Chern-Weil Theory | Itinerarium Mentis

Setting and Review

For $E\rightarrow M$ a smooth complex vector bundle over a mooth compact manifold $M$, put

$\Omega^\bullet(M,E):= \Gamma(\wedge^\bullet(T^\ast M)\otimes E)$.
$\nabla:\Gamma(E)\rightarrow \Omega^1(M;E)$ is a connection on $E$.
- It is $\mathbb{C}$-linear. $\nabla(fp)=(df)p+f\nabla p$.
$\nabla_X:\Gamma(E)\rightarrow \Gamma(E)$ is the covariant derivative of $\nabla$ along $X\in \Gamma(TM)$ (induced by contraction between $TM$ and $T^\ast M$).
- $\nabla_X(fp) = df(X) p + f\cdot\nabla p(X)$
The extended map on the differential forms $\nabla:\Omega^\bullet(M;E)\rightarrow \Omega^{\bullet+1}(M;E)$ (which we also denote as $\nabla$)
- $\nabla \omega p = (d\omega)p + (-1)^{\deg\omega}\omega\wedge \nabla p$.
- It would be better to understand the domain of the original $\nabla$ as $\Omega^0(M;E)$.
The curvature of $\nabla$ is $R = \nabla\circ\nabla:\Gamma(E)\rightarrow \Omega^2(M;E)$ which we write as $R=\nabla^2$.
- It is $C^\infty(M)$ linear.
- $R\in \Omega^2(M;\text{End}(E)) = \Gamma((\wedge^2 T^\ast M )\otimes\text{End}(M))$ where $\text{End}(M)$ is the bundle of fiberwise endomorphisms of $E$.
- $R(X,Y) = [\nabla_X,\nabla_Y] - \nabla_{[X,Y]}$. Where $[\cdot,\cdot]$ is understood either as a commutator or as a Lie bracket.
- $R^k:\Gamma(E)\rightarrow \Omega^{2k}(M,E)$ is well defined.

The trace function $tr:\Omega^\bullet(M;\text{End}(M))\rightarrow \Omega^\bullet(M)$ is simply $\omega T \mapsto \omega tr[T]$.

We extend the Lie bracket on $\text{End(E)}$ to $\Omega^\bullet(M;\text{End}(E))$ through defining $[\omega A,\eta B] = (\omega A)\wedge(\eta B) - (-1)^{|\omega||\eta|}(\eta B)\wedge(\omega A)$.

Hence we have obviously that $tr([A,B])=0$ for any $A,B\in \Omega^\bullet(M;\text{End}(M))$.

Given $\nabla,\nabla’$ two connections, we have $\nabla-\nabla’ \in \omega^1(M,\text{End}(M))$, hence $tr([\nabla-\nabla’,A])=0$ for any $A\in\Omega^\bullet(M,\text{End}(E))$. $tr([\nabla,A])$ is then independent of the choice of $\nabla$. Locally we can take a trivial connection $d$, then $tr[(\nabla-d),A]=0$, and hence $d tr [A] = tr([\nabla,A])$ (note that $d p$ vanishes where $p\in \Gamma(E)$). By the independence of the choice of the connection this holds globally.

Chern-Weil Theorem

Let $f(x) = a_0 + a_1 x + …$ be a power series in one variable,

The form $tr[f(R)]$ is closed, where $R$ is the curvature associated to a connection $\nabla$.
Given $(\nabla,R)$ and $(\nabla’,R’)$, there is an $\omega \in \Omega^\bullet(M)$, s.t. $$ tr[f(R)] - tr[f(R’)] = d\omega. $$

In fact $\omega = -d \int_0^1 tr[\frac{d\nabla_t}{dt}f’(R_t)]dt$ where $\nabla_t = (1-t)\nabla + t\nabla’$.

Thus the characteristic form $tr[f(\frac{\sqrt{-1}}{2\pi)}R]$ is a closed form whose cohomology class doesn’t depend on the choice of $\nabla$. We denote the form itself by $f(E,\nabla)$, and the class $f(E)$.

Obviously the product and sum of characteristic forms are characteristic forms. On an oriented manifold the top form component of a characteristic form can be integrated to the characteristic number associated, which is obviously independent on the choices of the connection.

Examples

Chern forms and classes. On a complex vector bundle with connection $(E\rightarrow M,\nabla)$, the total Chern form is $c(E,\nabla) = \det(1 + \frac{\sqrt{-1}}{2\pi} R)$ (Note that $\det(A) = \exp tr(\log A)$), which can be decomposed into $1+\sum_i c_i(E,\nabla)$ where $c_i\in \Omega^{2i}(M)$ is the $i$-th Chern form.
- We see that $c_1(M) = \frac{1}{2\pi i} [R]$
On a real vector bundle the Pontrjagin form $p(E,\nabla) = \det([1 - (\frac{\sqrt{-1}}{2\pi} R)^2]^{1/2})$. $i$-th Pontrjagin forms $p_i(E,\nabla) \in \Omega^{4i}(M)$ are defined in a similar way to $i$-th Chern form.
For complex vector bundle $E\rightarrow M$ on a compact manifold $M$, the Chern character form $ch(E,\nabla) = tr[\exp(\frac{\sqrt{-1}}{2\pi}R)] \in \Omega^{\text{even}}(M)$.
- Note that for the whitney $E\oplus F$ the curvature form is the direct sum $R^E \oplus R^F$, hence $ch(E\oplus F) = ch(E) + ch(F)$. This means $ch$ is an abelian group homomorphism $ch:K(M)\rightarrow H^{\text{even}}(M;\mathbb{C})$. Atiyah and Hirzebruch proved that upon ignoring the torsions in $K(M)$ we have an isomorphism, i.e. $ch:K(M)\otimes \mathbb{C} \rightarrow H^\text{even}(M;\mathbb{C})$ is an isomorphism.
- For the tensor product $E\otimes F$, the connection is $\nabla^E \otimes 1 + 1\otimes \nabla^F$, with the curvature $R^E\otimes 1 + 1\otimes R^F$, hence $ch(E\otimes F) = ch(E)ch(F)$.

Chern-Simons Transgressed term and form

We had, for $f$ a power series $$ tr[f(R)] - tr[f(R’)] = -d\int_0^1 tr[\frac{d\nabla_t}{dt}f’(R_t)]dt=d\omega $$ where $\nabla_t = (1-t)\nabla + t\nabla’$. The transgressed term $\omega$ is called a Chern-Simons term. Sometimes it is closed and induces a de Rham cohomology class. For example, when $\nabla$ and $\nabla’$ are flat, so $R=R’=0$.

On a tangent bundle $TM$ over a smooth compact oriented 3-manifold $TM$ (which by a theorem of Stiefel is trivial), this becomes precisely the Chern-Simons form. $$ tr[A\wedge d A + \frac{2}{3}A\wedge A \wedge A] $$ where $d$ is the connection on $TM$ defined by $d(f_i e_i) = df_i e_i$ where ${e_i}$ is a global basis, and $A\in \Omega^1(M;\text{End}(TM))$.

Atiyah Sequence

This quote is from Lie Groupoids and Lie Algebroids in Differential Geometry

The first advantage of the Atiyah sequence concept is that it allows the standard definitions and basic properties of infinitesimal connections and their curvature forms to be presented quickly and clearly, in an algebraically natural manner. The correspondence between the two standard definitions of a connection is seen to be a particular case of the correspondence between right- and left-split maps in an exact sequence; curvature is seen to measure precisely the extent to which a connection fails to preserve Lie brackets; associated connections, the Bianchi identities and the structural equation appear in a clear and natural algebraic manner. This approach also allows that infinitesimal connection theory should be ragarded not so much as a theory about principal bundles as about their first-order approimations- the Atiyah sequence or Lie algebroid.

Definition(Atiyah algebroid) . For a principal $G$-bundle $\pi:P\to M$, the bundle $A(P)=TP/G\to M$ is a Lie algebroid, called the Atiyah algebroid of $P$. The bracket on $\Gamma(A)$ is induced from its identification with $G$-invariant vector fields on $P$. The anchor map is induced from the bundle projection $T\pi:TP\to TM$.

Recall that a Lie algebroid is just a vector bundle $A\to M$ with a Lie bracket on $\Gamma(A)$ s.t. the anchor map $\alpha: A\to TM$ satisfies the Leibniz rule $$ [\sigma,f\xi] = f[\sigma,\xi] + (\alpha(\sigma)f)\xi $$ where $\sigma,\xi\in \Gamma(A)$ and $f\in C^\infty(M)$.

Here we see that algebraically $[\sigma,\cdot]$ is similar to $\nabla$. For a vector bundle $E\to M$, $$ \begin{gather*} \nabla_X fp = f \nabla_X p+(df)p \\ [\sigma, f\xi] = f [\sigma,\xi] + (\alpha(\sigma)f)\xi \end{gather*} $$ with $p\in \Gamma(E), f\in C^\infty(M)$, $X\in \Gamma(TM)$ and $\sigma, \xi\in \Gamma(E)$.

Indeed, the Atiyah algebroid fits into an exact sequence that encodes the principal connection and curvature.

Definition(Atiyah Sequence) . For a principal $G$-bundle $P\to M$, there is an exact sequence of Lie algebroids, $$ 0\to \mathfrak{gau}(P) \to A(P) \to TM\to 0 $$ where $\mathfrak{gau}(P) = (P\times \mathfrak{g})/G$ is the quotient of the vertical bundle $T_v P \cong P\times \mathfrak{g}$ by $G$ and thus is the bundle of infinitesimal gauge transformations, and $A(P)$ is the Atiyah algebroid. This is called the Atiyah sequence of $P$.

A splitting $j: TM\to A(P)$ of the Atiyah sequence is equivalent to a principal bundle connection.
Given vector fields $X,Y\in \Gamma(TM)$, since $[j(X),j(Y)]$ is a lift of $[X,Y]$, the difference $[j(X),j(Y)] - j([X,Y]) \in \Gamma(\mathfrak{gau}(O))$. The 2-form $F\in \Omega^2(M,\mathfrak{gau}(P))$ with $F(X,Y) = [j(X),j(Y)]-j([X,Y])$ is the curvature form of the connection.