Let $n\ge 1$ be an integer, let $M$ be a smooth manifold, let $G\subset GL(n,\mathbb C)$ be a matrix Lie group with identity matrix $I_n$ and [Lie algebra](/page/Lie%20Algebra) $\mathfrak g\subset \mathfrak{gl}(n,\mathbb C)$, and let $A\in\Omega^1(M;\mathfrak g)$. Products of matrix-valued differential forms are taken using wedge product of differential forms and matrix multiplication of coefficients, and $\operatorname{tr}:\mathfrak{gl}(n,\mathbb C)\to\mathbb C$ denotes the ordinary matrix trace. Define the matrix Chern-Simons three-form by

\begin{align*} \operatorname{CS}(A):=\operatorname{tr}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right)\in\Omega^3(M;\mathbb C). \end{align*}

For every smooth map $g:M\to G$, define

\begin{align*} \theta:=g^{-1}dg\in\Omega^1(M;\mathfrak g) \end{align*}

and

\begin{align*} A^g:=g^{-1}Ag+g^{-1}dg=g^{-1}Ag+\theta\in\Omega^1(M;\mathfrak g). \end{align*}

Then

\begin{align*} \operatorname{CS}(A^g)-\operatorname{CS}(A)=-d\operatorname{tr}\left(\theta\wedge g^{-1}Ag\right)-\frac{1}{3}\operatorname{tr}(\theta\wedge\theta\wedge\theta). \end{align*}

Equivalently, since $\operatorname{tr}(A\wedge dg\,g^{-1})=-\operatorname{tr}(\theta\wedge g^{-1}Ag)$, one has

\begin{align*} \operatorname{CS}(A^g)-\operatorname{CS}(A)=d\operatorname{tr}\left(A\wedge dg\,g^{-1}\right)-\frac{1}{3}\operatorname{tr}(\theta\wedge\theta\wedge\theta). \end{align*}