/05/21
UR  - https://scipost.org/SciPostPhys.16.5.128
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 16
IS  - 5
SP  - 128
A1  - Pace, Salvatore D.
AU  - Liu, Yu Leon
AB  - 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 ${k≥ n}$ 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.
ER  -

@Article{10.21468/SciPostPhys.16.5.128,
	title={{Topological aspects of brane fields: Solitons and higher-form symmetries}},
	author={Salvatore D. Pace and Yu Leon Liu},
	journal={SciPost Phys.},
	volume={16},
	pages={128},
	year={2024},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.16.5.128},
	url={https://scipost.org/10.21468/SciPostPhys.16.5.128},
}