nLab
curvature characteristic form

Contents

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

\infty-Chern-Weil 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.

Definition

Of connection 1-forms

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

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

This form is always an exact form. The (2n1)(2 n -1)-form trivializing it is called a Chern-Simons form.

Notably if GG is a Lie group with Lie algebra 𝔤\mathfrak{g}, PP is the total space of a GG-principal bundle π:PX\pi : P \to X, and AΩ 1(P,𝔤)A \in \Omega^1(P,\mathfrak{g}) is an Ehresmann connection 1-form on PP then by the very definition of the GG-equivariance of AA and the invariance of \langle \cdots \rangle it follows that the curvature form is invariant under the GG-action on PP and is therefore the pullback along π\pi of a 2n2 n-form P nΩ 2n(X)P_n \in \Omega^{2 n}(X) down on XX. 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 XX. 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 PXP \to X a GG-principal bundle write TXT X, TPT P and T vertPT_{vert} P for the tangent Lie algebroid of XX, of PP and the vertical tangent Lie algebroid of PP, respectively. Write inn(𝔤)inn(\mathfrak{g}) for the Lie 2-algebra given by the differential crossed module 𝔤Id𝔤\mathfrak{g}\stackrel{Id}{\to} \mathfrak{g} and finally ib n i\prod_i b^{n_i} \mathbb{R} for the L-∞-algebra with one abelian generator for each generating invariant polynomial of 𝔤\mathfrak{g}

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

Recall that a morphism of ∞-Lie algebroids

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

is equivalently a closed nn-form on XX. The data of an Ehresmann connection on PP then induces the following diagram of ∞-Lie algebroids

T vertP A vert 𝔤 flatverticalform firstEhresmanncondition TP A inn(𝔤) formontotalspace secondEhresmanncondition TX (P n) ib n i curvaturecharacteristicforms. \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

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 XX be a smooth manifold.

Write

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

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

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

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

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

(2)End (V)Mat n×n(). 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×nn \times n matrix algebra with coefficients in the complexification of even-degree differential forms:

Ω 2(X) End (V)Ω 2(X) (Mat n×n())Mat n×n(Ω 2(X) ). \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)IMat n×n(Ω 2(X) ) 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×nn \times n identity matrix and independent of tt.

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

(4)F Ω 2(X) 𝔤ρΩ 2(X) End (V)Ω 2(X) End (V)[t]Mat n×n( Ω 2(X)). 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)[t] \;\simeq\; Mat_{n \times n} \Big( \mathbb{C} \otimes_{\mathbb{R}} \Omega^{2}(X) \Big) \,.

The formulas

Chern forms

The total Chern form c()c(\nabla) is the determinant of the sum of the unit (3) with the curvature form (4), and its component in degree 2k2k, for kk \in \mathbb{N}, is the kkth Chern form c k()c_k(\nabla):

c() kc k()deg=2kdet(I+tiF 2π). c(\nabla) \;\; \coloneqq \;\; \sum_k \underset{ \mathclap{ deg = 2k } }{ \underbrace{ c_k(\nabla) } } \;\; \coloneqq \;\; det \left( I + t \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+iF 2πI + \frac{i F_\nabla}{2\pi}, this being the exponential series in the trace of the Mercator series in iF 2π\frac{i F_\nabla}{2\pi}:

(5)c() =det(I+tiF 2π) =exptrln(I+iF 2π) =exptr(k +1k(F 2πi) k) =exp(k +1k((i) k(2π) ktr(F k))) =1 =+11(itr(F )2π+12tr((F ) 2)(2π) 2i13tr((F ) 3)(2π) 314tr((F ) 4)(2π) 4+) =+12(itr(F )2π+12tr((F ) 2)(2π) 2i13tr((F ) 3)(2π) 314tr((F ) 4)(2π) 4+) 2 =+16(itr(F )2π+12tr((F ) 2)(2π) 2i13tr((F ) 3)(2π) 314tr((F ) 4)(2π) 4+) 3 =+124(itr(F )2π+12tr((F ) 2)(2π) 2i13tr((F ) 3)(2π) 314tr((F ) 4)(2π) 4+) 4 =+ =1 =+itr(F )2π =+12tr((F ) 2)(2π) 2+12(itr(F )2π) 2 =i13tr((F ) 3)(2π) 3+12(2(itr(F )2π)(12tr((F ) 2)(2π) 2))+16((itr(F )2π) 3) =14tr((F ) 4)(2π) 4+12(12tr((F ) 2)(2π) 2) 2+124(itr(F )2π) 4 =+ =1 =+itr(F )2π=c 1() =+tr((F ) 2)(tr(F )) 28π 2=c 2() =+i2tr((F ) 3)+3tr(F )tr((F ) 2)(tr(F )) 348π 3=c 3() =+6tr((F ) 4)+3tr((F ) 2) 2+(tr(F )) 4384π 4=c 4() =+ \begin{aligned} c(\nabla) & \;=\; det \left( I + t \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 )=0tr(F_\nabla) = 0 in these expressions (5) yields the total Pontrjagin form p()p(\nabla) with degree=4k4k-components the Pontrjagin forms p k()p_{k}(\nabla):

p() k(1) kp k()deg=4k =kc 2k()deg=4k =1 =+tr((F ) 2)8π 2=p 1() =+2tr((F ) 4)+tr((F ) 2) 2128π 4=p 2() =+ \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

p 1() =tr((F ) 2)8π 2 p 2() =tr((F ) 2) 22tr((F ) 4)128π 4. \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=2kn = 2k and with the curvature form again regarded as a 2-form valued (2k)×(2k)(2k) \times (2k)-square matrix

F =((F ) a b) 1a,b,2k 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 σSym(2k)\sigma \in Sym(2k) with summands signed by the the signature sgn(σ){±1}sgn(\sigma) \in \{\pm 1\}:

χ 2k()=(1) k(4π) kk!σsgn(σ)(F ) σ(1)σ(2)(F ) σ(3)σ(4)(F ) σ(2k1)σ(2k). \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:

χ 4()=ϵ abcd(F ) ab(F ) cd32π 2 \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)

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 BGB 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(𝔤)H (BG)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:

With an eye towards applications in mathematical physics:

See also in:

Last revised on July 25, 2021 at 11:28:13. See the history of this page for a list of all contributions to it.