Topological Field Theory and Nonlinear $\sigma$-Models on Symmetric Spaces

TL;DR

This paper explores the connection between topological field theories, specifically Chern-Simons actions, and nonlinear sigma models on symmetric spaces, revealing their relation to integrable systems and condensed matter physics.

Contribution

It establishes a link between non-abelian Chern-Simons theories and nonlinear sigma models on Hermitian symmetric spaces, including their classical and quantum properties.

Findings

Relation between Chern-Simons actions and nonlinear sigma models.

Connection to integrable and non-integrable evolution systems.

Implications for condensed matter physics.

Abstract

We show that the classical non-abelian pure Chern-Simons action is related to nonrelativistic models in (2+1)-dimensions, via reductions of the gauge connection in Hermitian symmetric spaces. In such models the matter fields are coupled to gauge Chern-Simons fields, which are associated with the isotropy subgroup of the considered symmetric space. Moreover, they can be related to certain (integrable and non-integrable) evolution systems, as the Ishimori and the Heisenberg model. The main classical and quantum properties of these systems are discussed in connection with the topological field theory and the condensed matter physics.