Solitonic symmetry beyond homotopy: Invertibility from bordism and noninvertibility from topological quantum field theory

Abstract

Solitonic symmetry has been believed to follow the homotopy-group classification of topological solitons. Here, we point out a more sophisticated algebraic structure when solitons of different dimensions coexist in the spectrum. We uncover this phenomenon in a concrete quantum field theory, the $4$d $\mathbb{C}{P}^1$ model. This model has two kinds of solitonic excitations, vortices and hopfions, which would follow two $U(1)$ solitonic symmetries according to homotopy groups. Nevertheless, we demonstrate the nonexistence of the hopfion $U(1)$ symmetry by evaluating the hopfion charge of vortex operators. We clarify that what conserves hopfion numbers is a non-invertible symmetry generated by 3d spin topological quantum field theories (TQFTs). Its invertible part is just ${\mathbb{Z}}_2$, which we recognize as a spin bordism invariant. Compared with the 3d $\mathbb{C}{P}^1$ model, our work suggests a unified description of solitonic symmetries and couplings to topological phases.

6 pages, 1 figure

References in corpus (14)

• Generalized Global Symmetries
• Duality and defects in rational conformal field theory
• Kramers-Wannier-like duality defects in (3+1)d gauge theories
• Non-Invertible Duality Defects in 3+1 Dimensions
• Symmetries and Strings of Adjoint QCD${}_2$
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• Non-Invertible Higher-Categorical Symmetries
• Interferometry of non-Abelian Anyons
• Non-Invertible Symmetries of $\mathcal{N}=4$ SYM and Twisted Compactification
• Non-invertible Time-reversal Symmetry
• Non-invertible 1-form symmetry and Casimir scaling in 2d Yang-Mills theory
• Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly
• Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond $N$-ality
• Topological terms of (2+1)d flag-manifold sigma models

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