Classification of symmetry protected states of quantum spin chains for continuous symmetry groups

Symmetry protected states (SPT's) of quantum spin systems were studied by several authors. For one-dimensional systems (spin chains), there is an essentially complete and rigorous understanding: SPT's corresponding to finite on-site symmetry groups G are classified by the second cohomology group H² (G, U (1) ), as established by Kapustin et al. J. Math. Phys. (2021). We extend this result to the case of compact topological symmetry groups G. We also strengthen the existing results in the sense that our classification results holds within the class of spin chains with locally bounded on-site dimensions.