Transgression form

Let $\pi:P\to M$ be a principal G-bundle with Lie algebra $\mathfrak g$, and let $\omega_0,\omega_1$ be two principal connections on $P$ with corresponding curvature 2-forms $\Omega_0,\Omega_1\in\Omega^2(P;\mathfrak g)$.

Fix an $\operatorname{Ad}$-invariant symmetric multilinear map

i.e. an invariant polynomial datum as used to build a Chern--Weil form . Define the affine path of connections

and set $\eta:=\omega_1-\omega_0\in\Omega^1(P;\mathfrak g)$. (By tensoriality of the difference of two connections , $\eta$ is horizontal and $\operatorname{Ad}$-equivariant.) Let $\Omega_t$ be the curvature of $\omega_t$.

Definition (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 .

Examples

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

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

2. 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 .

3. 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 $dT = dA_1-dA_0$.