∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
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.
For a Lie algebra, an invariant polynomial of arguments on the Lie algebra and a Lie-algebra-valued 1-form with curvature 2-form , the curvature characteristic form of with respect to is the differential form
This form is always an exact form. The -form trivializing it is called a Chern-Simons form.
Notably if is a Lie group with Lie algebra , is the total space of a -principal bundle , and is an Ehresmann connection 1-form on then by the very definition of the -equivariance of and the invariance of it follows that the curvature form is invariant under the -action on and is therefore the pullback along of a -form down on . 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 . This establishes the Weil homomorphism from invariant polynomials to de Rham cohomology
The above description of curvature characteristic forms may be formulated in terms of ∞-Lie theory as follows.
For a -principal bundle write , and for the tangent Lie algebroid of , of and the vertical tangent Lie algebroid of , respectively. Write for the Lie 2-algebra given by the differential crossed module and finally for the L-∞-algebra with one abelian generator for each generating invariant polynomial of
From the discussion at invariant polynomial we have a canonical morphism that represents the generating invariant polynomials.
Recall that a morphism of ∞-Lie algebroids
is equivalently a closed -form on . The data of an Ehresmann connection on then induces the following diagram of ∞-Lie algebroids
We spell out the formulas for the images under the Chern-Weil homomorphism of the Chern classes, Pontrjagin classes and Euler classes as characteristic forms over smooth manifolds.
Let be a smooth manifold.
Write
for the commutative algebra over the real numbers of even-degree differential forms on , under the wedge product of differential forms. This is naturally a graded commutative algebra, graded by form degree, but since we consider only forms in even degree it is actually a plain commutative algebra, too, after forgetting the grading.
Let be a semisimple Lie algebra (such as or ) with Lie algebra representation over the complex numbers of finite dimension (for instance the adjoint representation or the fundamental representation), hence a homomorphism of Lie algebras
to the linear endomorphism ring , regarded here through its commutator as the endomorphism Lie algebra of .
When regarded as an associative ring this is isomorphic to the matrix algebra of square matrices
The tensor product of the -algebras (1) and (2)
is equivalently the matrix algebra with coefficients in the complexification of even-degree differential forms:
The multiplicative unit
in this algebra is the smooth function (differential 0-forms) which is constant on the identity matrix and independent of .
Given a connection on a -principal bundle, we regard its -valued curvature form as an element of this algebra
The total Chern form is the determinant of the sum of the unit (3) with the curvature form (4), and its component in degree , for , is the th Chern form :
By the relation between determinant and trace, this is equal to the exponential of the trace of the logarithm of , this being the exponential series in the trace of the Mercator series in :
Setting in these expressions (5) yields the total Pontrjagin form with degree=-components the Pontrjagin forms :
Hence the first couple of Pontrjagin forms are
(See also, e.g., Nakahara 2003, Exp. 11.5)
For and with the curvature form again regarded as a 2-form valued -square matrix
the Euler form is its Pfaffian of this matrix, hence the following sum over permutations with summands signed by the the signature :
The first of these is, using the Einstein summation convention and the Levi-Civita symbol:
(See also, e.g., Nakahara 2003, Exp. 11.7)
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:
It is the independence of this construction under the choice of connection which Chern 50 attributes (below equation 10) to the unpublished
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 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:
the fact that invariant polynomials constitute the real cohomology of the classifying space, , which is later expanded on in:
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”.
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 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)
Eckhard Meinrenken, Section 5 of: Group actions on manifolds, Lecture Notes 2003 (pdf, pdf)
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)
With an eye towards applications in mathematical physics:
Mikio Nakahara, Chapter 11.1 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)
Gerd Rudolph, Matthias Schmidt, Chapter 4 of: Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields, Springer 2017 (doi:10.1007/978-94-024-0959-8)
See also in:
Enhancement of the Chern-Weil homomorphism from ordinary cohomology-groups to dg-categories of -local systems:
Last revised on July 17, 2022 at 14:48:46. See the history of this page for a list of all contributions to it.