Homological classification of topological terms in sigma models on homogeneous spaces

TL;DR

This paper classifies topological terms in non-linear sigma models on homogeneous spaces, providing a new condition for their G-invariance and illustrating with examples from quantum mechanics and field theory.

Contribution

It introduces a homological classification of topological terms in sigma models on G/H and derives a necessary and sufficient G-invariance condition for these terms.

Findings

Derived a new G-invariance condition for topological terms.

Classified topological terms using singular homology cycles.

Discussed examples including Aharonov-Bohm effect and Dirac monopole.

Abstract

We classify the topological terms (in a sense to be made precise) that may appear in a non-linear sigma model based on maps from an arbitrary worldvolume manifold to a homogeneous space $G/H$ (where $G$ is an arbitrary Lie group and $H \subset G$). We derive a new condition for $G$-invariance of topological terms, which is necessary and sufficient (at least when $G$ is connected), and discuss a variety of examples in quantum mechanics and quantum field theory. In the present work we discuss only terms that may be written in terms of (possibly only locally-defined) differential forms on $G/H$, leading to an action that is manifestly local. Such terms come in one of two types, with prototypical quantum-mechanical examples given by the Aharonov-Bohm effect and the Dirac monopole. The classification is based on the observation that, for topological terms, the maps from the worldvolume to $G/H$ may be replaced by singular homology cycles on $G/H$. In a forthcoming paper we apply the results to phenomenological models in which the Higgs boson is composite.