Chern–Simons form

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, ${\mathbf {A}}$ over it, we can define a family of p-forms:

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

{\rm Tr} [ \bold{A} ].

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

{\rm Tr} \left[ \bold{F}\wedge\bold{A}-\frac{1}{3}\bold{A}\wedge\bold{A}\wedge\bold{A}\right].

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

{\rm Tr} \left[ \bold{F}\wedge\bold{F}\wedge\bold{A}-\frac{1}{2}\bold{F}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A} +\frac{1}{10}\bold{A}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A} \right]

where the curvature F is defined as

\bold{F} = d\bold{A}+\bold{A}\wedge\bold{A}.

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

d\omega_{2k-1}={\rm Tr} \left( F^{k} \right),

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. 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, S.-S.; Simons, J. (1974). "Characteristic forms and geometric invariants". Annals of Mathematics. Second Series. 99 (1): 48–69. 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.

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