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{Nleft(N-1right)}{2}-1 types of Chern-Simons terms, some of which can turn skyrmions into fermions, and we write down explicit forms of effective Lagrangians.
Highlights
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 skyrmions cannot become anyons with arbitrary statistics
Assuming quantum field theories (QFTs) can be defined on any spacetime manifolds, the principle of locality requires their partition functions to be consistent under cutting and gluing of spacetime manifolds. Along this line together with the unitarity, topological terms should be regarded as partition functions of the so-called invertible QFTs [7]
Summary
3d CP 1 sigma model gives a low-energy description of the 2d anti-ferromagnetic quantum spin systems. The Hopf term does not have a local expression in the original spin fields, but can be written as a Chern-Simons form using the U(1) gauge field in the gauged sigma-model description [38]. Starting from the (2 + 1)d SU(2) anti-ferromagnetic quantum Heisenberg spin system, it is known that the Hopf term does not appear in the low-energy theory [39]. In a recent paper [11], it has been elucidated that only these values, θ = 0 and θ = π, are consistent as local and unitary QFTs. k = θ/π behaves as the level of U(1) spin Chern-Simons term, and the magnetic skyrmion becomes fermion if k is odd. In the following of this section, we will give a review on these facts, as it is a basic ingredient in order to extend these results for general flag-manifold sigma models
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