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

TL;DR

This paper investigates the topological terms in (2+1)d flag-manifold sigma models, revealing the absence of Hopf-like terms and identifying specific Chern-Simons terms that influence skyrmion statistics.

Contribution

It provides a detailed classification of topological terms in (2+1)d flag sigma models using bordism groups, highlighting the absence of Hopf-like terms and identifying new Chern-Simons terms.

Findings

Hopf-like term is absent despite nontrivial homotopy group

Existence of ${N(N-1) rack 2}-1$ Chern-Simons terms

Some Chern-Simons terms can turn skyrmions into fermions

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 $\pi_3(U(N)/U(1)^N)=\mathbb{Z}$, and thus skyrmions cannot become anyons with arbitrary statistics. Instead, we find that there exist ${N(N-1)\over 2}-1$ types of Chern-Simons terms, some of which can turn skyrmions into fermions, and we write down explicit forms of effective Lagrangians.