Chern–Weil Forms and Abstract Homotopy Theory — The statement

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 GG-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 Shv(Man,S)\operatorname{Shv}(\mathsf{Man}, \mathcal{S}). For the purposes of this theorem, it actually suffices to work with sheaves of groupoids, i.e. Shv(Man,τ1S)\operatorname{Shv}(\mathsf{Man}, \tau _{\leq 1} \mathcal{S}). This only requires 22-category theory instead of \infty -category theory. However, working with \infty -categories presents no additional difficulty, and is what we shall do.

We now introduce the main characters of the story.

Example 1

Any MManM \in \mathsf{Man} defines a representable (discrete) sheaf, which we denote by MM again.

Example 2

Any sheaf of sets on Man\mathsf{Man} is in particular sheaf of (discrete) spaces. Thus, for p0p \geq 0, we have a discrete sheaf

ΩpShv(Man,S). \Omega ^ p \in \operatorname{Shv}(\mathsf{Man}, \mathcal{S}).

This is in fact a sheaf of vector spaces, and moreover, there are linear natural transformations d:ΩpΩp+1\mathrm{d}: \Omega ^ p \to \Omega ^{p + 1}. Thus, we get a sheaf of chain complexes Ω\Omega ^{\bullet }, and

[M,Ω]=Ω(M). [M, \Omega ^\bullet ] = \Omega ^\bullet (M).

In general, for any sheaf F\mathcal{F}, we can think of Ω(F)[F,Ω]\Omega ^\bullet (\mathcal{F}) \equiv [\mathcal{F}, \Omega ^{\bullet }] as the de Rham complex of F\mathcal{F}.

From now on, fix GG a Lie group.

Example 3

For MManM \in \mathsf{Man}, define BG(M)B_\nabla G(M) to be the groupoid of principal GG-bundles on MM with connection and isomorphisms, which we think of as a 1-truncated space. This defines BGShv(Man,S)B_\nabla G \in \operatorname{Shv}(\mathsf{Man}, \mathcal{S}).

The main theorem is
Theorem 4

The Chern–Weil homomorphism induces an isomorphism

(Symg)GΩ(BG). (\operatorname{Sym}^\bullet \mathfrak {g}^*)^ G \overset {\sim }{\to } \Omega ^\bullet (B_\nabla G).

To prove the theorem, we consider the universal principal GG-bundle EGBGE_\nabla G \to B_\nabla G. The point is that EGE_\nabla G admits a much more explicit description, and then we use BG=EG/ ⁣/GB_\nabla G = E_\nabla G /\! /G to understand BGB_\nabla G itself.

EGE_\nabla G can be described explicitly as follows:

Example 5

Define EG(M)E_\nabla G (M) to be the groupoid of trivialized GG-bundles on MM with connection. Equivalently, this is the groupoid of connections on the trivial GG-bundle M×GGM \times G \to G. So EGΩ1gE_\nabla G \cong \Omega ^1 \otimes \mathfrak {g}.

There is then a natural map EG(M)BG(M)E_\nabla G(M) \to B_\nabla G(M), which one can easily check is the universal principal GG-bundle. Our next claim is that BG(M)=EG(M)/ ⁣/GB_\nabla G(M) = E_\nabla G(M) /\! /G, which is clear once we know what the latter is.

Definition 6

Let FShv(Man,S)\mathcal{F} \in \operatorname{Shv}(\mathsf{Man}, \mathcal{S}), and let α:G×FF\alpha : G \times \mathcal{F} \to \mathcal{F} be an action by GG. Explicitly, for each MManM \in \mathsf{Man}, there is a group action

HomMan(M,G)×F(M)F(M) \operatorname{Hom}_{\mathsf{Man}}(M, G) \times \mathcal{F}(M) \to \mathcal{F}(M)

where HomMan(M,G)\operatorname{Hom}_{\mathsf{Man}}(M, G) is given the pointwise group structure. We can then define the action groupoid

(F/ ⁣/G)=G××FShv(Man,S)Δop. (\mathcal{F}/\! /G)_\bullet = G^{\times \bullet } \times \mathcal{F} \in \operatorname{Shv}(\mathsf{Man}, \mathcal{S})^{\Delta ^\mathrm{op}}.

The homotopy quotient of F\mathcal{F} by GG is then

F/ ⁣/G=(F/ ⁣/G). \mathcal{F}/\! /G = |(\mathcal{F}/\! /G)_\bullet |.

Note that this geometric realization is taken in the category Shv(Man,S)\operatorname{Shv}(\mathsf{Man}, \mathcal{S}). To compute this, one takes the geometric realization in the category of presheaves, then sheafify.

We then see that BG=EG/ ⁣/GB_\nabla G = E_\nabla G /\! /G. Explicitly, the action of the gauge group can be described as follows — given g:MGg: M \to G and αEG(M)=Ω1(M;g)\alpha \in E_\nabla G(M) = \Omega ^1(M; \mathfrak {g}), we have

gα=gθ+Adg1α. g \cdot \alpha = g^* \theta + \operatorname{Ad}_{g^{-1}} \alpha .

Remark 7

Formally, to prove that BG=EG/ ⁣/GB_\nabla G = E_\nabla G /\! /G, we first form the quotient of EGE_\nabla G by GG in the category of presheaves. Since EGE_\nabla G is discrete, this is given by (the nerve of) the action groupoid of the GG-action on EGE_\nabla G. This gives the presheaf of trivial principal GG-bundles. To show that the sheafification is BGB_\nabla G, observe that there is a natural map from this presheaf to BGB_\nabla G, and it is an equivalence on stalks since all principal GG-bundles on contractible spaces are trivial. So it induces an isomorphism after sheafification.

Our proof then naturally breaks into two steps. First, we compute Ω(EG)\Omega ^\bullet (E_\nabla G), and then we need to know how to compute Ω(F/ ⁣/G)\Omega ^\bullet (\mathcal{F}/\! /G) from Ω(F)\Omega ^\bullet (\mathcal{F}) for any discrete sheaf F\mathcal{F}.

We first do the second part.

Lemma 8

Let FShv(Man,S)\mathcal{F} \in \operatorname{Shv}(\mathsf{Man}, \mathcal{S}) be a discrete sheaf with a GG-action α:G×FF\alpha : G \times \mathcal{F} \to \mathcal{F}. Then Ω(F/ ⁣/G)\Omega ^\bullet (\mathcal{F} /\! /G) is the subcomplex of Ω(F)\Omega ^\bullet (\mathcal{F}) consisting of the ω\omega such that

  1. αω{g}×F=ω\alpha ^* \omega |_{\{ g\} \times \mathcal{F}} = \omega for all gGg \in G; and

  2. ιξω=0\iota _\xi \omega = 0 for all ξg\xi \in \mathfrak {g}.

The first condition says ω\omega should be GG-invariant, and the second condition says ω\omega is suitably “horizontal”.

Remark 9

Let us explain what we mean by ιξω\iota _\xi \omega . In general, for MM a manifold and XX is a vector field on MM, we can define ιX:Ωp(M×N)Ωp1(M×N)\iota _ X: \Omega ^ p(M \times N) \to \Omega ^{p - 1}(M \times N) for all manifolds NN. Then by left Kan extension, this induces a map ιX:Ωp(M×F)Ωp1(M×F)\iota _ X: \Omega ^ p(M \times \mathcal{F}) \to \Omega ^{p - 1}(M \times \mathcal{F}) for all FShv(Man,S)\mathcal{F} \in \operatorname{Shv}(\mathsf{Man}, \mathcal{S}).

Now if F\mathcal{F} has a GG-action and ξg\xi \in \mathfrak {g}, then ξ\xi induces an invariant vector field on GG, which we also call ξ\xi . We then define ιξ:Ωp(F)Ωp1(F)\iota _\xi : \Omega ^ p(\mathcal{F}) \to \Omega ^{p - 1}(\mathcal{F}) by the following composition

\begin{useimager} 
    \[
      \begin{tikzcd}
        \Omega^p(\mathcal{F}) \ar[r, "\alpha^*"] & \Omega^p(G \times \mathcal{F}) \ar[r, "\iota_\xi"] & \Omega^{p - 1}(G \times \mathcal{F}) \ar[r] & \Omega^{p - 1}(\{e\} \times \mathcal{F}) = \Omega^{p - 1}(\mathcal{F}),
      \end{tikzcd}
    \]
  \end{useimager}

where the last map is induced by the inclusion.

This gives us a very explicit method to compute the natural transformation ιξω\iota _\xi \omega for ωΩp(F)\omega \in \Omega ^ p(\mathcal{F}) and ξg\xi \in \mathfrak {g}. Given a test manifold MM and ϕF(M)\phi \in \mathcal{F}(M), which we think of as a natural transformation ϕ:MF\phi : M \to \mathcal{F}, we form the composiite

\begin{useimager} 
    \[
      \begin{tikzcd}
        G \times M \ar[r, "1 \times \phi"] & G \times \mathcal{F} \ar[r, "\alpha"] & \mathcal{F} \ar[r, "\omega"] & \Omega^p
      \end{tikzcd}
    \]
  \end{useimager}

This defines a differential form ηΩp(G×M)\eta \in \Omega ^ p(G \times M). Then we have

(ιξω)M(ϕ)=ιξη{e}×M. (\iota _\xi \omega )_ M(\phi ) = \iota _{\xi } \eta |_{\{ e\} \times M}.

Proof

We have

Ωp(F/ ⁣/G)=Ωp((F/ ⁣/G))=Tot(Ωp((F/ ⁣/G))). \Omega ^ p (\mathcal{F} /\! /G) = \Omega ^ p(|(\mathcal{F}/\! /G)_\bullet |) = \operatorname{Tot}(\Omega ^ p((\mathcal{F}/\! /G)_\bullet )).

Since (F/ ⁣/G)(\mathcal{F}/\! /\mathcal{G})_\bullet is a simplicial discrete sheaf, its totalization can be computed by

\begin{useimager} 
    \[
      \Omega^p(\mathcal{F}\modmod G) = \ker \left(\begin{tikzcd}\Omega^p(\mathcal{F}) \ar[r, "\pr^* - \alpha^*"] & \Omega^p(G \times \mathcal{F})\end{tikzcd}\right),
    \]
  \end{useimager}

where pr:G×FF\mathrm{pr}: G \times \mathcal{F} \to \mathcal{F} is the projection.

To prove the lemma, we have to show that prω=αω\mathrm{pr}^* \omega = \alpha ^* \omega iff the conditions in the lemma are satisfied. This follows from the more general claim below with η=αωprω\eta = \alpha ^* \omega - \mathrm{pr}^* \omega .

Claim

Let MM be a manifold and F\mathcal{F} a sheaf. Then ηΩp(M×F)\eta \in \Omega ^ p(M \times \mathcal{F}) is zero iff

  1. η{x}×F=0\eta |_{\{ x\} \times \mathcal{F}} = 0 for all xMx \in M

  2. ιXη=0\iota _ X \eta = 0 for any vector field XX on MM.

The conditions (1) and (1') match up exactly. Unwrapping the definition of ιξ\iota _\xi and noting that ιXprω=0\iota _ X \mathrm{pr}^* \omega = 0 always, the only difference between (2) and (2') is that in (2), we only test on invariant vector fields on GG, instead of all vector fields, and we only check the result is zero after restricting to a fiber {e}×F\{ e\} \times \mathcal{F}. The former is not an issue because the condition C(G)C^\infty (G)-linear and the invariant vector fields span as a C(G)C^\infty (G)-module. The latter also doesn't matter because we have assumed that αω\alpha ^* \omega is invariant.

To prove the claim, if F\mathcal{F} were a manifold, this is automatic, since the first condition says η\eta vanishes on vectors in the NN direction while the second says it vanishes on vectors in the MM direction.

If F\mathcal{F} were an arbitrary sheaf, we know η\eta is zero when pulled back along any map (1×ϕ):M×NM×F(1 \times \phi ): M \times N \to M \times \mathcal{F} where NN is a manifold, by naturality of the conditions. But since M×FM \times \mathcal{F} is a colimit of such maps, η\eta must already be zero on M×FM \times \mathcal{F}.

Now it remains to describe Ω(EG)=Ω(Ω1g)\Omega ^\bullet (E_\nabla G) = \Omega ^\bullet (\Omega ^1 \otimes \mathfrak {g}). More generally, for any vector space VV, we can calculate Ω(Ω1V)\Omega ^\bullet (\Omega ^1 \otimes V). We first state the result in the special case where V=RV = \mathbb {R}.

Theorem 10
Ωp(Ω1)R for all p0. \Omega ^ p(\Omega ^1) \cong \mathbb {R}\text{ for all }p \geq 0.

For p=2qp = 2q, it sends ω\omega to (dω)q(\mathrm{d}\omega )^ q. For p=2q+1p = 2q + 1, it sends ω\omega to ω(dω)q\omega \wedge (\mathrm{d}\omega )^ q.

The general case is no harder to prove, and the result is described in terms of the Koszul complex.

Definition 11

Let VV be a vector space. The Koszul complex KosV\operatorname{Kos}^\bullet V is a differential graded algebra whose underlying algebra is

KosV=VSymV. \operatorname{Kos}^\bullet V = {\textstyle \bigwedge }^\bullet V \otimes \operatorname{Sym}^\bullet V.

For vVv \in V, we write vv for the corresponding element in 1V{\textstyle \bigwedge }^1 V, and v~\tilde{v} for the corresponding element in Sym1V\operatorname{Sym}^1 V. We set v=1|v| = 1 and v~=2|\tilde{v}| = 2. The differential is then

d(v)=v~,d(v~)=0. \mathrm{d}(v) = \tilde{v},\quad d(\tilde{v}) = 0.

Theorem 12

For any vector space VV, we have an isomorphism of differential graded algebras

η:KosVΩ(Ω1V). \eta : \operatorname{Kos}^\bullet V^* \overset {\sim }{\to } \Omega ^\bullet (\Omega ^1 \otimes V).

In particular,

Ω(EG)=Kosg. \Omega ^\bullet (E_\nabla G) = \operatorname{Kos}^\bullet \mathfrak {g}^*.

Explicitly, for V=1V\ell \in V^* = {\textstyle \bigwedge }^1 V^*, the element η()Ω1(Ω1V)\eta (\ell ) \in \Omega ^1(\Omega ^1 \otimes V) is defined by

η()(αv)=v,α \eta (\ell )(\alpha \otimes v) = \langle v, \ell \rangle \, \alpha

for αΩ1\alpha \in \Omega ^1 and vVv \in V. This is then extended to a map of differential graded algebras.

In other words, the theorem says every natural transformation

ωM:Ω1(M;V)Ωp(M) \omega _ M: \Omega ^1(M; V) \to \Omega ^ p(M)

is (uniquely) a linear combination of transformations of the form

αiviI,JMI,J(vi1,,vik,vj1,,vj)αi1αikdαj1dαj \sum \alpha _ i \otimes v_ i \mapsto \sum _{I, J} M_{I, J}(v_{i_1}, \ldots , v_{i_ k}, v_{j_1}, \ldots , v_{j_\ell })\, \alpha _{i_1} \wedge \cdots \wedge \alpha _{i_ k} \wedge \mathrm{d}\alpha _{j_1} \wedge \cdots \wedge \mathrm{d}\alpha _{j_\ell }

where MI,JM_{I, J} is anti-symmetric in the first kk variables and symmetric in the last \ell .

Using this, we conclude

Theorem 13

The Chern–Weil homomorphism gives an isomorphism

(Symg)GΩ(BG), (\operatorname{Sym}^\bullet \mathfrak {g}^*)^ G \overset {\sim }{\to } \Omega ^\bullet (B_\nabla G),

and the differential on Ω(BG)\Omega ^\bullet (B_\nabla G) is zero.

Note that this Symg\operatorname{Sym}^\bullet \mathfrak {g}^* is different from that appearing in the Koszul complex.

Proof

We apply the criteria in Lemma 8. The first condition is the GG-invariance condition, and translates to the ()G(\cdots )^ G part of the statement. So we have to check that the forms satisfying the second condition are isomorphic to Symg\operatorname{Sym}^\bullet \mathfrak {g}^*.

To do so, we have to compute the action of ιξ\iota _\xi on EGE_\nabla G following the recipe in Remark 9. Fix ωΩp(EG)\omega \in \Omega ^ p(E_\nabla G) and ξg\xi \in \mathfrak {g}.

Let ϕ:MEG\phi : M \to E_\nabla G be a trivial principal GG-bundle with connection AΩ1(M;g)A \in \Omega ^1(M; \mathfrak {g}). The induced principal GG-bundle on G×MG \times M under the action then has connection θ+Adg1A\theta + \operatorname{Ad}_{g^{-1}} A. So by definition,

(ιξω)M(A)=ιξ(ω(θ+Adg1A)){e}×M. (\iota _\xi \omega )_ M(A) = \left.\iota _\xi \left(\omega (\theta + \operatorname{Ad}_{g^{-1}} A)\right)\right|_{\{ e\} \times M}.

To compute the action on Kosg\operatorname{Kos}^\bullet \mathfrak {g}^*, it suffices to compute it on 1g{\textstyle \bigwedge }^1 \mathfrak {g}^* and Sym1g\operatorname{Sym}^1 \mathfrak {g}^*.

  1. If λg=1g\lambda \in \mathfrak {g}^* = {\textstyle \bigwedge }^1 \mathfrak {g}^*, then λ(A)=A,λ\lambda (A) = \langle A, \lambda \rangle , and

    ιξθ+Adg1A,λ=ιξθ+ιξAdg1A,λ. \iota _\xi \langle \theta + \operatorname{Ad}_{g^{-1}} A, \lambda \rangle = \langle \iota _\xi \theta + \iota _\xi \operatorname{Ad}_{g^{-1}} A, \lambda \rangle .

    We know ιξθ=ξ\iota _\xi \theta = \xi , and ιξAdg1A=0\iota _\xi \operatorname{Ad}_{g^{-1}} A = 0 since Adg1A\operatorname{Ad}_{g^{-1}} A vanishes on all vectors in the GG direction. So we know

    ιξλ=ξ,λ0g. \iota _\xi \lambda = \langle \xi , \lambda \rangle \in {\textstyle \bigwedge }^0 \mathfrak {g}^*.
  2. Next, λ~(A)=dA,λ\tilde{\lambda }(A) = \langle \mathrm{d}A, \lambda \rangle . We compute

    ιξd(θ+Adg1A),λ{e}×M=ιξ12[θ,θ]+Addg1A+Adg1dA,λ{e}×M=AdξA,λ=A,Adξλ. \iota _\xi \langle \mathrm{d}(\theta + \operatorname{Ad}_{g^{-1}}A), \lambda \rangle |_{\{ e\} \times M} = \left.\iota _\xi \left\langle -\frac{1}{2}[\theta , \theta ] + \operatorname{Ad}_{\mathrm{d}g^{-1}} \wedge A + \operatorname{Ad}_{g^{-1}} \mathrm{d}A, \lambda \right\rangle \right|_{\{ e\} \times M} = \langle -\operatorname{Ad}_\xi A, \lambda \rangle = \langle A, -\operatorname{Ad}_\xi ^* \lambda \rangle .

    So

    ιξλ~=Adξλ1g. \iota _\xi \tilde{\lambda } = -\operatorname{Ad}^*_\xi \lambda \in {\textstyle \bigwedge }^1 \mathfrak {g}^*.

First observe that in g{\textstyle \bigwedge }^\bullet \mathfrak {g}^*, the only elements killed by ιξ\iota _\xi are those in 0gR{\textstyle \bigwedge }^0 \mathfrak {g}^* \cong \mathbb {R}. To take care of the Sym\operatorname{Sym} part, set

Ωλ=λ~+12[λ,λ]. \Omega _\lambda = \tilde{\lambda } + \frac{1}{2}[\lambda , \lambda ].

Since λ~(A)=dA,λ\tilde{\lambda }(A) = \langle \mathrm{d}A, \lambda \rangle , we see that Ωλ(A)=ΩA,λ\Omega _\lambda (A) = \langle \Omega _ A, \lambda \rangle , where ΩA\Omega _ A is the curvature, and one calculates ιξΩλ=0\iota _\xi \Omega _\lambda = 0. By a change of basis, we can identify

KosggSymΩλ:λg, \operatorname{Kos}^\bullet \mathfrak {g}^* \cong {\textstyle \bigwedge }^\bullet \mathfrak {g}^* \otimes \operatorname{Sym}^\bullet \langle \Omega _\lambda : \lambda \in \mathfrak {g}^*\rangle ,

and ιξ\iota _\xi vanishes on the second factor entirely. So we are done.

More generally, the same proof shows that

Theorem 14

If MM is a smooth manifold, the de Rham complex of M×(Ω1V)M \times (\Omega ^1 \otimes V) is Ω(M;KosV)\Omega (M; \operatorname{Kos}V^*)^\bullet (the total complex of Ω(M;KosV)\Omega ^\bullet (M; \operatorname{Kos}^\bullet V^*)).

In particular, if MM has a GG-action, then M×EG/ ⁣/GM \times E_\nabla G /\! /G is exactly the Cartan model for equivariant de Rham cohomology.

This would follow immediately if we had a result that says Ω(M×F)Ω(M)^Ω(F)\Omega ^\bullet (M \times \mathcal{F}) \cong \Omega ^\bullet (M) \hat{\otimes } \Omega ^\bullet (\mathcal{F}), and since Ω(EG)\Omega ^\bullet (E_\nabla G) is finite dimensional, the completed tensor product is the usual tensor product.