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

Abstract

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

16 + 15 pages, 0 + 6 figures. v2: minor changes

References in corpus (17)

• Generalized Global Symmetries
• "Deconfined" quantum critical points
• A Duality Web in 2+1 Dimensions and Condensed Matter Physics
• Generalized Symmetries in Condensed Matter
• Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond
• Lecture Notes on Generalized Symmetries and Applications
• Generalized Charges, Part I: Invertible Symmetries and Higher Representations
• Mean string field theory: Landau-Ginzburg theory for 1-form symmetries
• String Field Theory -- A Modern Introduction
• Exact emergent higher-form symmetries in bosonic lattice models
• Higher-group symmetry in finite gauge theory and stabilizer codes
• Higher representations for extended operators
• Vortex-boson duality in four space-time dimensions
• Condensing Nielsen-Olesen strings and the vortex-boson duality in 3+1 and higher dimensions
• Fermionic Higher-form Symmetries
• Solitonic symmetry as non-invertible symmetry: cohomology theories with TQFT coefficients
• Emergent 1-form symmetries

Cited by in corpus (3)

• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Higher-Matter and Landau-Ginzburg Theory of Higher-Group Symmetries
• 2-Form U(1) Spin Liquids: Classical Model and Quantum Aspects