As you say yourself, indeed every connection on a bundle is *locally* given by a Lie algebra valued 1-form and in general only locally.

Let's say this more in detail: for $X$ any manifold, a $G$-principal connection on it is (in "Cech data"):

a choice of good open cover $\{U_i \to X\}$;

on each patch a 1-form $A_i \in \Omega^1(U_i)\otimes \mathfrak{g}$;

on each double intersection of patches a gauge transformation function $g_{i j} \in C^\infty(U_i \cap U_j, G)$

such that

on each double intersectin $U_i \cap U_j$ we have the equation $A_j = g_{i j}^{-1} A g_{i j} + g_{i j}^{-1} \mathbf{d} g_{i j}$

on each triple intersection $U_i \cap U_j \cap U_k$ we have the equation $g_{i j} g_{j k} = g_{i k}$.

Okay, now you would like to form a Chern-Simons 3-form... something out of this. What you immediately get from the above data is a bunch of local differential 3-forms, one on each patch: $CS(A_i) \in \Omega^3(U_i)$.

To make these 3-forms globally glue together to what is called a *3-form connection* we need the evident data of higher gauge transformation

on each patch we have the local 3-form $CS(A_i)$;

on each double intersection there should be a 2-form $B_{i j} \in \Omega^2(U_i \cap U_j)$ which gauge transforms the respective CS-3-forms into each other, by $CS(A_j) = CS(A_i) + \mathbf{d} B_{i j}$;

on each triple intersection there should be a 1-form $\alpha_{i j k} \in \Omega^1(U_i \cap U_j \cap U_k)$ which exhibits a second-order gauge transformation ("ghosts of ghosts"!) between the first order gauge trasformations, in that $B_{i j} + B_{j k} = B_{i k} + \mathbf{d} \alpha_{ i j k}$

finally on each quadruple intersection there should be a smooth function $h_{i j k l} \in C^\infty(U_i \cap U_j \cap U_k \cap U_l, U(1))$ which gauge-of-gauge-of-gauge-transforms the gauge-of-gauge-transforms into each other, in that $\alpha_{i j k} + \alpha_{i k l} = \alpha_{j k l} + \alpha_{i j l} + h_{i j k l}^{-1}\mathbf{d}h_{i j k l}$.

That's the data that makes the local Chern-Simons 3-form into a globally well-defined 3-form field. (For instance the supergravity C-field is of this form, with some further twists and bells and whistles added, as we have discussed here).

In mathematical language one says that this kind of local gauge-of-gauge-of-gauge gluing data for global definition of higher form fields is a "degree-4 cocycle in Cech-Deligne cohomology". This is *precisely* the right data needed to have a well-defined 3-dimensional higher holonomy as is needed here for the definition, because the Chern-Simons action functional is nothing but the 3-dimensional higher holonomy of this 3-form connection.

If you can build it, that is. From the above it is not entirely obvious how to build the 3-form cocycle data $\{CS(A_i), B_{i j}, \alpha_{i j k}, h_{i j k}\}$ from the given gauge field data $\{A_i, g_{i j}\}$.

But this can be done. This is what Cheeger-Simons differential characters were discovered for. An explicit construction that is very natural for the application to Chern-Simons theory we have given in

- Fiorenza, Schreiber, Stasheff,
*Cech cocycles for differential characteristic classes*, Advances in Theoretical and Mathematical Phyiscs, Volume 16 Issue 1 (2012) (arXiv:1011.4735, web)

Based on this we give a detailed introduction to and discussion of Chern-Simons action functionals for globally non-trivial situations like above in

- Fiorenza, Sati, Schreiber,
*A higher stacky perspective on Chern-Simons theory* (arXiv:1301.2580, web)

That article gives the local formulas that apply generally, discusses the simplifications that occur when the 3-manifold can be assumed to be bounding, discusses what happens if not, and then explores various other properties of globally defined Chern-Simons theory, such as how to couple Wilson lines to the above story. If you just look at the first part, I think you should find what you need.

edit: In the comments below came up the question why a similar discussion is not also needed when writing down the Yang-Mills action functional, whose Lagrangian is the 4-form $\langle F_A \wedge \star F_A \rangle$ (where $\star$ is the Hodge star of a given metric (gravity) and $\langle -,-\rangle$ is an invariant polynomial, the "Killing form" or trace), or similarly the topological Yang-Mills action functional, whose Lagrangian is the 4-form $\langle F_A \wedge F_A \rangle$.

The reason is that these Lagrangians are built from *curvatures evaluated in an invariant polynomial*. The very invariance of these invariant polynomials under the adjoint action of the gauge group on its arguments ensures that if $\{U_i \to X\}$ is a good open cover of 4-dimensional space(-time) and if the gauge field is given by the Cech-cocycle data $\{A_i, g_{i j}\}$ with respect to these local patches, that then on double overlaps the two (topological or not) Yang-Mills Lagrangians coming from two patches are already *equal*

$$
\langle F_{A_i} \wedge F_{A_i}\rangle = \langle F_{A_j}\wedge F_{A_j}\rangle
\,.
$$

Hence if we write $\nabla = \{A_i, g_{i j}\}$ for the gauge field connection abstractly and denote the (topological) Yang-Mills Lagrangian globally by $\langle F_\nabla \wedge F_\nabla\rangle$, then this is already a globally defined 4-form. Mathematically, this statement is what is at the core of Chern-Weil theory.

Notice that there is nevertheless an intricate relation to the story of the Chern-Simons functional. Namely the local Chern-Simons form $CS(A_i)$ has the special property (essentially by definition) that its differential is the topological Yang-Mills Lagrangian:

$$
\mathbf{d}CS(A_i) = \langle F_{A_i} \wedge F_{A_i}\rangle
\,.
$$

This means that with the Chern-Simons Lagrangian regarded as a 3-form connection then the topological Yang-Mills Lagrangian is its *curvature 4-form*. Therefore the relation between the topological Yang-Mills Lagrangian 4-form and the Chern-Simons 3-form is precisely an analogue in higher gauge theory of the familiar relation two degrees down of how the electromagnetic potential 1-form -- which is not globally defined in general -- has a curvature 2-form that is globally well defined.

Mathematically this is why Chern-Simons functionals are called "secondary invariants"

Indeed, this is a bit more than just an analogy: the Chern-Simons 3-form is precisely a doubly higher analog of the electromagnetic field as we pass from the point, via the string, to the membrane.

I have some lecture notes with more along these lines at nLab:twisted smooth cohomology in string theory.

This post imported from StackExchange Physics at 2014-04-05 03:54 (UCT), posted by SE-user Urs Schreiber