nLab
curvature characteristic form

Contents

Context

\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

\infty-Chern-Weil theory


Contents

Idea

A curvature characteristic form is a differential form naturally associated to a Lie algebra-valued 1-form that is a measure for the non-triviality of the curvature of the 1-form.

More generally, there is a notion of curvature characteristic forms of L-∞-algebra-valued differential forms and ∞-Lie algebroid valued differential forms.

Of connection 1-forms

For 𝔤\mathfrak{g} a Lie algebra, ,,,\langle -,-, \cdots, -\rangle an invariant polynomial of nn arguments on the Lie algebra and AΩ 1(P,𝔤)A \in \Omega^1(P,\mathfrak{g}) a Lie-algebra-valued 1-form with curvature 2-form F A=d dRA+[AA]F_A = d_{dR} A + [A \wedge A], the curvature characteristic form of AA with respect to \langle \cdots \rangle is the differential form

F AF AΩ 2n(P). \langle F_A \wedge \cdots \wedge F_A \rangle \in \Omega^{2 n}(P) \,.

This form is always an exact form. The (2n1)(2 n -1)-form trivializing it is called a Chern-Simons form.

Notably if GG is a Lie group with Lie algebra 𝔤\mathfrak{g}, PP is the total space of a GG-principal bundle π:PX\pi : P \to X, and AΩ 1(P,𝔤)A \in \Omega^1(P,\mathfrak{g}) is an Ehresmann connection 1-form on PP then by the very definition of the GG-equivariance of AA and the invariance of \langle \cdots \rangle it follows that the curvature form is invariant under the GG-action on PP and is therefore the pullback along π\pi of a 2n2 n-form P nΩ 2n(X)P_n \in \Omega^{2 n}(X) down on XX. This form is in general no longer exaxt, but is always a closed form and hence represent a class in the de Rham cohomology of XX. This establishes the Weil homomorphism from invariant polynomials to de Rham cohomology

In terms of \infty-Lie algebroids

The above description of curvature characteristic forms may be formulated in terms of ∞-Lie theory as follows.

For PXP \to X a GG-principal bundle write TXT X, TPT P and T vertPT_{vert} P for the tangent Lie algebroid of XX, of PP and the vertical tangent Lie algebroid of PP, respectively. Write inn(𝔤)inn(\mathfrak{g}) for the Lie 2-algebra given by the differential crossed module 𝔤Id𝔤\mathfrak{g}\stackrel{Id}{\to} \mathfrak{g} and finally ib n i\prod_i b^{n_i} \mathbb{R} for the L-∞-algebra with one abelian generator for each generating invariant polynomial of 𝔤\mathfrak{g}

From the discussion at invariant polynomial we have a canonical morphism inn(𝔤) ib n iinn(\mathfrak{g}) \to \prod_i b^{n_i}\mathbb{R} that represents the generating invariant polynomials.

Recall that a morphism of ∞-Lie algebroids

TXb n T X \to b^n \mathbb{R}

is equivalently a closed nn-form on XX. The data of an Ehresmann connection on PP then induces the following diagram of ∞-Lie algebroids

T vertP A vert 𝔤 flatverticalform firstEhresmanncondition TP A inn(𝔤) formontotalspace secondEhresmanncondition TX (P n) ib n i curvaturecharacteristicforms. \array{ T_{vert} P &\stackrel{A_{vert}}{\to}& \mathfrak{g} &&& flat vertical form \\ \downarrow && \downarrow &&& first Ehresmann condition \\ T P &\stackrel{A}{\to}& inn(\mathfrak{g}) &&& form on total space \\ \downarrow && \downarrow &&& second Ehresmann condition \\ T X &\stackrel{(P_n)}{\to}& \prod_i b^{n_i} \mathbb{R} &&& curvature characteristic forms } \,.

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:

Last revised on October 30, 2020 at 04:09:47. See the history of this page for a list of all contributions to it.