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 F^k. 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 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

Further reading

• Chern, S.-S.; Simons, J. (1974). "Characteristic forms and geometric invariants". Annals of Mathematics. Second Series. 99 (1): 48–69. doi:10.2307/1971013. JSTOR 1971013.
• Bertlmann, Reinhold A. (2001). "Chern–Simons form, homotopy operator and anomaly". Anomalies in Quantum Field Theory (Revised ed.). Clarendon Press. pp. 321–341. ISBN 0-19-850762-3.