Transgression form

The transgression form associated to $p$ and the pair $(\omega_0,\omega_1)$ is the $(2n-1)$-form $T_p(\omega_0,\omega_1)$ on $M$ uniquely characterized by the requirement that its pullback to $P$ is

where $\Omega_t^{\,n-1}$ denotes the wedge product of $(n-1)$ copies of $\Omega_t$ and $p(\eta\wedge \Omega_t^{n-1})$ means the $\mathbb R$-valued form obtained by feeding the $\mathfrak g$-valued factors into $p$.

Because the integrand is basic (horizontal and $G$-invariant), $T_p(\omega_0,\omega_1)$ is well-defined on the base. It is designed so that the difference of the corresponding Chern--Weil forms is exact:

which is the content of the transgression theorem. Specializing $\omega_0$ to a reference connection produces the usual Chern--Simons transgression form.

1. Degree 1 (linear invariant polynomial). For $n=1$ and an $\operatorname{Ad}$-invariant linear functional $p:\mathfrak g\to\mathbb R$, the formula reduces to
   $$T_p(\omega_0,\omega_1)=p(\omega_1-\omega_0), \qquad dT_p = p(\Omega_1)-p(\Omega_0).$$

1. Degree 2 on a trivial bundle (the usual Chern--Simons 3-form). On a trivial bundle and in a global gauge, a connection is represented by a $\mathfrak g$-valued 1-form $A$ (see local connection 1-form). For $p(X,Y)=\operatorname{tr}(XY)$ (degree $2$), taking $\omega_0=0$ gives the standard 3-form
   $$\operatorname{CS}(A)=\operatorname{tr}\!\Big(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\Big),$$
   with $d\,\operatorname{CS}(A)=\operatorname{tr}(F\wedge F)$, where $F=dA+A\wedge A$ is the local curvature.

1. Abelian case. If $G$ is abelian (e.g. $U(1)$), then $\operatorname{Ad}$ is trivial and $A\wedge A=0$. For a degree 1 invariant polynomial, the transgression between two $U(1)$-connections $A_0,A_1$ is simply $T(A_0,A_1)=A_1-A_0$, and $dT = dA_1-dA_0$.