Chern-Simons Forms in Gravitation Theories

TL;DR

This paper reviews the role of Chern-Simons forms in gravitation theories, highlighting their mathematical origins and diverse applications in physics, including gauge theories, anomalies, and topological phenomena.

Contribution

It provides an introductory overview of how Chern-Simons forms are integrated into gravitation theories and their significance across various physical contexts.

Findings

Chern-Simons forms are crucial in gauge theories and gravity.

They relate to anomalies in quantum field theories.

CS terms appear in topological insulators and high Tc superconductivity.

Abstract

The Chern-Simons (CS) form evolved from an obstruction in mathematics into an important object in theoretical physics. In fact, the presence of CS terms in physics is more common than one may think: they seem to play an important role in high Tc superconductivity and in recently discovered topological insulators. In classical physics, the minimal coupling in electromagnetism and to the action for a mechanical system in Hamiltonian form are examples of CS functionals. CS forms are also the natural generalization of the minimal coupling between the electromagnetic field and a point charge when the source is not point-like but an extended fundamental object, a membrane. They are found in relation with anomalies in quantum field theories, and as Lagrangians for gauge fields, including gravity and supergravity. A cursory review of the role of CS forms in gravitation theories is presented at an introductory level.