Chern–Simons form

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.²³

Given a manifold and a Lie algebra valued 1-form ${\displaystyle \mathbf {A} }$ over it, we can define a family of p-forms:⁴

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

${\displaystyle \operatorname {Tr} [\mathbf {A} ].}$

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

${\displaystyle \operatorname {Tr} \left[\mathbf {F} \wedge \mathbf {A} -{\frac {1}{3}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right]=\operatorname {Tr} \left[d\mathbf {A} \wedge \mathbf {A} +{\frac {2}{3}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right].}$

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

${\displaystyle {\begin{aligned}&\operatorname {Tr} \left[\mathbf {F} \wedge \mathbf {F} \wedge \mathbf {A} -{\frac {1}{2}}\mathbf {F} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} +{\frac {1}{10}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right]\\[6pt]={}&\operatorname {Tr} \left[d\mathbf {A} \wedge d\mathbf {A} \wedge \mathbf {A} +{\frac {3}{2}}d\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} +{\frac {3}{5}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right]\end{aligned}}}$

where the curvature F is defined as

${\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} .}$

The general Chern–Simons form ${\displaystyle \omega _{2k-1}}$ is defined in such a way that

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

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 ${\displaystyle \mathbf {A} }$.

In general, the Chern–Simons p-form is defined for any odd p.⁵

In 1978, Albert Schwarz formulated Chern–Simons theory, early topological quantum field theory, using Chern-Simons forms.⁶

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.