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Chern-Weil homomorphism

Contents

Context

\infty-Chern-Weil theory

Differential cohomology

Contents

Idea

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 differenial 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) in terms of the universal connection on the universal principal bundle. We follow (HopkinsSinger, section 3.3).

Definition

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) \,.
Proposition

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.

Examples

References

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

Review of the Chern-Weil homomorphism:

With an eye towards applications in mathematical physics:

See also in:

Chern-Weil theory

See also the references at

Last revised on October 29, 2020 at 05:50:09. See the history of this page for a list of all contributions to it.