Chern–Simons form

Template:Short description 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 $𝐀$ over it, we can define a family of p-forms:

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

$\mathrm{Tr}[𝐀].$

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

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

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

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

where the curvature F is defined as

$𝐅=d𝐀+𝐀\wedge 𝐀.$

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 $𝐀$.

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

• Template:Cite book