Chern–Simons form - Alchetron, The Free Social Encyclopedia

In mathematics, the Chern–Simons forms are certain secondary characteristic classes. They have been found to be of interest in gauge theory, and they (especially the 3-form) define the action of Chern–Simons theory. 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. See Chern and Simons (1974)

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

T r [ A ] .

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

T r [ F ∧ A − 1 3 A ∧ A ∧ A ] .

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

T r [ F ∧ F ∧ A − 1 2 F ∧ A ∧ A ∧ A + 1 10 A ∧ A ∧ A ∧ A ∧ A ]

where the curvature F is defined as

F = d A + A ∧ A .

The general Chern–Simons form ω 2 k − 1 is defined in such a way that

d ω 2 k − 1 = T r ( 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. See also gauge theory for the definitions. Its integral over a p-dimensional manifold is a global geometric invariant, and is typically gauge invariant modulo addition of an integer.

References

Chern–Simons form Wikipedia

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