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 2-form $F_A$ . Fix an $\mathrm{Ad}$ -invariant homogeneous polynomial $P$ of degree $k$ on the Lie algebra $\mathfrak{g}$ .

Statement (name-level, with standard formula)

On a local trivialization (or globally after choosing a trivialization or reference datum), the Chern–Simons $(2k-1)$ -form $\mathrm{CS}_P(A)$ satisfies

d\,\mathrm{CS}_P(A)=P(F_A),

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

A^g = g^{-1}Ag + g^{-1}dg,

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

\mathrm{CS}_P(A^g)-\mathrm{CS}_P(A) \;=\; d\,\alpha_P(A,g)\;+\; g^*\eta_P,

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 a closed manifold with the extra data needed to define the Chern–Simons functional (for example, a trivial bundle and a chosen global trivialization): if $P$ corresponds (via Chern–Weil theory) to an integral characteristic class, then the number

\int_M \mathrm{CS}_P(A)

is invariant under gauge transformations modulo integers (with the conventional $2\pi$ -normalization built into $P$ ). For a nontrivial bundle, the corresponding global object requires additional differential-cohomological or extension data; the local form alone need not glue to a form on $M$ .

Examples

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.

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).

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.