# nLab curvature characteristic form

Contents

## Examples

### $\infty$-Lie algebras

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten 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 $\mathfrak{g}$ 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

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

## 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 $B G$ 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:

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:

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: