Let $G$ be a compact Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, let $k\ge 1$, and let $P\in (S^k\mathfrak g^*)^G$ be an invariant symmetric $k$-linear polynomial whose universal real Chern-Weil class is the image of a class $c_P\in H^{2k}(BG;\mathbb Z)$ under the coefficient homomorphism to real cohomology. Fix the natural Cheeger-Simons differential characteristic class $\widehat c_P$ refining $c_P$, so that for every smooth principal $G$-bundle $E\to X$ with smooth connection $A$, it assigns a differential character

\begin{align*} \widehat c_P(E,A)\in \widehat H^{2k}(X;\mathbb Z) \end{align*}

with curvature form $P(F_A)\in \Omega^{2k}(X)$.

Let $M$ be a closed oriented smooth manifold of dimension $2k-1$, let $E\to M$ be a smooth principal $G$-bundle, let $\mathcal A_{\mathrm{flat}}(E)$ denote the set of flat smooth connections on $E$, and let $\mathcal G(E)$ denote the gauge group of smooth principal $G$-bundle automorphisms of $E$ covering $\operatorname{id}_M$. Then the map

\begin{align*} \operatorname{CS}_{P,E}: \mathcal A_{\mathrm{flat}}(E) &\to \mathbb R/\mathbb Z \end{align*}

\begin{align*} A &\mapsto \widehat c_P(E,A)([M]) \end{align*}

is invariant under the action of $\mathcal G(E)$. Consequently it descends to a well-defined map

\begin{align*} \overline{\operatorname{CS}}_{P,E}: \mathcal A_{\mathrm{flat}}(E)/\mathcal G(E) &\to \mathbb R/\mathbb Z \end{align*}

\begin{align*} [A] &\mapsto \widehat c_P(E,A)([M]). \end{align*}