Topological term in the non-linear σ model of the SO(5) spin chains

We show that there is a topological (Berry phase) term in the non-linear σ model description of the SO(5) spin chain. It distinguishes the linear and projective representations of the SO(5) symmetry group, in exact analogy to the well-known θ-term of the SO(3) spin chain. The presence of the topological term is due to the fact that π2(SO(5)SO(3)×SO(2))=Z. We discuss the implication of our results on the spectra of the SO(5) spin chain, and connect it with a recent solvable SO(5) spin model which exhibits valence bond solid ground state and edge degeneracy.