The general Chern-Simons form is defined such that
where is defined by the outer product of differential forms.
If there is a parallelizable 2k-1 -dimensional manifold (e.g. an orientable 3-manifold), then there is a cut and the integral of over the manifold is a global invariant, which is well-defined modulo the addition of whole numbers. (For different cuts the integrals only differ by whole numbers.) The invariant defined in this way is the Chern-Simons invariant
.
General definition of principal bundles and invariant polynomials
Each invariant polynomial corresponds to a Chern-Simons form of principle bundles as follows.
Be a principal bundle with a structure group . Choose a form of connection and use to designate its form of curvature . Then the Chern-Simons form is defined by
with .
In the case of flat bundles, this formula is simplified to .
The equation applies
,
in the case of flat bundles .
As is well known, every characteristic class corresponds to an invariant polynomial, see Chern-Weil theory . If so , then according to Chern-Weil theory the corresponding characteristic class vanishes in real cohomology. The form is closed in this case and initially defines a class in the cohomology of . Withdrawal by means of a cut defines a cohomology class of which is well defined modulo of integers. The cohomology class defined in this way fits into the Bockstein sequence
,
where it is mapped to the characteristic class , the image of which vanishes in real cohomology.