nLab Chern-Weil homomorphism



\infty-Chern-Weil theory

Differential cohomology



For GG a Lie group with Lie algebra 𝔤\mathfrak{g}, a GG-principal bundle PXP \to X on a smooth manifold XX induces a collection of classes in the de Rham cohomology of XX: the classes of the curvature characteristic forms

F AF AΩ closed 2n(X) \langle F_A \wedge \cdots F_A \rangle \in \Omega^{2n}_{closed}(X)

of the curvature 2-form F AΩ 2(P,𝔤)F_A \in \Omega^2(P, \mathfrak{g}) of any connection on PP, and for each invariant polynomial \langle -\rangle of arity nn on 𝔤\mathfrak{g}.

This is a map from the first nonabelian cohomology of XX with coefficients in GG to the de Rham cohomology of XX

char:H 1(X,G) n iH dR 2n i(X) char : H^1(X,G) \to \prod_{n_i} H_{dR}^{2 n_i}(X)

where ii runs over a set of generators of the invariant polynomials. This is the analogy in nonabelian differential cohomology of the generalized Chern character map in generalized Eilenberg-Steenrod-differential cohomology.

Plain Chern-Weil homomorphism

This subsection is to give an outline of construction of Weil homomorphism as in Kobayashi-Nomizu 63

Let GG be a Lie group and 𝔤\mathfrak{g} be its Lie algebra. Given an element gGg\in G, the adjoint map Ad(g):GGAd(g):G\rightarrow G is defined as Ad(g)(h)=ghg 1Ad(g)(h)=ghg^{-1}. For gGg\in G, let ad(g):𝔤𝔤ad(g):\mathfrak{g}\rightarrow \mathfrak{g} be the differential of Ad(g):GGAd(g):G\rightarrow G at eGe\in G.

Let I k(G)I^k(G) denote the set of symmetric, multilinear maps

f:𝔤××𝔤 k~times f:\underbrace{\mathfrak{g}\times\cdots\times\mathfrak{g}}_{k ~\text{times}}\rightarrow \mathbb{R}

that are GG invariant in the sense that f(ad(g)(t 1),,ad(g)(t k))=f(t 1,,t k)f(ad(g)(t_1),\cdots,ad(g)(t_k))=f(t_1,\cdots,t_k) for all gGg\in G and t i𝔤t_i\in \mathfrak{g}. These I k(G)I^k(G) are vector spaces over \mathbb{R}. Let I(G)I(G) denote the \mathbb{R} algebra k=0 I k(G)\oplus_{k=0}^{\infty}I^k(G).

Let MM be a manifold and H *(M,)H^*(M,\mathbb{R}) be the deRham cohomology ring of MM.

Given a principal GG bundle over MM, say π:PM\pi:P\rightarrow M, Weil homomorphism gives a homomorphism I(G)H *(M,)I(G)\rightarrow H^*(M,\mathbb{R}). Though it does not depend on connection on P(M,G)P(M,G), the construction of this map is done after fixing a connection on P(M,G)P(M,G). Outline of the construction is as follows.

  1. Fix a connection Γ\Gamma on P(M,G)P(M,G). Let Ω\Omega denote the curvature of Γ\Gamma.

  2. Given an element fI k(G)f\in I^k(G), define a 2k2k-form f(Ω)f(\Omega)on PP. \item Prove that the 2k2k form f(Ω)f(\Omega) on PP projects uniquely to a 2k2k form on MM and call it f˜(Ω)\tilde{f}(\Omega) i.e., π *(f˜(Ω))=f(Ω)\pi^*(\tilde{f}(\Omega))=f(\Omega).

  3. Next step is to prove that f˜(Ω)\tilde{f}(\Omega) is closed 2k2k form on MM. To prove f˜(Ω)\tilde{f}(\Omega) is closed, it suffices to prove that f(Ω)f(\Omega) is closed.

  4. For a special kk-form φ\varphi on PP, the exterior differential dφd\varphi coincides with the exterior covariant differential DφD\varphi of φ\varphi i.e., dφ=Dφd\varphi=D\varphi. That special property is that φ=π *σ\varphi=\pi^*\sigma for some kk-form σ\sigma on MM.

  5. As f(Ω)f(\Omega) has that special property, we see that d(f(Ω))=D(f(Ω))d(f(\Omega))=D(f(\Omega)).

  6. By Bianchi’s identity, we have DΩ=0D\Omega=0. We then see that DΩ=0D\Omega=0 implies that D(f(Ω))=0D(f(\Omega))=0 i.e., d(f(Ω))=D(f(Ω))=0d(f(\Omega))=D(f(\Omega))=0 for fI k(G)f\in I^k(G) i.e., f(Ω)f(\Omega) is a closed 2k2k-form on PP. Thus, f˜(Ω)\tilde{f}(\Omega) is a closed 2k2k-form on MM, giving an element in the deRham cohomology H 2k(M,)H^{2k}(M,\mathbb{R}).

  7. Next step is to prove that, this assignment ff˜(Ω)f\mapsto \tilde{f}(\Omega) does not depend on the connection Γ\Gamma that we have started with i.e., for connections Γ 0\Gamma_0 (with curvature form Ω 0\Omega_0) and Γ 1\Gamma_1 (with curvature form Ω 1\Omega_1), the elements f˜(Ω 0)\tilde{f}(\Omega_0) and f˜(Ω 1)\tilde{f}(\Omega_1) are in the same equivalence class i.e., f˜(Ω 0)f˜(Ω 1)\tilde{f}(\Omega_0)-\tilde{f}(\Omega_1) is an exact form i.e., f˜(Ω 0)f˜(Ω 1)=dΦ˜\tilde{f}(\Omega_0)-\tilde{f}(\Omega_1)=d\tilde{\Phi} for some 2k12k-1 form Φ˜\tilde{\Phi} on MM.

  8. Using lemma , to prove f˜(Ω 0)f˜(Ω 1)=dΦ˜\tilde{f}(\Omega_0)-\tilde{f}(\Omega_1)=d\tilde{\Phi} for some 2k12k-1 form Φ˜\tilde{\Phi} on MM, it suffices to prove that f(Ω 0)f(Ω 1)=dΦf(\Omega_0)-f(\Omega_1)=d \Phi for some 2k12k-1 form Φ\Phi on PP that projects to a unique 2k12k-1 form Φ˜\tilde{\Phi} on MM.

  9. We then see that f(Ω 0)f(Ω 1)=dΦf(\Omega_0)-f(\Omega_1)=d \Phi for some 2k12k-1 form Φ\Phi on PP that projects to a unique 2k12k-1 form Φ˜\tilde{\Phi} on MM. This confirm that the assignment ff(Ω)f\mapsto f(\Omega) is independent of the connection Γ\Gamma that we have started with. We can extend this linearly to I(G)H *(M,)I(G)\rightarrow H^*(M,\mathbb{R}).

Given a principal bundle π:PM\pi:P\rightarrow M the morphism defined above I(G)H *(M,)I(G)\rightarrow H^*(M,\mathbb{R}) is called the Weil homomorphism.

Refined Chern–Weil homomorphism

We describe the refined Chern–Weil homomorphism (which associates a class in ordinary differential cohomology to a principal bundle with connection).

The modern construction is rather short and elegant, and appears in the work of Bunke–Nikolaus–Völkl and Schreiber. For an exposition, see, for example, Section 13.1 in Amabel–Debray–Haine [TODO: add reference].

The input data is an arbitrary Lie group GG, an invariant polynomial PP on the Lie algebra of GG, and a level cH k(BG,Z)c\in H^k(B G,\mathbf{Z}) whose image under the homomomorphism

H k(BG,Z)H k(BG,R)H^k(B G,\mathbf{Z}) \to H^k(B G,\mathbf{R})

equals the image of PP.

The output data is a morphism of (∞,1)-sheaves

B GB 2k1U(1)B_\nabla G\to B^{2k-1} \mathrm{U}(1)

whose image under the characteristic class map equals cc and under the curvature map equals the classical Chern–Weil homomorphism associated to PP.

Perhaps the quickest way to construct this refinement is to observe that B 2k1U(1)B^{2k-1} \mathrm{U}(1) is the homotopy pullback of Ω closed 2k\Omega^{2k}_{closed} and B 2kZB^{2k}\mathbf{Z} over B 2kRB^{2k}\mathbf{R}. Using the universal property of homotopy pullbacks, it suffices to construct a compatible pair of maps

B GB 2kZ,B GΩ closed 2k.B_\nabla G\to B^{2k} \mathbf{Z}, \qquad B_\nabla G\to \Omega^{2k}_{closed}.

The former map is given by cc, the latter is given by PP via the classical Chern–Weil homomorphism. They are compatible by assumption on the input data.

Refined Chern–Weil homomorphism: old construction

Here is a description of an older construction in terms of the universal connection on the universal principal bundle, following (HopkinsSinger, section 3.3). It makes an additional assumption that GG is compact, which is not necessary in the other approaches.


For XX a smooth manifold, PXP \to X a smoth GG-principal bundle with smooth classifying map f:XBGf : X \to B G and connection \nabla. Write CS(,f * univ)CS(\nabla, f^* \nabla_{univ}) for the Chern-Simons form for the interpolation between \nabla and the pullback of the universal connection along ff.

Then defined the cocycle in ordinary differential cohomology given by the function complex

c^:=(f *c,f *h+CS(,f * univ),w(F t))(c,h,w)C k(BG,)×C k1(BG,)×Ω cl k(X). \hat \mathbf{c} := (f^* c , f^* h + CS(\nabla, f^* \nabla_{univ}), w(F_{\nabla_t})) \in (c, h, w) \in C^k(B G, \mathbb{Z}) \times C^{k-1}(B G, \mathbb{R}) \times \Omega_{cl}^k(X) \,.

The above construction constitutes a map

c^:GBund (X) H diff k(X) \hat \mathbf{c} : G Bund_\nabla(X)_\sim \to H_{diff}^k(X)

from equivalence classes of GG-principal bundles with connection to degree kk ordinary differential cohomology.



Chern-Weil homomorphism

Original articles

The differential-geometric Chern-Weil homomorphism (evaluating curvature 2-forms of connections in invariant polynomials) first appears in print (Cartan's map) in:

  • Henri Cartan, Section 7 of: Cohomologie réelle d’un espace fibré principal différentiable. I : notions d’algèbre différentielle, algèbre de Weil d’un groupe de Lie, Séminaire Henri Cartan, Volume 2 (1949-1950), Talk no. 19, May 1950 (numdam:SHC_1949-1950__2__A18_0)

    Henri Cartan, Section 7 of: Notions d’algèbre différentielle; applications aux groupes de Lie et aux variétés où opère un groupe de Lie, in: Centre Belge de Recherches Mathématiques, Colloque de Topologie (Espaces Fibrés) Tenu à Bruxelles du 5 au 8 juin 1950, Georges Thon 1951 (GoogleBooks, pdf)

    reprinted in the appendix of:

(These two articles have the same content, with the same section outline, but not the same wording. The first one is a tad more detailed. The second one briefly attributes the construction to Weil, without reference.)

and around equation (10) of:

  • Shiing-shen Chern, Differential geometry of fiber bundles, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., (August-September 1950), vol. 2, pages 397-411, Amer. Math. Soc., Providence, R. I. (1952) (pdf, full proceedings vol 2 pdf)

It is the independence of this construction under the choice of connection which Chern 50 attributes (below equation 10) to the unpublished

  • André Weil, Géométrie différentielle des espaces fibres, unpublished, item [1949e] in: André Weil Oeuvres Scientifiques / Collected Papers, vol. 1 (1926-1951), 422-436, Springer 2009 (ISBN:978-3-662-45256-1)

The proof is later recorded, in print, in: Chern 51, III.4, Kobayashi-Nomizu 63, XII, Thm 1.1.

But the main result of Chern 50 (later called the fundamental theorem in Chern 51, XII.6) is that this differential-geometric “Chern-Weil” construction is equivalent to the topological (homotopy theoretic) construction of pulling back the universal characteristic classes from the classifying space BGB G along the classifying map of the given principal bundle.

This fundamental theorem is equation (15) in Chern 50 (equation 31 in Chern 51), using (quoting from the same page):

methods initiated by E. Cartan and recently developed with success by H. Cartan, Chevalley, Koszul, Leray, and Weil [13]

Here reference 13 is:

More in detail, Chern’s proof of the fundamental theorem (Chern 50, (15), Chern 51, III (31)) uses:

  1. the fact that invariant polynomials constitute the real cohomology of the classifying space, inv(𝔤)H (BG)inv(\mathfrak{g}) \simeq H^\bullet(B G), which is later expanded on in:

    Some authors later call this the “abstract Chern-Weil isomorphism”.

  2. existence of universal connections for manifolds in bounded dimension (see here), which is later developed in:


Review of the Chern-Weil homomorphism:

With an eye towards applications in mathematical physics:

See also in:

Enhancement of the Chern-Weil homomorphism from ordinary cohomology-groups to dg-categories of \infty -local systems:

Chern-Weil theory

See also the references at

Last revised on June 10, 2023 at 19:40:45. See the history of this page for a list of all contributions to it.