Chern-Simons term

Question

In the literature I can only find Chern-Simons terms ( i.e. for a 3-dimensional manifold $A \wedge dA + A \wedge A \wedge A$) for odd-dimensional manifolds. Why can't I write such forms for even-dimensional manifolds?

Answer 1

The starting points for the construction of the Chern-Simons terms are objects called Chern-Pontryagin densities. On a 2n dimensional manifold, these are of the form $$ {\mathcal{P}}^{2n} = \alpha\epsilon^{\mu_1\mu_2...\mu_{2n}}Tr F_{\mu_1\mu_2}...F_{\mu_{2n-1}\mu_{2n}} $$ where F is the curvature 2-form of some G-connection (G is the gauge group). These are gauge-invariant, closed, and their integral over the manifold M (compact, no boundary) is an integer which is a topological invariant. These sorts of invariants are examples of characteristic classes.

Now ${\mathcal{P}}^{2n}$ can be locally expressed as a differential $$ {\mathcal{P}}^{2n} = d({\mathcal{C}}^{2n-1})$$ of a 2n-1 form. $ \ {\mathcal{C}}^{2n-1}$ is the Chern Simons form. (It can be written in the familiar form in terms of the connection form A). It has the remarkable property that if I perform a G-gauge transformation, the action obtained by integrating $ \ {\mathcal{C}}^{2n-1}$ is gauge-invariant. At no point is a metric involved in this construction, so it's a topological theory.

Anyway, getting back to the question, the Chern Pontryagin density ${\mathcal{P}}^{2n}$ which we started with is only defined on a 2n (i.e even) dimensional manifold, so consquently the Chern-Simon's term is only defined on an odd dimensional one.

Edit: Example in which Chern Simon's term in 3d is produced:

The Chern-Pontryagin class is the integral of the C.P. density on a 4d manifold $$ {\mathcal{P}} \propto \int d^4xTr({^*F}^{\mu\nu}F_{\mu\nu})$$ $$ \propto \int d^4x \ \epsilon^{\mu\nu\rho\sigma}Tr( F_{\mu\nu}F_{\rho\sigma})$$ We can write the C.P. density as the divergence of a 4-current (the Chern Simons current) $$Tr({^*F}^{\mu\nu}F_{\mu\nu}) = \partial_{\mu}C^{\mu}$$ where $$C^{\mu} = \epsilon^{\mu\nu\rho\sigma} \ tr(A_{\nu}\partial_{\rho}A_{\sigma}+\frac{2}{3}f^{abc}A^a_{\nu}A^b_{\rho}A^b_{\sigma}) $$ If we now pick a local coordinate system (on the 4 manifold) such that $\frac{\partial}{\partial x^0}$ (say) is in the direction of the vector $C^{\mu}$, then we just look at the other components in the epsilon symbol (they're guaranteed to be 1,2,3), then we just freeze the $x^0$ dependence of the $A_{\nu}(x^{\mu})$. We've now got ourselves a 3 form on the three-manifold $x^0$ = constant.

Answer 2

When you look at the definition of the Chern–Simons form

$d\omega_{2k-1}={\rm Tr} \left( F^{k} \right)$

one needs to understand that $F$ is a $2$-form. So the right hand side $F^k$ will always be a $2k$-form (with $k \in Z^+$ ). So the exterior derivative $d$ of the Chern–Simons form $\omega$ needs then to be a $2k-1$-form (since the exterior derivative of a $n$-form is a $n+1$-form) and since $k$ is a positive integer $\omega$ needs to be an odd form by definition.