Completeness of the H^2(G,U(1)) SPT index for compact Lie groups

Prove that for any compact Lie group G, the H^2(G,U(1)) topological index associated to one-dimensional G-symmetric unique gapped ground states is a complete invariant of G-symmetry protected topological phases; that is, show that if two such states ω0 and ω1 have the same index, then there exists a smooth uniformly gapped path of G-invariant local interactions connecting their parent Hamiltonians, so the interactions lie in the same G-SPT phase (Φ0 ∼G Φ1).

Background

Ogata proved completeness of the H^2(G,U(1)) index for finite groups G in the one-dimensional, unique gapped ground state setting. Extending this result to compact Lie groups would unify the classification framework and cover physically important continuous symmetries (e.g., SO(n) and SU(n)).

In this thesis the index h_Φ is constructed for compact Lie groups and shown to be invariant along symmetric gapped paths, but the completeness direction (equivalence of indices implies phase equivalence) remains conjectural. Establishing completeness would close the classification for 1D G-SPT phases with continuous symmetries.