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

Shi Chen, Yuya Tanizaki

TL;DR

The paper shows that in the 4$d$ $\mathbb{C}P^1$ model, solitonic symmetry extends beyond the homotopy-classified $U(1)$ hopfion symmetry due to the presence of coexisting hopfions and vortices. The invertible part of the hopfion symmetry is captured by a spin-bordism invariant giving a $\mathbb{Z}_2$ symmetry, while the full hopfion symmetry is non-invertible and realized by coupling to 3$d$ spin TQFTs, such as spin Chern-Simons theories $\mathcal{A}^{N,p}$. This framework unifies solitonic symmetries with topological orders, showing that certain symmetry operators are TQFT partition functions and that fusion rules are non-invertible. The results indicate a broader, unified language for solitonic conservation laws and topological phases across spacetime dimensions.

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.

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

TL;DR

The paper shows that in the 4 model, solitonic symmetry extends beyond the homotopy-classified hopfion symmetry due to the presence of coexisting hopfions and vortices. The invertible part of the hopfion symmetry is captured by a spin-bordism invariant giving a symmetry, while the full hopfion symmetry is non-invertible and realized by coupling to 3 spin TQFTs, such as spin Chern-Simons theories . This framework unifies solitonic symmetries with topological orders, showing that certain symmetry operators are TQFT partition functions and that fusion rules are non-invertible. The results indicate a broader, unified language for solitonic conservation laws and topological phases across spacetime dimensions.

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 d model. This model has two kinds of solitonic excitations, vortices and hopfions, which would follow two solitonic symmetries according to homotopy groups. Nevertheless, we demonstrate the nonexistence of the hopfion 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 , which we recognize as a spin bordism invariant. Compared with the 3d model, our work suggests a unified description of solitonic symmetries and couplings to topological phases.
Paper Structure (6 sections, 20 equations, 1 figure)

This paper contains 6 sections, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Operators and solitonic charges
  3. Invertible solitonic symmetry from bordism
  4. Non-invertible solitonic symmetry from TQFT
  5. Summary and outlook
  6. Acknowledgements

Figures (1)

  • Figure 1: The range of the coordinates $\alpha,\beta$ defined in Eq. \ref{['eq:M_4']} and the function $\theta(\alpha,\beta)$ defined in Eq. \ref{['eq:theta']}. The solid quarter arc describes the point defect and the solid semicircle describes the line defect. On the $\alpha$-axis and the $\beta$-axis, $\theta$ is either $0$ or $\pi$, which ensures that the $\mathbb{C}P^1$ configuration is regular and single-valued.