nLab Chern- and Pontrjagin forms -- section

Chern-, Pontrjagin-, and Euler- characteristic forms

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)

Last revised on July 27, 2021 at 07:16:16. See the history of this page for a list of all contributions to it.