1 The statement
The main theorem of the Freed–Hopkins paper Chern–Weil forms and abstract homotopy theory is that Chern–Weil forms are the only natural way to get a differential form from a principal -bundle.
Theorems along these lines are of interest historically. It is an important ingredient in the heat kernel proof of the Atiyah–Singer index theorem. Essentially, the idea of the proof is to use the heat equation to show that there is some formula for the index of a vector bundle in terms of the derivatives of the metric, and then by invariant theory, this must be given by the Chern–Weil forms we know and love. One then computes this for sufficiently many examples to figure out exactly which characteristic class it is, as Hirzebruch originally did for his signature formula.
To state the theorem, we work in the category . For the purposes of this theorem, it actually suffices to work with sheaves of groupoids, i.e. . This only requires -category theory instead of -category theory. However, working with -categories presents no additional difficulty, and is what we shall do.
We now introduce the main characters of the story.
Any defines a representable (discrete) sheaf, which we denote by again.
Any sheaf of sets on is in particular sheaf of (discrete) spaces. Thus, for , we have a discrete sheaf
This is in fact a sheaf of vector spaces, and moreover, there are linear natural transformations . Thus, we get a sheaf of chain complexes , and
In general, for any sheaf , we can think of as the de Rham complex of .
From now on, fix a Lie group.
For , define to be the groupoid of principal -bundles on with connection and isomorphisms, which we think of as a 1-truncated space. This defines .
The Chern–Weil homomorphism induces an isomorphism
To prove the theorem, we consider the universal principal -bundle . The point is that admits a much more explicit description, and then we use to understand itself.
can be described explicitly as follows:
Define to be the groupoid of trivialized -bundles on with connection. Equivalently, this is the groupoid of connections on the trivial -bundle . So .
There is then a natural map , which one can easily check is the universal principal -bundle. Our next claim is that , which is clear once we know what the latter is.
Let , and let be an action by . Explicitly, for each , there is a group action
where is given the pointwise group structure. We can then define the action groupoid
The homotopy quotient of by is then
Note that this geometric realization is taken in the category . To compute this, one takes the geometric realization in the category of presheaves, then sheafify.
Formally, to prove that , we first form the quotient of by in the category of presheaves. Since is discrete, this is given by (the nerve of) the action groupoid of the -action on . This gives the presheaf of trivial principal -bundles. To show that the sheafification is , observe that there is a natural map from this presheaf to , and it is an equivalence on stalks since all principal -bundles on contractible spaces are trivial. So it induces an isomorphism after sheafification.
Our proof then naturally breaks into two steps. First, we compute , and then we need to know how to compute from for any discrete sheaf .
We first do the second part.
Let be a discrete sheaf with a -action . Then is the subcomplex of consisting of the such that
for all ; and
for all .
Let us explain what we mean by . In general, for a manifold and is a vector field on , we can define for all manifolds . Then by left Kan extension, this induces a map for all .
Now if has a -action and , then induces an invariant vector field on , which we also call . We then define by the following composition
where the last map is induced by the inclusion.
This gives us a very explicit method to compute the natural transformation for and . Given a test manifold and , which we think of as a natural transformation , we form the composiite
This defines a differential form . Then we have
Since is a simplicial discrete sheaf, its totalization can be computed by
where is the projection.
To prove the lemma, we have to show that iff the conditions in the lemma are satisfied. This follows from the more general claim below with .
Let be a manifold and a sheaf. Then is zero iff
for any vector field on .
The conditions (1) and (1') match up exactly. Unwrapping the definition of and noting that always, the only difference between (2) and (2') is that in (2), we only test on invariant vector fields on , instead of all vector fields, and we only check the result is zero after restricting to a fiber . The former is not an issue because the condition -linear and the invariant vector fields span as a -module. The latter also doesn't matter because we have assumed that is invariant.
To prove the claim, if were a manifold, this is automatic, since the first condition says vanishes on vectors in the direction while the second says it vanishes on vectors in the direction.
If were an arbitrary sheaf, we know is zero when pulled back along any map where is a manifold, by naturality of the conditions. But since is a colimit of such maps, must already be zero on .□
Now it remains to describe . More generally, for any vector space , we can calculate . We first state the result in the special case where .
For , it sends to . For , it sends to .
The general case is no harder to prove, and the result is described in terms of the Koszul complex.
Let be a vector space. The Koszul complex is a differential graded algebra whose underlying algebra is
For , we write for the corresponding element in , and for the corresponding element in . We set and . The differential is then
For any vector space , we have an isomorphism of differential graded algebras
Explicitly, for , the element is defined by
for and . This is then extended to a map of differential graded algebras.
In other words, the theorem says every natural transformation
is (uniquely) a linear combination of transformations of the form
where is anti-symmetric in the first variables and symmetric in the last .
Using this, we conclude
The Chern–Weil homomorphism gives an isomorphism
and the differential on is zero.
We apply the criteria in Lemma 8. The first condition is the -invariance condition, and translates to the part of the statement. So we have to check that the forms satisfying the second condition are isomorphic to .
To do so, we have to compute the action of on following the recipe in Remark 9. Fix and .
Let be a trivial principal -bundle with connection . The induced principal -bundle on under the action then has connection . So by definition,
To compute the action on , it suffices to compute it on and .
If , then , and
We know , and since vanishes on all vectors in the direction. So we know
Next, . We compute
First observe that in , the only elements killed by are those in . To take care of the part, set
Since , we see that , where is the curvature, and one calculates . By a change of basis, we can identify
and vanishes on the second factor entirely. So we are done.□
More generally, the same proof shows that
If is a smooth manifold, the de Rham complex of is (the total complex of ).
In particular, if has a -action, then is exactly the Cartan model for equivariant de Rham cohomology.