Chern–Simons theory

The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type developed by Edward Witten. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who introduced the Chern–Simons 3-form. In the Chern–Simons theory, the action is proportional to the integral of the Chern–Simons 3-form.

In condensed-matter physics, Chern–Simons theory describes the topological order in fractional quantum Hall effect states. In mathematics, it has been used to calculate knot invariants and three-manifold invariants such as the Jones polynomial.

Particularly, Chern–Simons theory is specified by a choice of simple Lie group G known as the gauge group of the theory and also a number referred to as the level of the theory, which is a constant that multiplies the action. The action is gauge dependent, however the partition function of the quantum theory is well-defined when the level is an integer and the gauge field strength vanishes on all boundaries of the 3-dimensional spacetime.

It is also the central mathematical object in theoretical models for topological quantum computers (TQC). Specifically, an SU(2) Chern–Simons theory describes the simplest non-abelian anyonic model of a TQC, the Yang–Lee–Fibonacci model.

The dynamics of Chern–Simons theory on the 2-dimensional boundary of a 3-manifold is closely related to fusion rules and conformal blocks in conformal field theory, and in particular WZW theory.