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 CP¹ 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 CP¹ model, our work suggests a unified description of solitonic symmetries and couplings to topological phases.