Chern–Simons and BF action data

Let $M$ be a compact oriented smooth three-manifold, possibly with smooth boundary, and fix a trivialization of a principal $G$-bundle over $M$, where $G$ has finite-dimensional real Lie algebra $\mathfrak{g}$. Give $\mathfrak{g}$ its Lie bracket $[,{]}_{\mathfrak{g}}$ and a nondegenerate symmetric bilinear form $⟨,{⟩}_{\mathfrak{g}}$ satisfying $⟨[X,Y{]}_{\mathfrak{g}},Z{⟩}_{\mathfrak{g}}=⟨X,[Y,Z{]}_{\mathfrak{g}}{⟩}_{\mathfrak{g}}$. Write ${\mathrm{\Omega }}^p(M;\mathfrak{g})=\mathrm{\Gamma }(M,{\wedge }^p{T}^{\ast }M\otimes \mathfrak{g})$ for smooth $\mathfrak{g}$-valued $p$-forms. Extend the Lie bracket and pairing to homogeneous simple tensors by $[X\otimes \alpha ,Y\otimes \beta ]=[X,Y{]}_{\mathfrak{g}}\otimes (\alpha \wedge \beta )$ and $⟨X\otimes \alpha ,Y\otimes \beta ⟩=⟨X,Y{⟩}_{\mathfrak{g}}\alpha \wedge \beta$, then extend bilinearly.

A connection in the chosen trivialization is a form $A\in {\mathrm{\Omega }}^1(M;\mathfrak{g})$. Its curvature and covariant exterior derivative are $F_A=dA+\frac{1}{2}[A,A]$ and $D_{A}X=dX+[A,X]$. Define the local Chern–Simons functional and the BF functional, for $B\in {\mathrm{\Omega }}^1(M;\mathfrak{g})$, by

$\mathrm{CS}(A)={\int }_M⟨A\wedge dA+\frac{1}{3}A\wedge [A,A]⟩,\mathrm{BF}(B,A)={\int }_M⟨B\wedge F_A⟩\mathrm{.}$

These are integrals of smooth top forms on a compact oriented manifold. The Chern–Simons formula is the local expression in the fixed trivialization; it does not define a globally gauge-invariant exponentiated action on arbitrary nontrivial bundles without additional characteristic-class and level data. No path-integral measure, quantization, or identification with a Reshetikhin–Turaev amplitude is part of this definition.