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 <math>\mathbf{A}</math> over it, we can define a family of p-forms:
In one dimension, the Chern–Simons 1-form is given by
:<math>\operatorname{Tr} [ \mathbf{A} ].</math>
In three dimensions, the Chern–Simons 3-form is given by
:<math>\operatorname{Tr} \left[ \mathbf{F} \wedge \mathbf{A}-\frac{1}{3} \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} \right] = \operatorname{Tr} \left[ d\mathbf{A} \wedge \mathbf{A} + \frac{2}{3} \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A}\right].</math>
In five dimensions, the Chern–Simons 5-form is given by
:<math>
\begin{align}
& \operatorname{Tr} \left[ \mathbf{F}\wedge\mathbf{F} \wedge \mathbf{A}-\frac{1}{2} \mathbf{F} \wedge\mathbf{A}\wedge\mathbf{A}\wedge\mathbf{A} +\frac{1}{10} \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} \wedge\mathbf{A} \right] \\[6pt]
= {} & \operatorname{Tr} \left[ d\mathbf{A}\wedge d\mathbf{A} \wedge \mathbf{A} + \frac{3}{2} d\mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} +\frac{3}{5} \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A}\wedge\mathbf{A}\wedge\mathbf{A} \right]
\end{align}
</math>
where the curvature F is defined as
:<math>\mathbf{F} = d\mathbf{A}+\mathbf{A}\wedge\mathbf{A}.</math>
The general Chern–Simons form <math>\omega_{2k-1}</math> is defined in such a way that
:<math>d\omega_{2k-1}= \operatorname{Tr}(F^k),</math>
where the wedge product is used to define F<sup>k</sup>. The right-hand side of this equation is proportional to the k-th Chern character of the connection <math>\mathbf{A}</math>.
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