nLab
Chern-Weil theory

Contents

Context

\infty-Chern-Weil theory

Bundles

Differential cohomology

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

Chern-Weil theory studies the refinement of characteristic classes of principal bundles in ordinary cohomology to de Rham cohomology and further to ordinary differential cohomology.

The central operation that models this refinement is the construction of the Chern-Weil homomorphism from GG-principal bundles to de Rham cohomology by choosing a connection \nabla and evaluating its curvature form F F_\nabla in the invariant polynomials \langle -\rangle of the Lie algebra 𝔤\mathfrak{g} to produce the curvature characteristic form F \langle F_\nabla \rangle. Its de Rham cohomology class refines a corresponding characteristic class in integral cohomology.

Concretely, the Chern-Weil homomorphism is presented by the following construction:

For

we get for each smooth manifold XX an assignment

[c]:GBund(X) H n(X,) [c] : G Bund(X)_\sim \to H^n(X,\mathbb{Z})

on integral cohomology classes of base space to equivalence classes of GG-principal bundles by sending a bundle classified by a map f:XBGf : X \to B G to the class [f *c][f^* c].

Let [c] H n(BG,)[c]_\mathbb{R} \in H^n(B G, \mathbb{R}) be the image of [c][c] in real cohomology, induced by the evident inclusion of coefficients \mathbb{Z} \hookrightarrow \mathbb{R}.

The first main statement of Chern-Weil theory is that there is an invariant polynomial

:=ϕ 1[c] \langle- \rangle := \phi^{-1} [c]_{\mathbb{R}}

on the Lie algebra 𝔤\mathfrak{g} of GG associated to [c] [c]_{\mathbb{R}}, given by an isomorphism (of real graded vector space)s

ϕ:inv(𝔤)H (BG,). \phi : inv(\mathfrak{g}) \stackrel{\simeq}{\to} H^\bullet(B G, \mathbb{R}) \,.

The second main statement is that this invariant polynomial serves to provide a differential (Lie integration) construction of [c] [c]_{\mathbb{R}}:

for any choice of connection \nabla on a GG-principal bundle PXP \to X we have the curvature 2-form F Ω 2(P,𝔤)F_\nabla \in \Omega^2(P, \mathfrak{g}) and fed into the invariant polynomial this yields an nn-form

F F Ω n(X). \langle F_\nabla \wedge \cdots \wedge F_\nabla \rangle \in \Omega^n(X) \,.

The statement is that under the de Rham theorem-isomorphism H dR (X)H (X,)H^\bullet_{dR}(X) \simeq H^\bullet(X, \mathbb{R}) this presents the class [c] [c]_{\mathbb{R}}.

The third main statement, says that this construction may be refined by combining integral cohomology and de Rham cohomology to ordinary differential cohomology: the nn-form F F \langle F_\nabla \wedge \cdots F_\nabla\rangle may be realized itself as the curvature nn-form of a circle n-bundle with connection c^\hat \mathbf{c}.

[c^]:GBund (X) U(1)nBund (X) H diff n(X). [\hat \mathbf{c}] : G Bund_\nabla(X)_\sim \to U(1) n Bund_\nabla(X)_\sim \simeq H^n_{diff}(X) \,.

In summary this yields the following picture:

[c^] [c] [F F ] [c] H diff n(X) H n(X,) H dR n(X) H n(X,). \array{ && [\hat \mathbf{c}] \\ & \swarrow && \searrow \\ [c] && && [\langle F_\nabla \wedge \cdots F_\nabla\rangle] \\ & \searrow && \swarrow \\ && [c]_{\mathbb{R}} } \;\;\;\;\;\;\;\;\; \in \;\;\;\;\;\;\;\;\; \array{ && H_{diff}^n(X) \\ & \swarrow && \searrow \\ H^n(X,\mathbb{Z}) && && H_{dR}^n(X) \\ & \searrow && \swarrow \\ && H^n(X, \mathbb{R}) } \,.

A central implication of the last step is that with the refinement from curvatures in de Rham cohomology to circle n-bundles with connection in differential cohomology is that these come with a notion of higher parallel transport and higher holonomy:

Specifically

So the refined Chern-Weil homomorphism provides a large family of gauge quantum field theories of Chern-Simons type in odd dimensions whose field configurations are always connections on principal bundles and whose Lagrangians are Chern-Simons elements on a Lie algebra.

But the notion of invariant polynomials and Chern-Simons elements naturally exists much more generally for L-∞ algebras, and even more generally for L-∞ algebroids. We claim here that in this fully general case there is still a natural analog of the Chern-Weil homomorphism – which we call the ∞-Chern-Weil homomorphism . Accordingly this gives rise to a wide class of action functionals for gauge quantum field theories, which may be called ∞-Chern-Simons theories.

Generalizations

To principal \infty-bundles

The notions of Lie group, Lie algebra, principal bundle and all the other ingredients of ordinary Chern-Weil theory generalize to notions in higher category theory such as ∞-Lie group, ∞-Lie algebra, principal ∞-bundle etc. The generalization of Chern-Weil theory to this context is discussed at

In noncommutative geometry

There is a noncommutative analogue discussed in (AlekseevMeinrenken2000).

Examples and applications

History

(following notes provided by Jim Stasheff)

The beginnings of the rational homotopy theory of Lie groups GG and hence their dg-algebra-description in terms of the Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) originate in the first half of the 20th century.

In his survey of what was known in 1936 on the homology of compact Lie groups

  • Eli Cartan, La topologie des espaces représentatifs des groupes de Lie Act. Sci. Ind., No. 358, Hermann, Paris, (1936).

    reprinted in Cartan’s Complete Works vol I 2I_2 pp. 1307-1330

E. Cartan conjectured that there should be a general result implying that the homology of the classical Lie groups is the same as the homology of a product of odd-dimensional spheres. In particular, he lists the Poincare polynomial?s for classical simple compact Lie groups.

In

  • Heinz Hopf, Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen (German) Ann. of Math. (2) 42, (1941). 22–52.

Hopf showed that such a characterization in terms of homology groups as intersection pairing algebras holds for any compact finite dimensional connected orientable manifold with a map m:M×MMm:M\times M\to M such that left and right translation have non-zero degrees.

Later in

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

with the development of cohomology, especially de Rham cohomology, this was stated as H (G)H^\bullet(G) being isomorphic to an exterior algebra on odd dimensional generators: the generating Lie algebra cohomology cocycles μCE(𝔤)\mu \in CE(\mathfrak{g}), d CE(𝔤)μ=0d_{CE(\mathfrak{g})} \mu = 0.

Henri Cartan in

  • Henri Cartan, Notions d’algébre différentielle; applications aux groupes de Lie et aux variétés où opère un groupe de Lie , Coll. Topologie Algébrique Bruxelles (1950)

    15-28

    section 7, titled Classes caracteristiques (reelles) d’un espace fibre principal

at the end (1951) of an era of deRham cohomology dominence (prior to Serre’s thesis) abstracted the differential geometric approach of Chern-Weil and the Weil algebra W(𝔤)W(\mathfrak{g}) to the dg-algebra context with his notion of 𝔤\mathfrak{g}-algebras AA . This involves what is known sometimes as the Cartan calculus. In addition to the differential dd of differential forms on a principal bundle, Cartan abstracts the inner product aka contraction of differential forms with vector fields XX and the Lie derivative X\mathcal{L}_X with respect to vector fields. that is, he posits 3 operators on a differential-graded-commutative alggebra (dgca):

dd of degree 1, i Xi_X of degree -1 and L XL_X of degree 0 for X in 𝔤\mathfrak{g} subject to the relations:

  • [ι X,ι Y]=ι [X,Y][\iota_X,\iota_Y] = \iota_{[X,Y]}

  • [ X,ι Y]=ι [X,Y] [\mathcal{L}_X,\iota_Y]= \iota_{[X,Y]}

and perhaps most useful

  • X=dι X+iι Xd\mathcal{L}_X = d \iota_X + i\iota_X d

This is what he terms a 𝔤\mathfrak{g}-algebra.

For Cartan, an infinitesimal connection on a principal bundle PXP \to X are projectors (at each point pp of PP) ϕ p:T pPT p vert\phi_p: T_p P\to T_p^{vert} equivariant with repect to the GG-action. This can be abstracted to a morphism

Ω (P)𝔤 *:A \Omega^\bullet(P) \leftarrow \mathfrak{g}^* : A

of graded vector spaces of degree 1 – equivalently a Lie-algebra valued 1-form AΩ 1(P,𝔤)A \in \Omega^1(P,\mathfrak{g}) – such that the two Ehresmann conditions hold:

  1. restricted to the fibers the 1-form AA is the Maurer-Cartan form

    ι XA(h)=ι Xh\iota_X A(h) = \iota_X h

  2. the form is equivariant in that

    XA(h)=A( Xh)\mathcal{L}_X A (h) = A(\mathcal{L}_X h)

for all X𝔤X\in \mathfrak{g} and h𝔤 *h\in \mathfrak{g}^*. This data Cartan calls an algebraic connection .

He then extends such an AA to a homomorphism of graded algebras

Ω (P)CE(𝔤):A \Omega^\bullet(P) \leftarrow CE(\mathfrak{g}) : A

from the Chevalley-Eilenberg algebra 𝔤 *\wedge^\bullet \mathfrak{g}^*.

In general, this will not respect the differentials, hence not be a morphism of dg-algebras. In fact, the deviation gives the curvature of the connection: the curvature tensor is the map hd dRA(h)A(d CEh)h\mapsto d_{dR} A(h)-A(d_{CE} h).

Jim: HAVE TO BREAK OFF NOW - WHAT WILL COME NEXT IS the Weil algebra W(𝔤)W(\mathfrak{g}) as a Cartan 𝔤\mathfrak{g}-algebra

  • Weil, Geometrie differentielle des espaces fibres (1949, unpublished) appears in Vol. 1, pp. 422-436, of his Collected Papers.

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:

More review:

  • Fei Han, Chern-Weil theory and some results on classic genera (pdf)

Some standard monographs are

  • Johan Louis Dupont, Fibre bundles and Chern-Weil theory, Lecture Notes Series 69, Dept. of Math., University of Aarhus, Aarhus, 2003, 115 pp. pdf

  • Johan Louis Dupont, Curvature and characteristic classes, Lecture Notes in Math. 640, Springer-Verlag, Berlin-Heidelberg-New York, 1978.

  • Mikhail Postnikov, Лекции по геометрии. Семестр 4, Дифференциальная геометрия — М.: Наука, 1988

  • Raoul Bott, Loring Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer 1982. xiv+331 pp.

  • Loring Tu, Differential Geometry – Connections, Curvature, and Characteristic Classes, Springer 2017 (ISBN:978-3-319-55082-4)

  • Victor Guillemin, Shlomo Sternberg, Supersymmetry and equivariant de Rham theory, Springer, 1999.

Lecture notes with an eye on Morse theory in terms of supersymmetric quantum mechanics are in

  • Weiping Zhang, Lectures on Chern-Weil theory and Witten deformations , Nankai Tracts in Mathematics - Vol. 4 (web)

Chern-Weil theory in the context of noncommutative geometry is discussed in

  • A. Alekseev, E. Meinrenken, The non-commutative Weil algebra, Invent. Math. 139, n. 1, 135-172, 2000, doi

Last revised on March 24, 2021 at 10:08:49. See the history of this page for a list of all contributions to it.