nLab
curvature characteristic form

• ∞-Lie algebroid valued differential forms

• connection on an ∞-bundle

• curvature

• Bianchi identity

• curvature characteristic form

• Chern-Simons form

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

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

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

Notably if $G$ is a Lie group with Lie algebra $\mathfrak{g}$, $P$ is the total space of a $G$-principal bundle $\pi : P \to X$, and $A \in \Omega^1(P,\mathfrak{g})$ is an Ehresmann connection 1-form on $P$ then by the very definition of the $G$-equivariance of $A$ and the invariance of $\langle \cdots \rangle$ it follows that the curvature form is invariant under the $G$-action on $P$ and is therefore the pullback along $\pi$ of a $2 n$-form $P_n \in \Omega^{2 n}(X)$ down on $X$. 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 $X$. 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 $P \to X$ a $G$-principal bundle write $T X$, $T P$ and $T_{vert} P$ for the tangent Lie algebroid of $X$, of $P$ and the vertical tangent Lie algebroid of $P$, respectively. Write $inn(\mathfrak{g})$ for the Lie 2-algebra given by the differential crossed module $\mathfrak{g}\stackrel{Id}{\to} \mathfrak{g}$ and finally $\prod_i b^{n_i} \mathbb{R}$ for the L-∞-algebra with one abelian generator for each generating invariant polynomial of $\mathfrak{g}$

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

Recall that a morphism of ∞-Lie algebroids

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

is equivalently a closed $n$-form on $X$. The data of an Ehresmann connection on $P$ then induces the following diagram of ∞-Lie algebroids

$$\array{ T_{vert} P &\stackrel{A_{vert}}{\to}& \mathfrak{g} &&& flat vertical form \\ \downarrow && \downarrow &&& first Ehresmann condition \\ T P &\stackrel{A}{\to}& inn(\mathfrak{g}) &&& form on total space \\ \downarrow && \downarrow &&& second Ehresmann condition \\ T X &\stackrel{(P_n)}{\to}& \prod_i b^{n_i} \mathbb{R} &&& curvature characteristic forms } \,.$$

Examples

• The single curvature characteristic form of a complex line bundle/$U(1)$-principal bundle is the curvature 2-form itself.

• A sum of all curvature characteristic forms of a complex vector bundle/U(n)-principal bundle gives the Chern character of a vector bundle.

Related concepts

• characteristic class

• Chern-Weil 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, 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)

• Mike Hopkins, Isadore Singer, Section 3.3 of: Quadratic Functions in Geometry, Topology,and M-Theory

  J. Differential Geom. Volume 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, Volume 16 Issue 1 (2012), pages 149-250 (arXiv:1011.4735, euclid:1358950853, doi:10.1007/BF02104916)

  (in generalization to principal ∞-bundles)

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

  (using the stacky language of FSS 10)

• Adel Rahman, Chern-Weil theory, 2017 (pdf)

See also in:

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