nLab curvature characteristic form

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.

Definition

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/principal U(1)-bundle is the curvature 2-form itself.

Chern-, Pontrjagin-, and Euler- characteristic forms

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.

Preliminaries

Let $X$ be a smooth manifold.

Write

(1)$$\Omega^{2\bullet}(X) \;\; \in \; CAlg_{\mathbb{R}}$$

for the commutative algebra over the real numbers of even-degree differential forms on $X$, 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 $\mathfrak{g}$ be a semisimple Lie algebra (such as $\mathfrak{su}(d)$ or $\mathfrak{so}(d)$) with Lie algebra representation $V \,\in\, Rep_{\mathbb{C}}(\mathfrak{g})$ over the complex numbers of finite dimension $dim_{\mathbb{C}}(V) \,=\, n \,\in\, \mathbb{N}$ (for instance the adjoint representation or the fundamental representation), hence a homomorphism of Lie algebras

$$\mathfrak{g} \xrightarrow{\;\;\rho\;\;} End_{\mathbb{C}}(V)$$

to the linear endomorphism ring $End_{\mathbb{C}}(V)$, regarded here through its commutator as the endomorphism Lie algebra of $V$.

When regarded as an associative ring this is isomorphic to the matrix algebra of $n \times n$ square matrices

(2)$$End_{\mathbb{C}}(V) \;\; \simeq \;\; Mat_{n \times n}(\mathbb{C}) \,.$$

The tensor product of the $\mathbb{C}$-algebras (1) and (2)

is equivalently the $n \times n$ matrix algebra with coefficients in the complexification of even-degree differential forms:

$$\Omega^{2\bullet} \big(X\big) \otimes_{\mathbb{R}} End_{\mathbb{C}}(V) \;\simeq\; \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} \big( Mat_{n \times n}( \mathbb{R} ) \big) \;\; \simeq \;\; Mat_{n \times n} \big( \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} \mathbb{C} \big) \,.$$

The multiplicative unit

(3)$$I \;\in\; Mat_{n \times n} \big( \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} \mathbb{C} \big)$$

in this algebra is the smooth function (differential 0-forms) which is constant on the $n \times n$ identity matrix.

Given a connection on a $G$-principal bundle, we regard its $\mathfrak{g}$-valued curvature form as an element of this algebra

(4)$$F_\nabla \,\in\, \Omega^2(X) \otimes_{\mathbb{R}} \mathfrak{g} \xrightarrow{\; \rho \;} \Omega^2(X) \otimes_{\mathbb{R}} End_{\mathbb{C}}(V) \xhookrightarrow{\;\;\;} \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} End_{\mathbb{C}}(V) \;\simeq\; Mat_{n \times n} \Big( \mathbb{C} \otimes_{\mathbb{R}} \Omega^{2}(X) \Big) \,.$$

The formulas

Chern forms

The total Chern form $c(\nabla)$ is the determinant of the sum of the unit (3) with the curvature form (4), and its component in degree $2k$, for $k \in \mathbb{N}$, is the $k$th Chern form $c_k(\nabla)$:

$$c(\nabla) \;\; \coloneqq \;\; \sum_k \underset{ \mathclap{ deg = 2k } }{ \underbrace{ c_k(\nabla) } } \;\; \coloneqq \;\; det \left( I + \frac{i F_\nabla}{2\pi} \right) \,.$$

By the relation between determinant and trace, this is equal to the exponential of the trace of the logarithm of $I + \frac{i F_\nabla}{2\pi}$, this being the exponential series in the trace of the Mercator series in $\frac{i F_\nabla}{2\pi}$:

(5)$$\begin{aligned} c(\nabla) & \;=\; det \left( I + \frac{i F_\nabla}{2\pi} \right) \\ & \;=\; \exp \circ tr \circ ln \left( I + \frac{i F_\nabla}{2\pi} \right) \\ & \;=\; \exp \circ tr \left( - \underset {k \in \mathbb{N}_+} {\sum} \tfrac{1}{k} \left( \frac{F_\nabla}{2\pi i} \right)^k \right) \\ & \;=\; \exp \left( \underset {k \in \mathbb{N}_+} {\sum} \tfrac{1}{k} \left( \frac { - (-i)^k } {(2\pi)^k} tr\big( F_\nabla^{\wedge_k} \big) \right) \right) \\ & \;=\; 1 \\ & \phantom{\;=\;} + \phantom{\frac{1}{1}} \left( i \tfrac{ tr\big(F_\nabla\big) }{2 \pi} + \tfrac{1}{2} \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2} -i \tfrac{1}{3} \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3} - \tfrac{1}{4} \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4} + \cdots \right) \\ & \phantom{\;=\;} + \frac{1}{2} \left( i \tfrac{ tr\big(F_\nabla\big) }{2 \pi} + \tfrac{1}{2} \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2} -i \tfrac{1}{3} \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3} - \tfrac{1}{4} \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4} + \cdots \right)^2 \\ & \phantom{\;=\;} + \frac{1}{6} \left( i \tfrac{ tr\big(F_\nabla\big) }{2 \pi} + \tfrac{1}{2} \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2} -i \tfrac{1}{3} \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3} - \tfrac{1}{4} \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4} + \cdots \right)^3 \\ & \phantom{\;=\;} + \frac{1}{24} \left( i \tfrac{ tr\big(F_\nabla\big) }{2 \pi} + \tfrac{1}{2} \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2} -i \tfrac{1}{3} \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3} - \tfrac{1}{4} \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4} + \cdots \right)^4 \\ & \phantom{\;=\;} + \cdots \\ & \;=\; 1 \\ & \phantom{\;=\;} + i \frac { tr\big(F_\nabla\big) } { 2 \pi } \\ & \phantom{\;=\;} + \tfrac{1}{2} \frac { tr\big( (F_\nabla)^2 \big) } { (2 \pi)^2 } + \frac{1}{2} \left( i \frac { tr\big( F_\nabla \big) } { 2\pi } \right)^2 \\ & \phantom{\;=\;} - i \tfrac{1}{3} \frac { tr\big( (F_\nabla)^3 \big) } { (2 \pi)^3 } + \frac{1}{2} \left( 2 \left( i \frac { tr\big( F_\nabla \big) } { 2 \pi } \right) \left( \tfrac{1}{2} \frac { tr\big( (F_\nabla)^2 \big) } { (2 \pi)^2 } \right) \right) + \frac{1}{6} \left( \left( i \frac { tr\big(F_\nabla\big) } { 2\pi } \right)^3 \right) \\ & \phantom{\;=\;} - \tfrac{1}{4} \frac {tr\big( (F_\nabla)^4 \big)} { (2 \pi)^4 } + \frac{1}{2} \left( \tfrac{1}{2} \frac {tr\big( (F_\nabla)^2 \big)} { (2 \pi)^2 } \right)^2 + \frac{1}{24} \left( i \frac {tr\big( F_\nabla \big)} { 2\pi } \right)^4 \\ & \phantom{\;=\;} + \cdots \\ & \;=\; 1 \\ & \phantom{\;=\;} + \underset{ \color{blue} = c_1(\nabla) }{ \underbrace{ i \frac { tr\big(F_\nabla\big) } { 2 \pi } }} \\ & \phantom{\;=\;} + \underset{ \color{blue} = c_2(\nabla) }{ \underbrace{ \frac {\tr\big( (F_\nabla)^2 \big) - \big( tr(F_\nabla) \big)^2 } { 8 \pi^2 } }} \\ & \phantom{\;=\;} + \underset{ \color{blue} = c_3(\nabla) }{ \underbrace{ i \frac { - 2 \cdot tr\big( (F_\nabla)^3 \big) + 3 \cdot tr(F_\nabla) \cdot tr\big( (F_\nabla)^2 \big) - \big( tr(F_\nabla ) \big)^3 } {48 \pi^3} }} \\ & \phantom{\;=\;} + \underset{ \color{blue} = c_4(\nabla) }{ \underbrace{ \frac { -6 \cdot tr\big( (F_\nabla)^4 \big) + 3 \cdot tr\big( (F_\nabla)^2 \big)^2 + \big( tr(F_\nabla) \big)^4 } {384 \pi^4} }} \\ & \phantom{\;=\;} + \cdots \end{aligned}$$

Pontrjagin forms

Setting $tr(F_\nabla) = 0$ in these expressions (5) yields the total Pontrjagin form $p(\nabla)$ with degree=$4k$-components the Pontrjagin forms $p_{k}(\nabla)$:

$$\begin{aligned} p(\nabla) & \;\coloneqq\; \underset{k \in \mathbb{N}}{\sum} \underset{ deg = 4k }{ \underbrace{ (-1)^{k} p_{k}(\nabla) } } \\ & \;=\; \underset{k \in \mathbb{N}}{\sum} \underset{ deg = 4k }{ \underbrace{ c_{2k}(\nabla) } } \\ & \;=\; 1 \\ & \phantom{\;=\;} + \underset{ \color{blue} = - p_1(\nabla) }{ \underbrace{ \frac {\tr\big( (F_\nabla)^2 \big) } { 8 \pi^2 } }} \\ & \phantom{\;=\;} + \underset{ \color{blue} = p_2(\nabla) }{ \underbrace{ \frac { - 2 \cdot tr\big( (F_\nabla)^4 \big) + tr\big( (F_\nabla)^2 \big)^2 } {128 \pi^4} }} \\ \phantom{\;=\;} + \cdots \end{aligned}$$

Hence the first couple of Pontrjagin forms are

$$\begin{aligned} p_1(\nabla) & \;=\; - \frac {\tr\big( (F_\nabla)^2 \big) } { 8 \pi^2 } \\ p_2(\nabla) & \;=\; \frac { tr\big( (F_\nabla)^2 \big)^2 - 2 \cdot tr\big( (F_\nabla)^4 \big) } {128 \pi^4} \,. \end{aligned}$$

(See also, e.g., Nakahara 2003, Exp. 11.5)

Euler forms

For $n = 2k$ and with the curvature form again regarded as a 2-form valued $(2k) \times (2k)$-square matrix

$$F_{\nabla} \;=\; \big( (F_{\nabla})^a{}_b \big)_{1 \leq a,b, \leq 2k}$$

the Euler form is its Pfaffian of this matrix, hence the following sum over permutations $\sigma \in Sym(2k)$ with summands signed by the the signature $sgn(\sigma) \in \{\pm 1\}$:

$$\chi_{2k}(\nabla) \;=\; \frac {(-1)^k} { (4 \pi)^k \cdot k! } \underset{\sigma}{\sum} sgn(\sigma) \cdot (F_{\nabla})_{\sigma(1)\sigma(2)} \wedge (F_{\nabla})_{\sigma(3)\sigma(4)} \wedge \cdots \wedge (F_{\nabla})_{\sigma(2k-1)\sigma(2k)} \,.$$

The first of these is, using the Einstein summation convention and the Levi-Civita symbol:

$$\chi_4(\nabla) \;=\; \frac { \epsilon^{ a b c d} (F_{\nabla})_{a b} \wedge (F_\nabla)_{c d} } {32 \pi^2}$$

(See also, e.g., Nakahara 2003, Exp. 11.7)

Related concepts

• characteristic class

• Chern-Weil homomorphism

• ∞-Lie algebroid valued differential forms

• adjusted Weil algebra

• connection on an ∞-bundle

• curvature

• Bianchi identity

• Chern-Simons form

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, but 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)

• Eckhard Meinrenken, Section 5 of: Group actions on manifolds, Lecture Notes 2003 (pdf, pdf)

  (via equivariant de Rham cohomology)

• Mike Hopkins, Isadore Singer, Section 3.3 of: Quadratic Functions in Geometry, Topology,and M-Theory, J. Differential Geom. 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 16 1 (2012) 149-250 [doi:10.1007/BF02104916, arXiv:1011.4735, euclid:1358950853]

  (in generalization to principal ∞-bundles)

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

  (using the stacky language of FSS 10)

• Adel Rahman: Chern-Weil theory (2017 $[$pdf]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber: Section 8 of: The Character Map in Non-Abelian Cohomology, World Scientific (2023) [doi:10.1142/13422]

  (as a special case of the character map on generalized nonabelian cohomology)

• Araminta Amabel, Arun Debray, Peter Haine, Part II in: Differential Cohomology: Categories, Characteristic Classes, and Connections, based on Fall 2019 talks at MIT’s Juvitop seminar [arXiv:2109.12250]

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:

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

Enhancement of the Chern-Weil homomorphism from ordinary cohomology-groups to dg-categories of $\infty$-local systems:

• Camilo Arias Abad, Santiago Pineda Montoya, Alexander Quintero Velez, Chern-Weil theory for $\infty$-local systems [arXiv:2105.00461]