Chern–Simons form

Short description: Secondary characteristic classes of 3-manifolds

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

$\mathrm{Tr}[\mathbf{𝐀}].$

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

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

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

$\begin{array}{rl}& \mathrm{Tr}[\mathbf{𝐅}\wedge \mathbf{𝐅}\wedge \mathbf{𝐀}-\frac{1}{2}\mathbf{𝐅}\wedge \mathbf{𝐀}\wedge \mathbf{𝐀}\wedge \mathbf{𝐀}+\frac{1}{10}\mathbf{𝐀}\wedge \mathbf{𝐀}\wedge \mathbf{𝐀}\wedge \mathbf{𝐀}\wedge \mathbf{𝐀}]\\ =& \mathrm{Tr}[d\mathbf{𝐀}\wedge d\mathbf{𝐀}\wedge \mathbf{𝐀}+\frac{3}{2}d\mathbf{𝐀}\wedge \mathbf{𝐀}\wedge \mathbf{𝐀}\wedge \mathbf{𝐀}+\frac{3}{5}\mathbf{𝐀}\wedge \mathbf{𝐀}\wedge \mathbf{𝐀}\wedge \mathbf{𝐀}\wedge \mathbf{𝐀}]\end{array}$

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

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

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. 
• 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. https://books.google.com/books?id=FC_DRRUHFXEC&pg=PA321.