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

\eta := \omega_1-\omega_0 \in \Omega^1(P;\mathfrak g).

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

\omega_t := \omega_0 + t\,\eta,\qquad t\in[0,1],

with curvature

\Omega_t := d\omega_t + \tfrac12[\omega_t\wedge \omega_t]

(compare Cartan's second structure equation).

Change-of-connection (transgression) formula

Define the transgression form

T_p(\omega_0,\omega_1) := k \int_0^1 p\bigl(\eta \wedge \Omega_t^{k-1}\bigr)\,dt,

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

p(\Omega_1) - p(\Omega_0) \;=\; d\,T_p(\omega_0,\omega_1),

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.

Remarks

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

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).

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.

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.

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Change of connection formula for Chern Weil characteristic forms

Change-of-connection (transgression) formula

Local form

Curvature

Change of section (local gauge transformation)

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