nLab Chern-Weil homomorphism

Idea

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

$$\langle F_A \wedge \cdots \wedge F_A \rangle \;\in\; \Omega^{2n}_{clsd}(X)$$

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

This is a map from the first nonabelian cohomology of $X$ with coefficients in $G$ to the de Rham cohomology of $X$

$$char \;\colon\; H^1(X,G) \longrightarrow \textstyle{\prod_{n_i}} H_{dR}^{2 n_i}(X) \,,$$

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

Plain Chern-Weil homomorphism

We give an outline of construction of Weil homomorphism following Kobayashi & Nomizu 1963.

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Given an element $g\in G$, the adjoint action $Ad(g) \,\colon\, G \longrightarrow G$ is defined as $Ad(g)(h)=g h g^{-1}$. For $g\in G$, let $ad(g) \,\colon\,\mathfrak{g}\rightarrow \mathfrak{g}$ be the differential of $Ad(g) \,\colon\, G\rightarrow G$ at $e\in G$.

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

$$f \,\colon\, \underset{ k \; \text{factors} }{ \underbrace{ \mathfrak{g}\times\cdots\times\mathfrak{g} } } \longrightarrow \mathbb{R}$$

that are $G$-invariant in that

$$f\big( ad(g)(t_1),\cdots,ad(g)(t_k) \big) \;=\; f(t_1,\cdots,t_k)$$

for all $g\in G$ and $t_i\in \mathfrak{g}$. These $I^k(G)$ are real vector spaces. Let $I(G)$ denote the $\mathbb{R}$ algebra $\oplus_{k=0}^{\infty}I^k(G)$.

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

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

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

2. Given an element $f\in I^k(G)$, define a $2k$-form $f(\Omega)$ on $P$.

   Prove that the $2k$ form $f(\Omega)$ on $P$ projects uniquely to a $2k$ form on $M$ and call it $\tilde{f}(\Omega)$ i.e., $\pi^*\big(\tilde{f}(\Omega)\big) = f(\Omega)$.

3. Next step is to prove that $\tilde{f}(\Omega)$ is closed $2k$ form on $M$. To prove $\tilde{f}(\Omega)$ is closed, it suffices to prove that $f(\Omega)$ is closed.

4. For a special $k$-form $\varphi$ on $P$, the exterior differential $d\varphi$ coincides with the exterior covariant differential $D\varphi$ of $\varphi$ i.e., $d\varphi=D\varphi$. That special property is that $\varphi=\pi^*\sigma$ for some $k$-form $\sigma$ on $M$.

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

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

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

8. Using lemma , to prove $\tilde{f}(\Omega_0)-\tilde{f}(\Omega_1)=d\tilde{\Phi}$ for some $2k-1$ form $\tilde{\Phi}$ on $M$, it suffices to prove that $f(\Omega_0)-f(\Omega_1)=d \Phi$ for some $2k-1$ form $\Phi$ on $P$ that projects to a unique $2k-1$ form $\tilde{\Phi}$ on $M$.

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

Given a principal bundle $\pi \,\colon\, P\rightarrow M$ the morphism defined above $I(G)\rightarrow H^*(M,\mathbb{R})$ 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 $G$, an invariant polynomial $P$ on the Lie algebra of $G$, and a level $c\in H^k(B G,\mathbf{Z})$ whose image under the homomomorphism

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

equals the image of $P$.

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

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

whose image under the characteristic class map equals $c$ and under the curvature map equals the classical Chern–Weil homomorphism associated to $P$.

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

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

The former map is given by $c$, the latter is given by $P$ 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 $G$ is compact, which is not necessary in the other approaches.

• Let $G$ be a compact Lie group

• with Lie algebra $\mathfrak{g}$;

• and write $inv(\mathfrak{g})$ for the dg-algebra of invariant polynomials on $\mathfrak{g}$ (which has trivial differential).

• Write $B^{(n)}G$ for the smooth level $n$ classifying space

• and $B G := {\lim_\to}_n B^{(n)}G$ for the colimit, a smooth model of the classifying space of $G$.

• Write $\nabla_{univ}$ for the universal connection on $E G \to B G$.

• Let $[c] \in H^k(B G, \mathbb{Z})$ be a characteristic class

• and choose a refinement $[\hat \mathbf{c}] \in H_{diff}^k(B G)$ in ordinary differential cohomology represented by a differential function

  $$(c, h, w) \in C^k(B G, \mathbb{Z}) \times C^{k-1}(B G, \mathbb{R}) \times (W(\mathfrak{g}) \simeq C^k(B G, \mathbb{R}))^k \,.$$

Definition

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

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

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

Proposition

The above construction constitutes a map

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

from equivalence classes of $G$-principal bundles with connection to degree $k$ ordinary differential cohomology.

Examples

• Gauss-Bonnet theorem

Related concepts

• Sullivan model of classifying space

• Cartan's map in equivariant de Rham cohomology

• Cheeger-Simons homomorphism

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:

  • Victor Guillemin, Shlomo Sternberg, Supersymmetry and equivariant de Rham theory, Springer 1999 (doi:10.1007/978-3-662-03992-2)

(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, but 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 $B G$ 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:

• Jean-Louis Koszul, Homologie et cohomologie des algebres de Lie, Bull. Soc. Math. France vol. 78 (1950) pp. 65-127 (numdam:BSMF_1950__78__65_0)

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, $inv(\mathfrak{g}) \simeq H^\bullet(B G)$, which is later expanded on in:

   • Chern 51, III.5

   • Raoul Bott, On the Chern-Weil homomorphism and the continuous cohomology of Lie-groups, Advances in Mathematics Volume 11, Issue 3, December 1973, Pages 289-303 (doi:10.1016/0001-8708(73)90012-1)

   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:

   • Mudumbai Narasimhan, Sundararaman Ramanan, Existence of Universal Connections, American Journal of Mathematics Vol. 83, No. 3 (Jul., 1961), pp. 563-572 (jstor:2372896)

   • Mudumbai Narasimhan, Sundararaman Ramanan, Existence of Universal Connections II, American Journal of Mathematics Vol. 85, No. 2 (Apr., 1963), pp. 223-231 (jstor:2373211)

   • Roger Schlafly, Universal connections, Invent Math 59, 59–65 (1980) (doi:10.1007/BF01390314)

   • Roger Schlafly, Universal connections: the local problem, Pacific J. Math. Volume 98, Number 1 (1982), 157-171 (euclid:pjm/1102734394)

Review

Review of the Chern-Weil homomorphism:

• Shiing-Shen Chern, Chapter III of: Topics in Differential Geometry, Institute for Advanced Study (1951) (pdf)

• Shoshichi Kobayashi, Katsumi Nomizu, Chapter XII in: Foundations of Differential Geometry, Volume 1, Wiley 1963 (web, ISBN:9780471157335, Wikipedia)

• Shiing-Shen Chern, James Simons, Section 2 of: Characteristic Forms and Geometric Invariants, Annals of Mathematics Second Series, Vol. 99, No. 1 (Jan., 1974), pp. 48-69 (jstor:1971013)

  (in the context of Chern-Simons forms)

• John Milnor, Jim Stasheff, Appendix C of: Characteristic classes, Princeton Univ. Press (1974) (ISBN:9780691081229)

• Eckhard Meinrenken, Section 5 of: Group actions on manifolds, Lecture Notes 2003 (pdf, pdf)

  (via equivariant de Rham cohomology)

• Mike Hopkins, Isadore Singer, Section 3.3 of: Quadratic Functions in Geometry, Topology,and M-Theory, J. Differential Geom. 70 Number 3 (2005), 329-452 (arXiv:math.AT/0211216, euclid:1143642908)

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Section 2.1 in: Cech Cocycles for Differential characteristic Classes, Advances in Theoretical and Mathematical Physics 16 1 (2012) 149-250 [doi:10.1007/BF02104916, arXiv:1011.4735, euclid:1358950853]

  (in generalization to principal ∞-bundles)

• Daniel Freed, Michael Hopkins: Chern-Weil forms and abstract homotopy theory, Bull. Amer. Math. Soc. 50 (2013) 431-468 [doi:10.1090/S0273-0979-2013-01415-0, arXiv:1301.5959]

  (using the stacky language of FSS 10)

• Adel Rahman: Chern-Weil theory (2017 $[$pdf]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber: Section 8 of: The Character Map in Non-Abelian Cohomology, World Scientific (2023) [doi:10.1142/13422]

  (as a special case of the character map on generalized nonabelian cohomology)

With an eye towards applications in mathematical physics:

• Mikio Nakahara, Chapter 11.1 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)

• Gerd Rudolph, Matthias Schmidt, Chapter 4 of: Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields, Springer 2017 (doi:10.1007/978-94-024-0959-8)

See also:

• Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology, Academic Press (1973)

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

• Camilo Arias Abad, Santiago Pineda Montoya, Alexander Quintero Velez, Chern-Weil theory for $\infty$-local systems [arXiv:2105.00461]

Chern-Weil theory

See also the references at

• Chern-Weil theory

• Chern-Simons forms