Gauge transformation behavior of Chern–Simons forms

Let $\pi:P\to M$ be a principal G-bundle with structure group a Lie group $G$, and let $A$ be a principal connection on $P$ with curvature $F_A$. Fix an $\mathrm{Ad}$-invariant homogeneous polynomial $P$ of degree $k$ on the Lie algebra $\mathfrak{g}$.

Statement (name-level, with standard formula)

There is a canonical Chern–Simons $(2k-1)$-form $\mathrm{CS}_P(A)$ on $M$ (defined locally from $A$) satisfying

where $d$ is the exterior derivative .

Under a gauge transformation $g$ (locally, a smooth map $g:U\to G$ in a trivialization), the transformed connection is

and the Chern–Simons form changes by a universal transgression formula of the form

where:

• $\alpha_P(A,g)$ is an explicit $(2k-2)$-form built from $A$ and $g^{-1}dg$, and
• $\eta_P$ is a closed $(2k-1)$-form on $G$ determined by $P$ (the “group term”), expressed using the Lie bracket and wedge products on $\mathfrak{g}$-valued forms.

Consequences on closed manifolds: if $P$ corresponds (via Chern–Weil theory) to an integral characteristic class, then for every closed oriented $(2k-1)$-manifold $M$ the number

is invariant under gauge transformations modulo integers (with the conventional $2\pi$-normalization built into $P$).

Examples

1. Abelian case $G=\mathrm{U}(1)$ (degree $k=1$).
   Here $A$ is an ordinary real $1$-form (in a trivialization) and gauge transformations act by $A\mapsto A + d\phi$ for a circle-valued function. The Chern–Simons “form” is just $A$, and its change is exact, so the integral over a closed loop depends only on the winding of the gauge function.

2. Three-dimensional Chern–Simons for $k=2$.
   When $k=2$ (so $\mathrm{CS}_P$ is a $3$-form), the group term is the classical $3$-form on $G$ built from $g^{-1}dg$. In particular, on a trivial bundle with $A=0$ one has

   $$\mathrm{CS}_P(A^g)=g^*\eta_P,$$

   and the integral over a closed $3$-manifold measures the homotopy class of $g$ (an integer for integral normalizations).

3. Gauge invariance modulo integers of the Chern–Simons functional.
   For compact $G$ and integral $P$, the Chern–Simons functional on a closed $(2k-1)$-manifold is a well-defined element of $\mathbb{R}/\mathbb{Z}$; different representatives differ by an integer coming from the group term.