Suppose $E$ is a complex vector bundle of rank $n$ on a compact oriented manifold (both assumed smooth). Let $h$ be a Hermitian metric on $E$, and let $A$ be a Hermitian connection on $E$ and $F_A$ its curvature form, an element of $\Omega^2(M,\text{End }E)$. Chern-Weil theory produces a closed even differential form $c(A) = \det(1 + \frac i{2\pi}F_A) = c_0(A)+c_1(A) + \cdots + c_n(A)$.

These classes have the property that for all compact oriented submanifolds $\Sigma\subset M$ of dimension $2k$, the expression $\int_\Sigma c(A)$ is an integer. The usual way to prove this seems to be to first show that the de Rham cohomology class of $c(A)$ is independent of the connection $A$ and functorial under pullbacks. We then check that it has the desired integrality property for the tautological bundles on Grassmannians by direct computation (which says this class lies in singular cohomology with $\mathbb Z$ coefficients) and then use the fact that every bundle on $M$ is a pullback of the tautological bundle from a sufficiently large Grassmannian. Alternatively, we show that the above definition satisfies the axiomatic characterization of Chern classes and then we identify it with the image in $H^\bullet(-,\mathbb C)$ of Chern classes constructed in $H^\bullet(-,\mathbb Z)$ by using a computation of the cohomology ring of infinite Grassmannians.

Is there a direct proof that $\int_\Sigma$ c(A) is an integer for all compact oriented submanifolds $\Sigma\subset M$?

By a direct proof, I mean something which only uses only local reasoning about curvature and connections and doesn't have to pass through much algebraic topology. For instance for the case $n=1$ (i.e., a line bundle) and taking $\Sigma = M$ to be a compact oriented surface, we note by elementary local computations (using Stokes' theorem) that for any disc $D$ inside $\Sigma$ with oriented boundary circle $\gamma$, the number $e^{\pm\int_D F_A}\in U(1)$ measures the holonomy of the parallel transport along the loop $\gamma$. From this, it easily follows that we can extend the above statement to the case when $D$ is a more general region with smooth oriented boundary $\gamma$. As $\Sigma$ has no boundary, applying this to $D = \Sigma$ gives $\int_\Sigma F_A\in \frac{2\pi}i\mathbb Z$ as desired. (This proof is similar to the moral proof of Stokes' theorem given by checking it at the infinitesimal level -- essentially the definition of the exterior derivative -- and then triangulating the domain of integration and noting that boundary terms with opposite orientations cancel.) The $c_1$ case seems to be simple because of $U(1)$ being abelian.

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    $\begingroup$ Is the splitting principle too indirect for you? It should reduce the general case to the abelian case. $\endgroup$ – Mike Miller Eismeier Jan 24 '20 at 20:28
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    $\begingroup$ This seems to be related to finding a "direct proof" of the following assertion: given $n$ line bundles $L_i$ ($1\le i\le n$) with connections $A_i$, how do we show that $\int_\Sigma\prod_i c(A_i)\in\mathbb Z$ for each compact oriented $\Sigma\subset M$? I'd like to avoid using singular (co)homology with $\mathbb Z$ coefficients in the proof. $\endgroup$ – Mohan Swaminathan Jan 24 '20 at 20:49
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    $\begingroup$ This seems surprisingly non-trivial to me, essentially requiring the Kunneth formula mod torsion for integral homology... $\endgroup$ – Mike Miller Eismeier Jan 26 '20 at 4:16

Yes, the Chern–Weil homomorphism lifts to differential cohomology, which guarantees that periods are integral. See the original paper by Cheeger and Simons, or the paper by Hopkins and Singer. The (modernized) construction of such a refinement relies on the computation of the de Rham complex of the stack B_∇(G) of principal G-bundles with connection and their isomorphisms, which is isomorphic (and not just quasi-isomorphic) to the algebra of G-invariant polynomials on the Lie algebra of G. This is proved by Freed and Hopkins, and their argument is a local argument that uses computations with differential forms. There is a bit of linear algebra and invariant theory at the very end.

This argument is essentially an abstract higher-dimensional version of the n=1 argument above with the Stokes theorem.

  • $\begingroup$ The lift to differential cohomology is already explained in the original paper by Cheeger and Simons. But they use a result of Narasimhan-Ramanan, that every vector bundle with a metric connection can be pulled back (as a vector bundle with metric connection) from a sufficiently large Grassmannian. So their proof is not much better than those the OP mentions. Is the paper by Hopkins and Singer in any way more of a "local" nature? $\endgroup$ – Sebastian Goette Jan 25 '20 at 21:17
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    $\begingroup$ @SebastianGoette: There is a modernized version of this argument, due to Freed and Hopkins (Chern–Weil forms and abstract homotopy theory), which computes instead the de Rham complex of the stack B_∇(G) of principal G-bundles with connections and their isomorphisms, and show that it is isomorphic (and not just quasi-isomorphic) to the G-invariant polynomials on the Lie algebra g. Their proof consists mostly of local constructions with differential forms, like the OP wanted. There is a small bit of linear algebra and invariant theory at the very end. $\endgroup$ – Dmitri Pavlov Jan 25 '20 at 21:57
  • $\begingroup$ I'm not at all familiar with differential cohomology and I think I will need some time to take a look at this paper to see if it is what I want. Thanks for the reference! $\endgroup$ – Mohan Swaminathan Jan 26 '20 at 16:26

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