Chern–Simons forms and transgression

Let $A_0,A_1$ denote local expressions for two connections on the same principal bundle, and set $a=A_1-A_0$. Their difference is an adjoint-valued one-form, because the inhomogeneous gauge terms cancel. The affine path $A_t=A_0+t a$ consists of connections with curvatures $F_t$. For a symmetric adjoint-invariant polynomial of degree $k$, differentiating its curvature evaluation gives an exterior derivative. The Bianchi identity removes the covariant derivatives of the remaining curvature factors, and invariance replaces the final covariant derivative by an ordinary derivative.

Integrating this identity in the path parameter yields the global relative transgression below. Both connections and their difference transform together, so its local expressions glue on the base. The formula compares two characteristic representatives of the same class, rather than supplying a primitive of either representative separately. Taking a nonexistent global zero connection on a nontrivial bundle would invalidate that inference. A zero local potential can be chosen in a trivialization, but its values in other frames include an inhomogeneous term.

For the matrix invariant $\operatorname{Tr}(F\wedge F)$, a potential on a trivializing chart gives the cubic expression below. Its derivative is the characteristic four-form. Products mean matrix multiplication combined with wedge products, and trace cyclicity has graded signs when forms are moved past one another. In an abelian line group the cubic term vanishes, but the quadratic term $A\wedge dA$ need not vanish: a one-form can wedge nontrivially with its exterior derivative.

The corresponding form can be defined globally on the principal total space using the principal connection form, but generally it is not basic and therefore does not descend as a single form on the base. Local primitives on different charts can disagree by gauge terms. This is entirely compatible with a globally nonexact characteristic form. In a globally trivial bundle with a chosen trivialization, one does obtain a global base expression; the choice is then part of the construction, and its behavior under changing that choice must be analyzed.

On a closed oriented three-manifold, an integrally normalized Chern–Simons construction can define an invariant modulo integers under suitable compact-group and characteristic-class hypotheses. Large gauges may shift a chosen real-valued representative by a period. For example, the standard compact simply connected theory uses an integral invariant bilinear form, or equivalently the appropriate degree-four integral characteristic data. An arbitrary real scaling of a trace polynomial does not automatically have integer periods, and general groups can require additional integral refinements.

Boundaries introduce further data and boundary terms, so a closed-manifold assertion cannot simply be transferred to a manifold with boundary. Likewise a vanishing primary form in dimension three does not force a Chern–Simons three-form to vanish: its derivative is zero there for dimensional reasons. Relative transgression, local primitives, and differential refinements each retain different information. Keeping their domains and normalizations explicit makes the secondary theory useful without turning a local algebraic identity into an incorrect global exactness claim.