Fig. 1: Schematic of high-dimensional non-Abelian holonomy.

a Schematic of four waveguides structure in a two-layer SNOI platform for realizing SO(2) holonomy. The upper inset shows the focused ion beam (FIB) image in a cross-section view, and the lower inset demonstrates the holonomy evolving in the \({\mathbf{\kappa }}\) sphere. For better observation, the coupling coefficient space is normalized as κ/|κ| to a 2-sphere. The direction of rotation determines the sign of \(\theta\). b The upper panel shows the details of waveguide parameters in a cross-section view. The lower panel shows an example of FDTD simulation of light magnitude distribution with different inputs when \(\theta\)=π/12. c Schematic of SO(3). The system with M = 1 and m = 3 protects three degenerate eigenstates. The initial/final states are input/output from three waveguides labeled with numbers. Three optional \(\theta\) of two types of non-coaxial rotations generate a complete set of SO(3) group. The panels below illustrate the cross-sections of two non-coaxial rotations, as well as \({\mathbf{\kappa }}=[{\kappa }_{1},\,{\kappa }_{2},\,{\kappa }_{3},\,{\kappa }_{4}]\) for Fig. 1e. d Schematic of SO(4). The complete SO(4) group can be generated through the cascade of M = 2 system (pink blocks) and M = 1 system (orange blocks), providing full six degrees of freedom for SO(4). e SO(3) holonomy travels through four-dimensional \({\mathbf{\kappa }}\) space. \({\mathbf{\kappa }}\) and \({\mathbf{\theta }}\) correspond to Fig. 1c. f SO(4) holonomy travels through six-dimensional \({\mathbf{\kappa }}\) space. \({\mathbf{\kappa }}\) and \({\mathbf{\theta }}\) correspond to Fig. 1d. The orange dashed line represents the projection of the orange line (six-dimensional curve) onto the inner sphere (three-dimensional), encircling the solid angle \({\theta }_{1}\) or \({\theta }_{5}\). g Infinite dimensions of \({\mathbf{\kappa }}\) space for SO(m) scheme in Fig. 1h. h Expandable schematic for SO(m) holonomy.