# 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, $\mathbf {A}$ over it, we can define a family of p-forms:

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

$\operatorname {Tr} [\mathbf {A} ].$ In three dimensions, the Chern–Simons 3-form is given by

$\operatorname {Tr} \left[\mathbf {F} \wedge \mathbf {A} -{\frac {1}{3}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right]=\operatorname {Tr} \left[d\mathbf {A} \wedge \mathbf {A} +{\frac {2}{3}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right].$ In five dimensions, the Chern–Simons 5-form is given by

{\begin{aligned}&\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{aligned}} where the curvature F is defined as

$\mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} .$ The general Chern–Simons form $\omega _{2k-1}$ is defined in such a way that

$d\omega _{2k-1}=\operatorname {Tr} (F^{k}),$ 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 $\mathbf {A}$ .

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 form.

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.