Topological terms of (2+1)d flag-manifold sigma models

Abstract

Abstract We examine topological terms of (2 + 1)d sigma models and their consequences in the light of classifications of invertible quantum field theories utilizing bordism groups. In particular, we study the possible topological terms for the U(N)/U(1)N flag-manifold sigma model in detail. We argue that the Hopf-like term is absent, contrary to the expectation from a nontrivial homotopy group π3(U(N)/U(1)N) = ℤ, and thus skyrmions cannot become anyons with arbitrary statistics. Instead, we find that there exist $$ \frac{N\left(N-1\right)}{2}-1 $$ N N − 1 2 − 1 types of Chern-Simons terms, some of which can turn skyrmions into fermions, and we write down explicit forms of effective Lagrangians.