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 $G$-principal bundles to de Rham cohomology by choosing a connection $\nabla$ and evaluating its curvature form $F_\nabla$ in the invariant polynomials $\langle -\rangle$ of the Lie algebra $\mathfrak{g}$ to produce the curvature characteristic form $\langle F_\nabla \rangle$. 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
$B G \in$ Top the classifying space of (the topological group underlying) a compact Lie group $G$
and $[c] \in H^n(B G, \mathbb{Z})$ a class in its integral cohomology – which we may call a characteristic class for $G$-principal bundles
we get for each smooth manifold $X$ an assignment
on integral cohomology classes of base space to equivalence classes of $G$-principal bundles by sending a bundle classified by a map $f : X \to B G$ to the class $[f^* c]$.
Let $[c]_\mathbb{R} \in H^n(B G, \mathbb{R})$ be the image of $[c]$ 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
on the Lie algebra $\mathfrak{g}$ of $G$ associated to $[c]_{\mathbb{R}}$, given by an isomorphism (of real graded vector space)s
The second main statement is that this invariant polynomial serves to provide a differential (Lie integration) construction of $[c]_{\mathbb{R}}$:
for any choice of connection $\nabla$ on a $G$-principal bundle $P \to X$ we have the curvature 2-form $F_\nabla \in \Omega^2(P, \mathfrak{g})$ and fed into the invariant polynomial this yields an $n$-form
The statement is that under the de Rham theorem-isomorphism $H^\bullet_{dR}(X) \simeq H^\bullet(X, \mathbb{R})$ this presents the class $[c]_{\mathbb{R}}$.
The third main statement, says that this construction may be refined by combining integral cohomology and de Rham cohomology to ordinary differential cohomology: the $n$-form $\langle F_\nabla \wedge \cdots F_\nabla\rangle$ may be realized itself as the curvature $n$-form of a circle n-bundle with connection $\hat \mathbf{c}$.
In summary this yields the following picture:
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:
the local connection form of $\hat \mathbf{c}$ is the Chern-Simons form $cs(\nabla)$ of a Chern-Simons element $cs$ of the invariant polynomial $\langle- \rangle$ evaluated on the given $G$-connection;
the corresponding higher parallel transport as an assignment
of $(n-1)$-dimensional manifolds in $X$ to the circle group is the action functional of the corresponding Chern-Simons theory.
Specifically
for $\langle -,-\rangle$ the Killing form invariant polynomial on a semisimple Lie algebra, one calls $\hat \mathbf{c}$ the Chern-Simons circle 3-bundle; whose higher holonomy is the action functional of ordinary Chern-Simons theory;
for $\langle -,-,-,-\rangle$ the next higher invariant polynomial on a semisimple Lie algebra, $\hat \mathbf{c}$ is a Chern-Simons circle 7-bundle, and so on.
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 [[schreiber:∞-Chern-Simons theories].
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
There is a noncommutative analogue discussed in (AlekseevMeinrenken2000).
(following notes provided by Jim Stasheff)
The beginnings of the rational homotopy theory of Lie groups $G$ and hence their dg-algebra-description in terms of the Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ 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 $I_2$ 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
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 $m:M\times M\to M$ such that left and right translation have non-zero degrees.
Later in
with the development of cohomology, especially de Rham cohomology, this was stated as $H^\bullet(G)$ being isomorphic to an exterior algebra on odd dimensional generators: the generating Lie algebra cohomology cocycles $\mu \in CE(\mathfrak{g})$, $d_{CE(\mathfrak{g})} \mu = 0$.
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 $W(\mathfrak{g})$ to the dg-algebra context with his notion of $\mathfrak{g}$-algebras $A$ . This involves what is known sometimes as the Cartan calculus. In addition to the differential $d$ of differential forms on a principal bundle, Cartan abstracts the inner product aka contraction of differential forms with vector fields $X$ and the Lie derivative $\mathcal{L}_X$ with respect to vector fields. that is, he posits 3 operators on a differential-graded-commutative alggebra (dgca):
$d$ of degree 1, $i_X$ of degree -1 and $L_X$ of degree 0 for X in $\mathfrak{g}$ subject to the relations:
$[\iota_X,\iota_Y] = \iota_{[X,Y]}$
$[\mathcal{L}_X,\iota_Y]= \iota_{[X,Y]}$
and perhaps most useful
This is what he terms a $\mathfrak{g}$-algebra.
For Cartan, an infinitesimal connection on a principal bundle $P \to X$ are projectors (at each point $p$ of $P$) $\phi_p: T_p P\to T_p^{vert}$ equivariant with repect to the $G$-action. This can be abstracted to a morphism
of graded vector spaces of degree 1 – equivalently a Lie-algebra valued 1-form $A \in \Omega^1(P,\mathfrak{g})$ – such that the two Ehresmann conditions hold:
restricted to the fibers the 1-form $A$ is the Maurer-Cartan form
$\iota_X A(h) = \iota_X h$
the form is equivariant in that
$\mathcal{L}_X A (h) = A(\mathcal{L}_X h)$
for all $X\in \mathfrak{g}$ and $h\in \mathfrak{g}^*$. This data Cartan calls an algebraic connection .
He then extends such an $A$ to a homomorphism of graded algebras
from the Chevalley-Eilenberg algebra $\wedge^\bullet \mathfrak{g}^*$.
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 $h\mapsto d_{dR} A(h)-A(d_{CE} h)$.
Jim: HAVE TO BREAK OFF NOW - WHAT WILL COME NEXT IS the Weil algebra $W(\mathfrak{g})$ as a Cartan $\mathfrak{g}$-algebra
Early original references are
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
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)
An overview with a collection of references is
A classical textbook reference is
A review of much of the theory and comments on applications to elliptic genera is in
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.
М. М. Постников, Лекции по геометрии. Семестр 4, Дифференциальная геометрия — М.: Наука, 1988
R. Bott, L. W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer 1982. xiv+331 pp.
V. Guillemin, S. 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
Chern-Weil theory in the context of noncommutative geometry is discussed in