Change of connection formula for Chern Weil characteristic forms

Let $\pi:P\to M$ be a principal G-bundle , and let $\omega_0,\omega_1$ be two principal connections on $P$ with connection 1-forms (also denoted) $\omega_0,\omega_1\in \Omega^1(P;\mathfrak g)$ as in connection 1-forms on principal bundles . Write their curvature 2-forms as $\Omega_0,\Omega_1\in \Omega^2(P;\mathfrak g)$ (see curvature 2-forms of principal connections ).

Let $p\in \operatorname{Sym}^k(\mathfrak g^*)^G$ be an $\operatorname{Ad}$-invariant homogeneous polynomial of degree $k$, so that the associated Chern Weil form $p(\Omega)$ is a closed $2k$-form on $M$ (via basicness on $P$), as in the Chern Weil theorem .

Define the difference

By tensoriality of differences of principal connections , $\eta$ is horizontal and equivariant (hence “tensorial”), so it corresponds to an $\operatorname{ad}(P)$-valued 1-form on $M$.

Consider the straight-line path of connections

with curvature

(compare Cartan's second structure equation ).

Change-of-connection (transgression) formula

Define the transgression form

a $(2k-1)$-form on $P$ which is basic and therefore descends to a $(2k-1)$-form on $M$ (often also denoted $T_p(\omega_0,\omega_1)$).

Then the characteristic forms satisfy

so the two forms differ by an exact form on $M$. In particular, the de Rham class of $p(\Omega)$ does not depend on the chosen connection, which is the mechanism behind Chern Weil characteristic classes being invariants of the principal bundle .

This construction is the standard transgression mechanism (see transgression forms and the transgression theorem ), and in low degrees it produces the usual Chern Simons forms .

Examples

1. Degree 1 (abelian-style) case.
   If $k=1$ and $p:\mathfrak g\to\mathbb R$ is an $\operatorname{Ad}$-invariant linear functional, then

   $$T_p(\omega_0,\omega_1)=p(\eta),\qquad p(\Omega_1)-p(\Omega_0)=d\,p(\eta).$$

   For $G=U(1)$ this recovers the familiar fact that changing a connection 1-form changes the curvature 2-form by an exact 2-form (compare the local picture in local connection 1-forms ).

2. Degree 2 and the Chern Simons 3-form.
   For $k=2$ and an invariant quadratic polynomial $p$ (for instance a suitably normalized trace form on a matrix Lie algebra), the transgression is a 3-form

   $$T_p(\omega_0,\omega_1) = 2\int_0^1 p(\eta\wedge \Omega_t)\,dt,$$

   and the formula says $p(\Omega_1)-p(\Omega_0)$ is exact with this primitive. On a trivial bundle, taking $\omega_0=0$ and $\omega_1=A$ yields the usual Chern Simons expression in terms of $A$ and $dA$, matching the standard Chern Simons form on a chart.

3. Pontryagin forms from different connections.
   Let $E\to M$ be a real vector bundle with two vector bundle connections $\nabla^0,\nabla^1$. Using the induced connections on the frame bundle (via the induced principal connection construction ), the change-of-connection formula implies that the associated Pontryagin forms differ by an exact form. Hence the resulting Pontryagin classes are independent of the chosen connection.