The invariant 3-form on a compact Lie group

Question

Let $G$ be a compact Lie group. We have the well-known Maurer-Cartan left-invariant and right-invariant $1$-forms $\theta$ and $\bar\theta$ in $\Omega^1(G, \mathfrak{g})$, probably discussed in every Lie theory lectures.

However the canonical bi-invariant closed $3$-form $\chi = \frac{1}{12} (\theta, [\theta, \theta]) = \frac{1}{12} (\bar\theta, [\bar\theta, \bar\theta])$ in $\Omega^3(G)$ may be a little bit less-known. And when I heard of it I had some questions in mind...

1) Are there canonical invariant $5$-forms and higher invariant forms on a $G$?

2) How are they related to Lie algebra cohomology and equivariant cohomology?

3) The construction looks a little bit like $tr(A \wedge dA)$, if we regard $\theta$ as a connection $A$ on the frame bundle of $G$, and use the Maurer-Cartan equation $d\theta = -\frac{1}{2}[\theta, \theta]$.

Now there is another famous $3$-form: the Chern-Simons form $tr(A \wedge dA + \frac{2}{3} A \wedge A \wedge A)$, and I wonder if these two are somehow related. Does the $tr(A \wedge A \wedge A)$ part vanish here, due to Jacobi identity?

(and, note there are Chern-Simons 5-forms etc.)

Thank you very much.

Answer 1

The relation between the three dimensional Chern-Simon term and the invariant 3-form is a special case of the BRST bicomplex which acts on the Lie algebra valued equivariant forms of a principal bundle. Locally, these forms depend polynomially on the gauge connection of the principal bundle and on the Maurer-Cartan one form of the Lie group of the gauge transformations.

One starts from the Chern-Simons term (in any dimension), and through a series of descent equations (involving the BRS operator and the exterior derivative) reaches an invariant odd degree form of the group manifold. At every stage, the descent equations (Sometimes called the Stora-Zumino descent equations) increase the polynomial degree of the Maurer-Cartan by 1 and decreases the degree of the gauge connection by one.

Here is a reference with the explicit construction starting from the five dimensional Chern-Simons term.

The intermediate terms in the descent are actually cocycles of an Abelian extentsion of the gauge group Lie algebra known as the Fadeev-Mickelsson extension.

Please, see also the following Quantum field theory books (first), (second) treating this subject.

Answer 2

The 3-form in question is the extension to de Rham cohomology of the group of the canonical semisimple Lie algebra 3-cocycle $\langle - [-,-] \rangle$.

A Lie algebra cocycle is the same as a closed element in the Chevalley-Eilenberg algebra. If any Lie algebra cocycle is in transgression with an invariant polynomial on the Lie algebra, then this is witnessed by the existence of a corresponding Chern-Simons element in the Weil algebra. The detailed construction is recalled here.

From this simple cohomological construction result most of the phenomena in higher Chern-Simons theory relating these algebraic entities. I have once written up some notes on this with Jim Stasheff and Hisham Sati (from section 6.3 on it discusses what I just mentioned).

Answer 3

The answers to your questions form part of a somewhat long and beautiful story which is explained, e.g., in Greub, Halperin and Vanstone's Connections, Curvature, and Cohomology. Vol. 2: Lie Groups, Principal Bundles, and Characteristic Classes. There is no Google Books preview available, but you can at least read a review here.