8
$\begingroup$

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends to a characteristic form on the base $M$ from the curvature $F_\omega$. This form is exact when considered as a form on $P$, and the Chern-Simons form $\text{cs}(\omega)$ on $P$ satisfies $p(F_\omega) = d \text{cs}(\omega)$.

I have seen it mentioned that Chern-Simons forms are related to boundary terms when computing characteristic numbers, but I have not been able to find a reference explaining this. In particular, I have heard that the Chern-Simons forms are somehow involved in the generalization of the Chern-Gauss-Bonnet theorem to manifolds with boundary.

Could anyone shed some light on this?

$\endgroup$
  • 3
    $\begingroup$ may be $p(F_{\omega}) = d cs(\omega)$ ? $\endgroup$ – valeri Feb 11 '16 at 19:11
  • $\begingroup$ I was hoping for something more substantial than that. Since the Chern-Simons form does not project down to the base manifold, the best you can do with this is use Stoke's theorem locally. $\endgroup$ – ಠ_ಠ Feb 12 '16 at 9:34
3
$\begingroup$

You might have heard the following. I am actually referring to the second meaning of Chern-Simons classes $\tilde p(\nabla^0,\nabla^1)\in\Omega^\bullet(M)$ satisfying $d\tilde p(\nabla^0,\nabla^1)=p((\nabla^1)^2)-p((\nabla^0)^2)$. They can of course be recovered from the classes you described.

Given a vector bundle $(V,\nabla^V)$ with connection over $M$, you get the Chern-Weil form $p(F)\in\Omega^\bullet(M)$ that you described. A lift to relative forms would be a pair $(p(F),\alpha)$ with $\alpha\in\Omega^\bullet(U)$ satisfying $d\alpha=p(F)|_U$ on a collar neighbourhood $U$ of $\partial M$. With $\alpha$, you could easily replace $p(F)$ by a cohomologous form with compact support.

To construct $\alpha$, you need more information than just $(V,\nabla^V)$ - you need an explicit reason for $p(V)$ to vanish on $U$. In this context, a plausible reason is the existence of another connection $\nabla'$ on $V|_U$ such that $F'=(\nabla')^2=0$. This could come from a trivialisation of $V|_U$, but there might be other possibilities. Now you get the relative term $\alpha=\tilde p(\nabla',\nabla^V|_U)$, and you can compute a characteristic number $$\langle(p(F),\tilde p(\nabla',\nabla^V|_U)),[M,\partial M]\rangle =\int_Mp(F)-\int_{\partial M}\tilde p(\nabla',\nabla^V|_U)\;.$$

For Gauss-Bonnet-Chern, you need a "reason" for the Euler class of $TM$ to be trivial in a neighbourhood of $\partial M$. This is given by the fact that $TM|_{\partial M}\cong T\partial M\oplus\underline{\mathbb R}$. But the Levi-Civita connection does not respect this decomposition unless $M$ has totally geodesic boundary. If it does not, then you use a Chern-Simons form for the Euler form as a correction term. In fact, this is also related to Mathai-Quillen currents, which can be used to reduce Gauss-Bonnet-Chern to Poincaré-Hopf.

$\endgroup$
  • $\begingroup$ Thanks! Do you happen to know of a reference article or text discussing these ideas? $\endgroup$ – ಠ_ಠ Feb 18 '16 at 21:47
  • 1
    $\begingroup$ @ಠ_ಠ Not really, but this question says something on relative de Rham cohomology that might be helpful $\endgroup$ – Sebastian Goette Feb 19 '16 at 11:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.