Chern-Simons for 2n-dimensional manifolds In the literature I can only find Chern-Simons terms  for odd-dimensional manifolds. For example, for a $G$-bundle over a 3-dimensional manifold we have $A \wedge dA + 2/3 * A \wedge A \wedge A$ with $A$ being a $\mathfrak{g}$-valued 1-form. Why can't I write such forms for even-dimensional manifolds?
 A: It has to do with the fact that the characteristic classes (over the reals) of a principal  $G$-bundle have even degree. We can associate  Chern-Simons-like theory  to each characteristic class of  degree $2k$ together with a $G$-bundle  $P$   over  a manifold of dimension  $2k-1$.
To be a bit more technical a Chern-Simons-like form is asssociated to the following data
1. A homogeneous polynomial $\Phi$ of degree $k$ on the Lie algebra  of $G$ invariant  under the action of $G$ by conjugation.
2. A  principal $G$-bundle  $P\to M$ over $M$.
3. A pair of connections $\nabla^0, \nabla^1$ on $P\to M$.
The Chern-Weil theory  produces two closed forms
$$ \Phi(\nabla^0),\Phi(\nabla^1)\in \Omega^{2k}(M) $$
and  a form 
$$ T\Phi(\nabla^1,\nabla^0)\in \Omega^{2k-1}(M), $$
such that 
$$ d T\Phi(\nabla^1,\nabla^0)=  \Phi(\nabla^1)-\Phi(\nabla^0). $$
(For details see Chapter 8 of these  notes.)
The  transgression form $T\Phi(\nabla^1,\nabla^0)$ is the one used in Chern-Simons  theories. It depends on two connections, but usually $\nabla^0$ is some fixed  connection.
