nLab Chern class





Special and general types

Special notions


Extra structure





The ordinary Chern classes are the integral characteristic classes

c i:BUB 2i c_i : B U \to B^{2 i} \mathbb{Z}

of the classifying space BUB U of the unitary group.

Accordingly these are characteristic classes in ordinary cohomology of U-principal bundles and hence of complex vector bundle

The first Chern class is the unique characteristic class of circle group-principal bundles.

The analogous classes for the orthogonal group are the Pontryagin classes.

More generally, there are generalized Chern classes for any complex oriented cohomology theory (Adams 74, Lurie 10).



For n1n \geq 1 the universal Chern classes

c iH 2i(BU(n),) c_i \;\in\; H^{2i} \big( B U(n), \mathbb{Z} \big)

of the classifying space B U ( n ) B U(n) of the unitary group are the cohomology classes of B U ( n ) B U(n) in integral cohomology that are characterized as follows:

  1. c 0=1c_0 = 1 and c i=0c_i = 0 if i>ni \gt n;

  2. for n=1n = 1, c 1c_1 is the canonical generator of H 2(BU(1),)H^2(B U(1), \mathbb{Z})\simeq \mathbb{Z};

  3. under pullback along the inclusion i:BU(n)BU(n+1)i : B U(n) \to B U(n+1) we have i *c i (n+1)=c i (n)i^* c_i^{(n+1)} = c_i^{(n)};

  4. under the inclusion BU(k)×BU(l)BU(k+l)B U(k) \times B U(l) \to B U(k+l) we have i *c i= j=0 ic ic jii^* c_i = \sum_{j = 0}^i c_i \cup c_{j-i}.

The corresponding total Chern class is the formal sum

c1+c 1+c 2+kH 2k(BU(n)) c \;\coloneqq\; 1 + c_1 + c_2 + \cdots \;\in\; \underset{k}{\prod} H^{2k} \big( B U(n) \big)




The cohomology ring of the classifying space B U ( n ) B U(n) (for the unitary group U(n)U(n)) is the polynomial ring on generators {c k} k=1 n\{c_k\}_{k = 1}^{n} of degree 2, called the Chern classes

H (BU(n),)[c 1,,c n]. H^\bullet(B U(n), \mathbb{Z}) \simeq \mathbb{Z}[c_1, \cdots, c_n] \,.

Moreover, for Bi:BU(n 1)BU(n 2)B i \colon B U(n_1) \longrightarrow BU(n_2) the canonical inclusion for n 1n 2n_1 \leq n_2 \in \mathbb{N}, then the induced pullback map on cohomology

(Bi) *:H (BU(n 2))H (BU(n 1)) (B i)^\ast \;\colon\; H^\bullet(B U(n_2)) \longrightarrow H^\bullet(B U(n_1))

is given by

(Bi) *(c k)={c k for1kn 1 0 otherwise. (B i)^\ast(c_k) \;=\; \left\{ \array{ c_k & for \; 1 \leq k \leq n_1 \\ 0 & otherwise } \right. \,.

(e.g. Kochman 96, theorem 2.3.1)


For n=1n = 1, in which case BU(1)P B U(1) \simeq \mathbb{C}P^\infty is the infinite complex projective space, we have (prop)

H (BU(1))[c 1], H^\bullet(B U(1)) \simeq \mathbb{Z}[ c_1 ] \,,

where c 1c_1 is the first Chern class. From here we proceed by induction. So assume that the statement has been shown for n1n-1.

Observe that the canonical map BU(n1)BU(n)B U(n-1) \to B U(n) has as homotopy fiber the (2n-1)sphere (prop.) hence there is a homotopy fiber sequence of the form

S 2n1BU(n1)BU(n). S^{2n-1} \longrightarrow B U(n-1) \longrightarrow B U(n) \,.

Consider the induced Thom-Gysin sequence.

In odd degrees 2k+1<2n2k+1 \lt 2n it gives the exact sequence

H 2k(BU(n1))H 2k+12n(BU(n))0H 2k+1(BU(n))(Bi) *H 2k+1(BU(n1))0, \cdots \to H^{2k}(B U(n-1)) \longrightarrow \underset{\simeq 0}{\underbrace{H^{2k+1-2n}(B U(n))}} \longrightarrow H^{2k+1}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} \underset{\simeq 0}{\underbrace{H^{2k+1}(B U(n-1))}} \to \cdots \,,

where the right term vanishes by induction assumption, and the middle term since ordinary cohomology vanishes in negative degrees. Hence

H 2k+1(BU(n))0for2k+1<2n H^{2k+1}(B U(n)) \simeq 0 \;\;\; for \; 2k+1 \lt 2n

Then for 2k+1>2n2k+1 \gt 2n the Thom-Gysin sequence gives

H 2k+12n(BU(n))H 2k+1(BU(n))(Bi) *H 2k+1(BU(n1))0, \cdots \to H^{2k+1-2n}(B U(n)) \longrightarrow H^{2k+1}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} \underset{\simeq 0}{\underbrace{H^{2k+1}(B U(n-1))}} \to \cdots \,,

where again the right term vanishes by the induction assumption. Hence exactness now gives that

H 2k+12n(BU(n))H 2k+1(BU(n)) H^{2k+1-2n}(B U(n)) \overset{}{\longrightarrow} H^{2k+1}(B U(n))

is an epimorphism, and so with the previous statement it follows that

H 2k+1(BU(n))0 H^{2k+1}(B U(n)) \simeq 0

for all kk.

Next consider the Thom Gysin sequence in degrees 2k2k

H 2k1(BU(n1))0H 2k2n(BU(n))H 2k(BU(n))(Bi) *H 2k(BU(n1))H 2k+12n(BU(n))0. \cdots \to \underset{\simeq 0}{\underbrace{H^{2k-1}(B U(n-1))}} \longrightarrow H^{2k-2n}(B U(n)) \longrightarrow H^{2k}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} H^{2k}(B U(n-1)) \longrightarrow \underset{\simeq 0}{\underbrace{H^{2k +1 - 2n}(B U(n))}} \to \cdots \,.

Here the left term vanishes by the induction assumption, while the right term vanishes by the previous statement. Hence we have a short exact sequence

0H 2k2n(BU(n))H 2k(BU(n))(Bi) *H 2k(BU(n1))0 0 \to H^{2k-2n}(B U(n)) \longrightarrow H^{2k}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} H^{2k}(B U(n-1)) \to 0

for all kk. In degrees 2n\bullet\leq 2n this says

0c n()H 2n(BU(n))(Bi) *([c 1,,c n1]) 2n0 0 \to \mathbb{Z} \overset{c_n \cup (-)}{\longrightarrow} H^{\bullet \leq 2n}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} (\mathbb{Z}[c_1, \cdots, c_{n-1}])_{\bullet \leq 2n} \to 0

for some Thom class c nH 2n(BU(n))c_n \in H^{2n}(B U(n)), which we identify with the next Chern class.

Since free abelian groups are projective objects in Ab, their extensions are all split (the Ext-group out of them vanishes), hence the above gives a direct sum decomposition

H 2n(BU(n)) ([c 1,,c n1]) 2n2n ([c 1,,c n]) 2n. \begin{aligned} H^{\bullet \leq 2n}(B U(n)) & \simeq (\mathbb{Z}[c_1, \cdots, c_{n-1}])_{\bullet \leq 2n} \oplus \mathbb{Z}\langle 2n\rangle \\ & \simeq (\mathbb{Z}[c_1, \cdots, c_{n}])_{\bullet \leq 2n} \end{aligned} \,.

Now by another induction over these short exact sequences, the claim follows.

First Chern class

Top Chern class

For 𝒱 X\mathcal{V}_X a complex vector bundle of complex rank nn, the highest degree Chern class that may generally be non-vanishing is c nc_n. This is hence often called the top Chern class of the vector bundle.


The top Chern class of a complex vector bundle 𝒱 X\mathcal{V}_X equals the Euler class ee of the underlying real vector bundle 𝒱 X \mathcal{V}^{\mathbb{R}}_X:

𝒱 Xhas complex ranknc n(𝒱 X)=e(𝒱 X )H 2n(X;). \mathcal{V}_X \; \text{has complex rank}\;n \;\;\;\;\; \Rightarrow \;\;\;\;\; c_n \big( \mathcal{V}_X \big) \;\; = \;\; e \big( \mathcal{V}^{\mathbb{R}}_X \big) \;\;\;\; \in H^{2n} \big( X; \, \mathbb{Z} \big) \,.

(e.g. Bott-Tu 82 (20.10.6))


The top Chern class of a complex vector bundle 𝒱 X\mathcal{V}_X equals the pullback of any Thom class thH 2n(𝒱 X;)th \;\in\; H^{2n}\big( \mathcal{V}_X; \mathbb{Z} \big) on 𝒱 X\mathcal{V}_X along the zero-section:

𝒱 Xhas complex ranknc n(𝒱 X)=(0 X) *(th)H 2n(X;) \mathcal{V}_X \; \text{has complex rank}\;n \;\;\;\;\; \Rightarrow \;\;\;\;\; c_n \big( \mathcal{V}_X \big) \;\;\; = \;\;\; (0_X)^\ast (th) \;\; \in \; H^{2n} \big( X ; \, \mathbb{Z} \big)

(e.g. Bott-Tu 82, Prop. 12.4)

Splitting principle and Chern roots

Under the splitting principle all Chern classes are determined by first Chern classes:

Write i:TU(1) nU(n)i \colon T \simeq U(1)^n \hookrightarrow U(n) for the maximal torus inside the unitary group, which is the subgroup of diagonal unitary matrices. Then

H (BT,)H (BU(1) n,) H^\bullet(B T, \mathbb{Z}) \simeq H^\bullet(B U(1)^n, \mathbb{Z})

is the polynomial ring in nn generators (to be thought of as the universal first Chern classes c ic_i of each copy of BU(1)B U(1); equivalently as the weights of the group characters of U(n)U(n)) which are traditionally written x ix_i:

H (BU(1) n,)[x 1,,x n]. H^\bullet(B U(1)^n, \mathbb{Z}) \simeq \mathbb{Z}[x_1, \cdots, x_n] \,.


Bi:BU(1) nBU(n) B i \;\colon\; B U(1)^n \to B U(n)

for the induced map of deloopings/classifying spaces, then the kk-universal Chern class c kH 2k(BU(n),)c_k \in H^{2k}(B U(n), \mathbb{Z}) is uniquely characterized by the fact that its pullback to BU(1) nB U(1)^n is the kkth elementary symmetric polynomial σ k\sigma_k applied to these first Chern classes:

(Bi) *(c k)=σ k(x 1,,x n). (B i)^\ast (c_k) = \sigma_k(x_1, \cdots, x_n) \,.

Equivalently, for c= i=1 nc kc = \sum_{i = 1}^n c_k the formal sum of all the Chern classes, and using the fact that the elementary symmetric polynomials σ k(x 1,,k n)\sigma_k(x_1, \cdots, k_n) are the degree-kk piece in (1+x 1)(1+x n)(1+x_1) \cdots (1+x_n), this means that

(Bi) *(c)=(1+x 1)(1+x 2)(1+x n). (B i)^\ast (c) = (1+x_1) (1+ x_2) \cdots (1+ x_n) \,.

Since here on the right the first Chern classes x ix_i appear as the roots of the Chern polynomial, they are also called Chern roots.

See also at splitting principle – Examples – Complex vector bundles and their Chern roots.

(e.g. Kochman 96, theorem 2.3.2, tom Dieck 08, theorem 19.3.2)


For nn \in \mathbb{N} let Bι n:B(U(1) n)BU(n)B \iota_n \;\colon\; B (U(1)^n) \longrightarrow B U(n) be the canonical map. Then the induced pullback operation on ordinary cohomology

(Bι n):H (BU(n);)H (BU(1) n;) \left( B \iota_n \right) \;\colon\; H^\bullet( B U(n); \mathbb{Z} ) \longrightarrow H^\bullet( B U(1)^n; \mathbb{Z} )

is a monomorphism.

A proof of lemma , via analysis of the Serre spectral sequence of U(n)/U(1) nBU(1) nBU(n)U(n)/U(1)^n \to B U(1)^n \to B U(n) is indicated in (Kochman 96, p. 40). A proof via transfer of the Euler class of U(n)/U(1) nU(n)/U(1)^n, following (Dupont 78, (8.28)), is indicated at splitting principle (here).


For knk \leq n \in \mathbb{N} let Bι n:B(U(1) n)BU(n)B \iota_n \;\colon\; B (U(1)^n) \longrightarrow B U(n) be the canonical map. Then the induced pullback operation on ordinary cohomology is of the form

(Bi n) *:[c 1,,c k][(c 1) 1,(c 1) n] (B i_n)^\ast \;\colon\; \mathbb{Z}[c_1, \cdots, c_k] \longrightarrow \mathbb{Z}[(c_1)_1,\cdots (c_1)_n]

and sends the kkth Chern class c kc_k (def. ) to the kkth elementary symmetric polynomial in the nn copies of the first Chern class:

(Bi n) *:c kσ k((c 1) 1,,(c 1) n). (B i_n)^\ast \;\colon\; c_k \mapsto \sigma_k( (c_1)_1, \cdots, (c_1)_n ) \,.

First consider the case n=1n = 1.

The classifying space BU(1)B U(1) is equivalently the infinite complex projective space P \mathbb{C}P^\infty. Its ordinary cohomology is the polynomial ring on a single generator c 1c_1, the first Chern class (prop.)

H (BU(1))[c 1]. H^\bullet(B U(1)) \simeq \mathbb{Z}[ c_1 ] \,.

Moreover, Bi 1B i_1 is the identity and the statement follows.

Now by the Künneth theorem for ordinary cohomology (prop.) the cohomology of the Cartesian product of nn copies of BU(1)B U(1) is the polynomial ring in nn generators

H (BU(1) n)[(c 1) 1,,(c 1) n]. H^\bullet(B U(1)^n) \simeq \mathbb{Z}[(c_1)_1, \cdots, (c_1)_n] \,.

By prop. the domain of (Bi n) *(B i_n)^\ast is the polynomial ring in the Chern classes {c i}\{c_i\}, and by the previous statement the codomain is the polynomial ring on nn copies of the first Chern class

(Bi n) *:[c 1,,c n][(c 1) 1,,(c 1) n]. (B i_n)^\ast \;\colon\; \mathbb{Z}[ c_1, \cdots, c_n ] \longrightarrow \mathbb{Z}[ (c_1)_1, \cdots, (c_1)_n ] \,.

This allows to compute (Bi n) *(c k)(B i_n)^\ast(c_k) by induction:

Consider n2n \geq 2 and assume that (Bi n1) n1 *(c k)=σ k((c 1) 1,,(c 1) (n1))(B i_{n-1})^\ast_{n-1}(c_k) = \sigma_k((c_1)_1, \cdots, (c_1)_{(n-1)}). We need to show that then also (Bi n) *(c k)=σ k((c 1) 1,,(c 1) n)(B i_n)^\ast(c_k) = \sigma_k((c_1)_1,\cdots, (c_1)_n).

Consider then the commuting diagram

BU(1) n1 Bi n1 BU(n1) Bj t^ Bi t^ BU(1) n Bi n BU(n) \array{ B U(1)^{n-1} &\overset{ B i_{n-1} }{\longrightarrow}& B U(n-1) \\ {}^{\mathllap{B j_{\hat t}}}\downarrow && \downarrow^{\mathrlap{B i_{\hat t}}} \\ B U(1)^n &\underset{B i_n}{\longrightarrow}& B U(n) }

where both vertical morphisms are induced from the inclusion

n1 n \mathbb{C}^{n-1} \hookrightarrow \mathbb{C}^n

which omits the ttth coordinate.

Since two embeddings i t^ 1,i t^ 2:U(n1)U(n)i_{\hat t_1}, i_{\hat t_2} \colon U(n-1) \hookrightarrow U(n) differ by conjugation with an element in U(n)U(n), hence by an inner automorphism, the maps Bi t^ 1B i_{\hat t_1} and B i^ t 2B_{\hat i_{t_2}} are homotopic, and hence (Bi t^) *=(Bi n^) *(B i_{\hat t})^\ast = (B i_{\hat n})^\ast, which is the morphism from prop. .

By that proposition, (Bi t^) *(B i_{\hat t})^\ast is the identity on c k<nc_{k \lt n} and hence by induction assumption

(Bi n1) *(Bi t^) *c k<n =(Bi n1) *c k<n =σ k((c 1) 1,,(c 1) t^,,(c 1) n). \begin{aligned} (B i_{n-1})^\ast (B i_{\hat t})^\ast c_{k \lt n} &= (B i_{n-1})^\ast c_{k \lt n} \\ = \sigma_k( (c_1)_1, \cdots, \widehat{(c_1)_t}, \cdots, (c_1)_n ) \end{aligned} \,.

Since pullback along the left vertical morphism sends (c 1) t(c_1)_t to zero and is the identity on the other generators, this shows that

(Bi n) *(c k<n)σ k<n((c 1) 1,,(c 1) t^,,(c 1) n)mod(c 1) t. (B i_n)^\ast(c_{k \lt n}) \simeq \sigma_{k\lt n}((c_1)_1, \cdots, \widehat{(c_1)_t}, \cdots, (c_1)_n) \;\; mod (c_1)_t \,.

This implies the claim for k<nk \lt n.

For the case k=nk = n the commutativity of the diagram and the fact that the right map is zero on c nc_n by prop. shows that the element (Bj t^) *(Bi n) *c n=0(B j_{\hat t})^\ast (B i_n)^\ast c_n = 0 for all 1tn1 \leq t \leq n. But by lemma the morphism (Bi n) *(B i_n)^\ast, is injective, and hence (Bi n) *(c n)(B i_n)^\ast(c_n) is non-zero. Therefore for this to be annihilated by the morphisms that send (c 1) t(c_1)_t to zero, for all tt, the element must be proportional to all the (c 1) t(c_1)_t. By degree reasons this means that it has to be the product of all of them

(Bi n) *(c n) =(c 1) 1(c 1) 2(c 1) n =σ n((c 1) 1,,(c 1) n). \begin{aligned} (B i_n)^{\ast}(c_n) & = (c_1)_1 \otimes (c_1)_2 \otimes \cdots \otimes (c_1)_n \\ & = \sigma_n( (c_1)_1, \cdots, (c_1)_n ) \end{aligned} \,.

This completes the induction step.

Whitney sum formula


For knk\leq n \in \mathbb{N}, consider the canonical map

μ k,nk:BU(k)×BU(nk)BU(n) \mu_{k,n-k} \;\colon\; B U(k) \times B U(n-k) \longrightarrow B U(n)

(which classifies the Whitney sum of complex vector bundles of rank kk with those of rank nkn-k). Under pullback along this map the universal Chern classes (prop. ) are given by

(μ k,nk) *(c t)=i=0tc ic ti, (\mu_{k,n-k})^\ast(c_t) \;=\; \underoverset{i = 0}{t}{\sum} c_i \otimes c_{t-i} \,,

where we take c 0=1c_0 = 1 and c j=0H (BU(r))c_j = 0 \in H^\bullet(B U(r)) if j>rj \gt r.

So in particular

(μ k,nk) *(c n)=c kc nk. (\mu_{k,n-k})^\ast(c_n) \;=\; c_k \otimes c_{n-k} \,.

e.g. (Kochman 96, corollary 2.3.4)


Consider the commuting diagram

H (BU(n)) μ k,nk * H (BU(k))H (BU(nk)) μ k * μ k *μ nk * H (BU(1) n) H (BU(1) k)H (BU(1) nk). \array{ H^\bullet( B U(n) ) &\overset{\mu_{k,n-k}^\ast}{\longrightarrow}& H^\bullet( B U(k) ) \otimes H^\bullet( B U(n-k) ) \\ {}^{\mathllap{\mu_k^\ast}}\downarrow && \downarrow^{\mathrlap{ \mu_{k}^\ast \otimes \mu_{n-k}^\ast }} \\ H^\bullet( B U(1)^n ) &\simeq& H^\bullet( B U(1)^k ) \otimes H^\bullet( B U(1)^{n-k} ) } \,.

This says that for all tt then

(μ k *μ nk *)μ k,nk *(c t) =μ n *(c t) =σ t((c 1) 1,,(c 1) n), \begin{aligned} (\mu_k^\ast \otimes \mu_{n-k}^\ast) \mu_{k,n-k}^\ast(c_t) & = \mu^\ast_n(c_t) \\ & = \sigma_t((c_1)_1, \cdots, (c_1)_n) \end{aligned} \,,

where the last equation is by prop. .

Now the elementary symmetric polynomial on the right decomposes as required by the left hand side of this equation as follows:

σ t((c 1) 1,,(c 1) n)=r=0tσ r((c 1) 1,,(c 1) nk)σ tr((c 1) nk+1,,(c 1) n), \sigma_t((c_1)_1, \cdots, (c_1)_n) \;=\; \underoverset{r = 0}{t}{\sum} \sigma_r((c_1)_1, \cdots, (c_1)_{n-k}) \cdot \sigma_{t-r}( (c_1)_{n-k+1}, \cdots, (c_1)_n ) \,,

where we agree with σ q((c 1) 1,,(c 1) p)=0\sigma_q((c_1)_1, \cdots, (c_1)_p) = 0 if q>pq \gt p. It follows that

(μ k *μ nk *)μ k,nk *(c t)=(μ k *μ nk *)(r=0tc rc tr). (\mu_k^\ast \otimes \mu_{n-k}^\ast) \mu_{k,n-k}^\ast(c_t) = (\mu_k^\ast \otimes \mu_{n-k}^\ast) \left( \underoverset{r=0}{t}{\sum} c_r \otimes c_{t-r} \right) \,.

Since (μ k *μ nk *)(\mu_k^\ast \otimes \mu_{n-k}^\ast) is a monomorphism by lemma , this implies the claim.

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.


Let XX be a smooth manifold.


(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)


Chern classes of linear representations

Under the Atiyah-Segal completion map linear representations of a group GG induce K-theory classes on the classifying space BGB G. Their Chern classes are hence invariants of the linear representations themselves.

See at characteristic class of a linear representation for more.

In Yang-Mills theory field configurations with non-vanishing second Chern class (and minimal energy) are called instantons. The second Chern class is the instanton number . For more on this see at SU(2)-instantons from the correct maths to the traditional physics story.


Original articles:

Textbook accounts:

With an eye towards mathematical physics:

A brief introduction is in chapter 23, section 7

For Conner-Floyd Chern classes in complex oriented cohomology theory:

See also

  • Dupont, Curvature and characteristic classes, Springer 1978

  • Timothy Hosgood, Chern classes of coherent analytic sheaves: a simplicial approach. Université d’Aix-Marseille (AMU), 2020. tel-02882140.

A MathOverflow question discussing references:

Last revised on March 4, 2024 at 22:43:16. See the history of this page for a list of all contributions to it.