In mathematics, the Chern–Simons forms are certain secondary characteristic classes.[1] 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.[2]
A
In one dimension, the Chern–Simons 1-form is given by
\operatorname{Tr}[A].
In three dimensions, the Chern–Simons 3-form is given by
\operatorname{Tr}\left[F\wedgeA-
| 1 | |
| 3 |
A\wedgeA\wedgeA\right]=\operatorname{Tr}\left[dA\wedgeA+
| 2 | |
| 3 |
A\wedgeA\wedgeA\right].
In five dimensions, the Chern–Simons 5-form is given by
\begin{align} &\operatorname{Tr}\left[F\wedgeF\wedgeA-
| 1 | |
| 2 |
F\wedgeA\wedgeA\wedgeA+
| 1 | |
| 10 |
A\wedgeA\wedgeA\wedgeA\wedgeA\right]\\[6pt] ={}&\operatorname{Tr}\left[dA\wedgedA\wedgeA+
| 3 | |
| 2 |
dA\wedgeA\wedgeA\wedgeA+
| 3 | |
| 5 |
A\wedgeA\wedgeA\wedgeA\wedgeA\right] \end{align}
where the curvature F is defined as
F=dA+A\wedgeA.
The general Chern–Simons form
\omega2k-1
d\omega2k-1=\operatorname{Tr}(Fk),
where the wedge product is used to define Fk. The right-hand side of this equation is proportional to the k-th Chern character of the connection
A
In general, the Chern–Simons p-form is defined for any odd p.[4]
In 1978, Albert Schwarz formulated Chern–Simons theory, early topological quantum field theory, using Chern-Simons forms.[5]
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.