Solitonic Symmetry beyond Homotopy: Invertibility from Bordism and Noninvertibility from Topological Quantum Field Theory

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 4D CP^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 noninvertible symmetry generated by 3D spin topological quantum field theories (TQFTs). Its invertible part is just Z₂, which we recognize as a spin bordism invariant. Compared with the 3D CP^1 model, our work suggests a unified description of solitonic symmetries and couplings to topological phases.