I’d like to finish the series I started a while back on Chern-Weil theory (and then get back to exponential sums).

So, in the discussion of the Cartan formalism a few days back, we showed that given a vector bundle $E$ with a connection on a smooth manifold, we can associate with it a curvature form, which is an $\hom(E, E)$-valued 2-form; this is a generalization of the Riemann curvature tensor (as some computations that I don’t feel like posting here will show). In the case of a line bundle, we saw that since $\hom(E, E)$ was canonically trivialized, we could interpret the curvature form as a plain old 2-form, and in fact it turned out to be a representative — in de Rham cohomology — of the first Chern class of the line bundle. Now we want to see what to do for a vector bundle, where there are going to be a whole bunch of Chern classes.

For a general vector bundle, the curvature ${\Theta}$ (of a connection) will not in itself be a form, but rather a differential form with coefficients in ${\hom(E, E)}$, which is generally not a trivial bundle. In order to get a differential form from this, we shall have to apply an invariant polynomial. In this post, I’ll describe the proof that one indeed gets well-defined characteristic classes (that are actually independent of the connection), and that they coincide with the usually defined topological Chern classes.

1. Invariant polynomials

To get ordinary differential forms from the curvature, we need a polynomial that can be evaluated at $\hom(V, V)$ for a finite-dimensional vector space $V$, for any such vector space (of fixed dimension).

Definition 1 An invariant ${k}$-linear form on ${n}$-by-${n}$ matrices is a multilinear map ${\phi(\cdot, \dots, \cdot) \rightarrow \mathbb{C}}$, whose ${k}$ inputs are elements of the ring of ${n}$-by-${n}$ complex matrices ${M_n(\mathbb{C})}$; this is required to satisfy

$\displaystyle \phi(A_1, \dots, A_k) = \phi( gA_1g^{-1}, \dots, g A_k g^{-1}), \quad A_1, \dots, A_k \in M_n(\mathbb{C}), g \in \mathrm{GL}_n(\mathbb{C}).$

We shall write ${\Phi}$ for the function ${M_n(\mathbb{C}) \rightarrow \mathbb{C}}$ given by ${\Phi(A) = \phi(A, A, \dots, A)}$.

Given such a form, and given a global section ${\tau}$ of ${\hom(E, E) \otimes \bigwedge^2 T_{\mathbb{C}}^*}$, we can define ${\Phi(\tau) \in \mathcal{A}^{2k}(M)}$. Indeed, ${\Phi}$ induces a canonical map (not linear) ${\hom(E, E) \rightarrow \mathbb{C}}$. The invariance with respect to conjugation assures this: it does not depend which basis we choose for a fiber of ${E}$. More generally, given ${\tau_1, \dots, \tau_k \in (\hom(E, E) \otimes \bigwedge^2 T_{\mathbb{C}}^*)(M)}$, we can define an ordinary differential form ${\phi(\tau_1, \dots, \tau_k) \in \mathcal{A}^{2k}(M)}$. Note the identity

$\displaystyle d \phi(\tau_1, \dots, \tau_k) = \sum_i \phi(\tau_1, \dots, d \tau_i, \dots, \tau_k),$

which follows from the identity for ${d(\omega \wedge \eta)}$ and the fact that everything has even degree. This will be how we use the curvature form to get differential forms.

Let us note that any invariant homogeneous polynomial of degree ${k}$, ${\Phi: M_n(\mathbb{C}) \rightarrow \mathbb{C}}$, can be obtained by restricting a multilinear map ${M_n(\mathbb{C})^k \rightarrow \mathbb{C}}$ to the diagonal. The construction is explicit combinatorics; in fact, ${\Phi}$ is enough to determine ${\phi}$. We omit the details (they won’t be necessary).

In the course of the next proof, we shall need:

Lemma 2 Let ${A_1, \dots, A_k \in M_n(\mathbb{C})}$, and let ${B \in M_n(\mathbb{C})}$. We have:

$\displaystyle \sum_{i=1}^k \phi( A_1, \dots, A_{i-1}, [B, A_i], A_{i+1}, \dots, A_k) = 0 .\ \ \ \ \ (1)$

Proof: Indeed, we have by invariance:

$\displaystyle \phi( \exp(tB) A_1 \exp(-tB), \dots, \exp(tB) A_k \exp(-tB)) = \phi(A_1, \dots, A_n).$

Differentiating with respect to ${t}$ now gives the result, because ${\frac{d}{dt}|_{t=0} \exp(tB) A \exp(-tB) = [B, A]}$.

$\Box$

2. The Chern-Weil homomorphism

Let ${M}$ be a smooth manifold, ${E \rightarrow M}$ a smooth vector bundle of dimension ${n}$. We saw in the last subsection that, given a global ${E}$-valued 2-form, the application of an invariant polynomial ${\Phi}$ allows one to obtain a global ${2k}$-form (with no twisting). The curvature of any connection is such a global ${E}$-valued 2-form.

Theorem 3 Let ${\phi}$ be an invariant ${k}$-linear form ${M_n(\mathbb{C})^k \rightarrow \mathbb{C}}$. For any connection ${\nabla}$ on ${E}$ with curvature form ${\Theta}$, the ${2k}$-form ${\Phi(\Theta)}$ is a closed form. The cohomology class of ${\Phi(\Theta)}$ is independent of the choice of connection.

Proof: The first assertion (closedness) is local, so let us work in a fixed frame ${\mathfrak{F}}$ where the connection matrix is ${\theta(\mathfrak{F})}$ and the curvature matrix is ${\Theta(\mathfrak{F})}$. Then ${\Phi(\Theta) = \Phi(\Theta(\mathfrak{F}))}$, locally, where ${\Theta(\mathfrak{F})}$ is considered as a matrix of forms and ${\Phi}$ a function on matrices.

We have $\Phi(\Theta(\mathfrak{F})) = \phi(\Theta(\mathfrak{F}), \dots, \Theta(\mathfrak{F})) ,$ so, because we are working with even-degree forms,

This proves that ${\Phi(\Theta)}$ is a closed form.

Now we need to show that the cohomology class is independent of the choice of connection. Suppose given two connections ${\nabla_0, \nabla_1}$ on ${E}$. The strategy will be to consider the one-parameter family of connections ${\nabla_t = (1-t) \nabla_0 + t \nabla_1}$ and the respective one-parameter family of curvature forms ${\Theta_t}$ (which will vary smoothly). We will show that the cohomology class of ${\Phi(\Theta_t)}$ is constant in ${t}$.

To do this, consider the vector bundle ${E' \rightarrow M \times [0, 1]}$ given by pull-back of ${E}$. We define a connection ${\nabla'}$ on ${E}$ by ${\nabla' = (1-t) \nabla_0 + t \nabla_1}$ on ${M \times \left\{t\right\}}$. Namely, we pull back ${\nabla_0}$ and ${\nabla_1}$ to ${E'}$ (we can pull back connections along with vector bundles), if ${t}$ is the projection on the second coordinate, we define ${\nabla'}$ as the convex combination ${\nabla' =(1- t) \nabla_0 + t \nabla_1}$.

We have two inclusions ${i_0, i_1: M \hookrightarrow M \times [0, 1]}$ given by inclusion on ${M \times \left\{0\right\}}$ and ${M \times \left\{1\right\}}$. Then clearly ${i_0^* E', i_1^* E'}$ are canonically isomorphic to ${E}$. Moreover, ${i_0^*}$ of ${\nabla'}$ is ${\nabla_0}$ while ${i_1^*}$ of ${\nabla'}$ is ${\nabla_1}$.

Let ${\Theta'}$ be the curvature of ${\nabla'}$, and let ${\Theta_0, \Theta_1}$ be the curvatures of ${\nabla_0, \nabla_1}$. By naturality of the curvature, we have

$\displaystyle i_0^* \Theta' = \Theta_0, \quad i_1^* \Theta' = \Theta_1.$

But by the “homotopy invariance” of de Rham cohomology, the ${i_0^*}$ and ${i_1^*}$ of a closed form on ${M \times [0, 1]}$ are cohomologous. This implies the result. $\Box$

3. The Chern classes

Now, we want to use the general theory of the previous section to describe the Chern classes: that is, we are going to fix polynomials ${\Phi_k}$, and then associate invariants ${c_k(E)}$ to a (smooth) complex vector bundle.

Namely, let ${\Phi_k: M_n(\mathbb{C})^k \rightarrow \mathbb{C}}$ be the invariant polynomial described by

$\displaystyle \Phi_k(A) = \mathrm{Tr}(\bigwedge^k A).$

The same ${\Phi_k}$ will be used for matrices of any degree. We shall use the following fact: if ${A, B}$ are matrices, then

$\displaystyle \Phi_k(A \oplus B) = \sum_{i+j=k} \Phi_i(A) \Phi_i(B). \ \ \ \ \ (2)$

This follows from the canonical decomposition ${\bigwedge^k (V \oplus W) = \bigoplus \sum_{i+j = k} \bigwedge^i V \otimes \bigwedge^j W}$ for vector spaces ${V, W}$.

Theorem 4 The Chern classes ${c_k(E)}$ of a complex vector bundle ${E \rightarrow M}$ can be calculated as follows: choose a connection on ${E}$ with curvature form ${\Theta}$, and then

$\displaystyle c_k(E) = \Phi_k\left( \frac{\sqrt{-1}}{2\pi} \Theta \right) \in H^{2k}(M; \mathbb{C}).$

Here, of course, we are identifying singular cohomology with de Rham cohomology.

Proof: It suffices to show that the above construction, which we denote by ${d_k(E)}$, satisfies the usual axioms for Chern classes. Then formal arguments, as given earlier, will imply that they coincide with the topological Chern classes (in particular, that they actually come from ${H^\bullet(M; \mathbb{Z})}$, which is not obvious here).

1. The construction ${E \mapsto d_k(E) = \Phi_k \left( \frac{\sqrt{-1}}{2\pi} \Theta \right) }$ is natural in ${E}$. Indeed, this follows because if given a map (say, smooth) ${f: N \rightarrow M}$, then we can pull back a connection ${\nabla}$ on ${E}$ to a connection ${f^* \nabla}$ on ${f^* E}$. The curvature also pulls back in the natural way.
2. If ${d(E) = \sum d_k(E)}$, then ${d(E)}$ is multiplicative: ${d(E \oplus E') = d(E) d(E')}$. Here we use the fact that if ${\nabla, \nabla'}$ are connections on ${E, E'}$, then there is a connection $\nabla + \nabla’$ on ${E \oplus E'}$. (The parallel transport for this corresponds to the direct sum of the parallel transports on ${E, E'}$ given by ${\nabla, \nabla'}$.) If ${\Theta, \Theta'}$ are the curvatures on ${E, E'}$, then the curvature on ${E \oplus E'}$ with respect to this new connection is the ${E \oplus E'}$-valued 2-form ${\Theta \oplus \Theta'}$. This follows by easy computation in a local frame: the ${\theta}$-matrices just add. Now the multiplicativity claim about ${d}$ follows from the result (2), which expresses the analogous multiplicativity on the functions ${\Phi_k}$.
3. ${d(E) = c(E)}$ if ${E}$ is a line bundle. This follows in view of the computation already done earlier for line bundles.

With these axioms verified, we can now conclude that the characteristic classes constructed by Chern-Weil theory are ordinary Chern classes. $\Box$

Note that since ${H^\bullet(M; \mathbb{R}) \subset H^\bullet(M; \mathbb{C})}$, one consequence is that the above construction provides real differential forms (or rather, cohomology classes represented by real forms).

What we have essentially done is, for a manifold ${M}$ with a smooth complex vector bundle ${E \rightarrow M}$ of dimension ${n}$, to give a homomorphism from the algebra of invariant polynomials on ${n}$-by-${n}$ matrices to the cohomology ring ${H^*(M; \mathbb{C})}$. In fact, an analog of this theory exists for principal ${G}$-bundles over any Lie group ${G}$. (Recall that there is an equivalence of categories between ${n}$-dimensional vector bundles and principal ${\mathrm{GL}_n(\mathbb{C})}$-bundles, or alternatively with ${U(n)}$-bundles.) That is, if ${\mathfrak{g}}$ is the Lie algebra of the compact Lie group ${G}$, then there is a morphism

$\displaystyle (\mathbf{Sym} \mathfrak{g}^{\vee})^G \rightarrow H^\bullet(M; \mathbb{R})$

defined for every ${M}$ with a principal ${G}$-bundle. So to every ${G}$-invariant polynomial function on the Lie algebra ${\mathfrak{g}}$, we can define a “characteristic class” of the principal ${G}$-bundle.

Moreover, with ${M = BG}$ and with the canonical bundle ${EG \rightarrow BG}$, this map is an isomorphism. (This should be taken with a pinch of salt, because ${BG}$ is not a manifold!) In other words, characteristic classes of principal ${G}$-bundles (that is to say, elements of ${H^\bullet(BG; \mathbb{R})}$) are the same thing ${G}$-invariant polynomials on the Lie algebra ${\mathfrak{g}}$.

This is quite a bit more than we have shown, even in the special case of ${G = U(n)}$: it implies that the Chern-Weil construction gives all possible (real-valued) characteristic classes. The actual construction of characteristic classes is given in a similar way, though; one chooses a connection on a principal ${G}$-bundle on a manifold, takes its curvature (a ${\mathfrak{g}}$-valued 2-form on the total space), applies an invariant polynomial, and projects down.