Transgression theorem (Chern–Simons)

Let $\pi:P\to M$ be a principal G-bundle . Let $\omega_0,\omega_1$ be two principal connections on $P$ with curvatures $\Omega_0,\Omega_1$. Let $P$ be an Ad-invariant homogeneous polynomial of degree $k$ on $\mathfrak g$, and let $\operatorname{cw}_P(\omega_i)$ be the descended Chern–Weil forms on $M$ as in the Chern–Weil theorem .

Write $\eta:=\omega_1-\omega_0$ (a tensorial $\mathfrak g$-valued $1$-form; see difference is tensorial ), and define a path of connections $\omega_t:=\omega_0+t\eta$ with curvature $\Omega_t$.

Theorem (Transgression / Chern–Simons). There exists a $(2k-1)$-form $\mathrm{CS}_P(\omega_0,\omega_1)\in\Omega^{2k-1}(M)$ such that

where $d$ is the exterior derivative . One explicit choice is the Chern–Simons transgression form

where $P(\eta,\Omega_t,\dots,\Omega_t)$ denotes the $(2k-1)$-form obtained by inserting one $1$-form $\eta$ and $(k-1)$ copies of the $2$-form $\Omega_t$ into the symmetric $k$-linear map $P$ and wedging.

In particular, the de Rham class $[\operatorname{cw}_P(\omega)]$ is independent of $\omega$.

Examples

1. Degree 1 (abelian case). For $k=1$ and $P(X)=X$ (e.g. $G=U(1)$), $\operatorname{cw}_P(\omega)$ is just the curvature $2$-form on the base, and the formula becomes $\operatorname{cw}_P(\omega_1)-\operatorname{cw}_P(\omega_0)=d(A_1-A_0)$ in a local gauge.
2. Degree 2 (classical 3D Chern–Simons). For a matrix group and $P(X)=\mathrm{tr}(X^2)$, the transgression gives the usual $3$-form on a trivialization: $\mathrm{CS}(A)=\mathrm{tr}\!\left(A\wedge dA+\tfrac23 A\wedge A\wedge A\right),$ whose exterior derivative is $\mathrm{tr}(F\wedge F)$.
3. Gauge-equivalent connections. If $\omega_1$ is obtained from $\omega_0$ by a gauge transformation, then $\operatorname{cw}_P(\omega_1)-\operatorname{cw}_P(\omega_0)$ is exact; the theorem produces an explicit primitive.