Chern–Simons form

In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose.

Definition

Given a manifold and a Lie algebra valued 1-form over it, we can define a family of p-forms:

In one dimension, the Chern–Simons 1-form is given by

In three dimensions, the Chern–Simons 3-form is given by

In five dimensions, the Chern–Simons 5-form is given by

where the curvature F is defined as

The general Chern–Simons form is defined in such a way that

where the wedge product is used to define F^k. The right-hand side of this equation is proportional to the k-th Chern character of the connection .

In general, the Chern–Simons p-form is defined for any odd p.

Application to physics

In 1978, Albert Schwarz formulated Chern–Simons theory, early topological quantum field theory, using Chern-Simons forms.

In the gauge theory, the integral of Chern-Simons form is a global geometric invariant, and is typically gauge invariant modulo addition of an integer.

See also

• Chern–Weil homomorphism
• Chiral anomaly
• Topological quantum field theory
• Jones polynomial

References

Further reading