Chern–Simons form
The Chern - Simons forms are in the definition of secondary characteristic classes using differential forms, which occur in mathematics in differential geometry and differential topology in different contexts, in particular in gauge theories. The Chern - Simons 3- form defines the functional effect of the Chern - Simons theory. They are named after Shiing - Shen Chern and James Harris Simons, the authors of the 1974 paper published in Characteristic Forms and Geometric Invariants.
Definition
Let M be a Riemannian manifold. The Riemannian connection
Is a Lie algebra -valued 1-form on the frame bundle.
The Chern - Simons 1- shape is defined by
Where Tr denotes the trace of matrices.
The Chern - Simons 3- shape is defined by
The Chern - Simons -5- shape is defined by
Wherein said curvature is defined by
The general Chern - Simons form is defined so that
Being defined by the exterior product of differential forms.
If a parallelizable 2k -1 dimensional manifold is (for example a 3- directional diversity ) then there is an intersection and of the integral of the manifold is a global invariants, which is well-defined modulo addition of integers. ( For different cuts, the integrals differ only by integers. ) Thus defined is invariant, the Chern Simons invariant
General definition for principal bundles and invariant polynomials
Let be a Lie group with Lie algebra and an invariant polynomial.
Each invariant polynomial corresponding to a Chern - Simons form of- principal bundles as follows.
Be a principal bundle with structure group. You choose a connection form and denote by its curvature form. Then Chern - Simons shape is defined by
With.
In the case of flat bundles, this formula simplifies to.
The equation applies
In the case of flat bundles so.
As is known, each corresponding to a characteristic class invariants polynomial, see Chern -Weil theory. If, then disappears after Chern -Weil theory, the corresponding characteristic class in real cohomology. The mold is closed in this case and initially defines a class in the cohomology of P. retracting means of a cut defines a cohomology class of M, which is well-defined modulo integers. The so- defined cohomology class in fits into the Bockstein sequence
Where it is mapped to the characteristic class vanishes whose image in real cohomology.