Transgression theorem (Chern–Simons)

Let $\pi:Q\to M$ be a principal $G$ -bundle, let $\omega_0,\omega_1$ be principal connections with curvatures $\Omega_0,\Omega_1$ , and let $p$ be an $\operatorname{Ad}$ -invariant real- or complex-valued homogeneous polynomial of degree $k\ge1$ on $\mathfrak g$ .

Set $\eta=\omega_1-\omega_0$ and $\omega_t=\omega_0+t\eta$ , and let $\Omega_t$ be the curvature of $\omega_t$ . Then

d\,\mathrm{CS}_p(\omega_0,\omega_1) =\operatorname{cw}_p(\omega_1)-\operatorname{cw}_p(\omega_0),

where

\pi^*\mathrm{CS}_p(\omega_0,\omega_1) =k\int_0^1p(\eta,\Omega_t,\dots,\Omega_t)\,dt.

Here $\mathrm{CS}_p$ is the transgression form on the base; the displayed basic form on $Q$ specifies its pullback uniquely. The symmetric polarization of $p$ is evaluated on one $1$ -form and $k-1$ curvature $2$ -forms. Consequently, the de Rham class of $\operatorname{cw}_p(\omega)$ is independent of $\omega$ .

Examples

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.
Degree 2 (classical 3D Chern–Simons). For a matrix group and $P(X)=\mathrm{tr}(X^2)$ , the transgression from the product flat reference connection (local potential zero) to the connection with local potential $A$ 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 $\mathrm{tr}(F\wedge F)$ .
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.

References

Hessel Posthuma, Notes on Chern–Simons Theory, §1.2, Theorem 1.10 and equation (1.3), p. 4. Author-hosted notes.