Chern–Weil form

Chern–Weil theory associates closed differential forms to a principal connection by applying invariant polynomials to its curvature. These are the differential-form representatives of characteristic classes .

Let $\pi:P\to M$ be a principal G-bundle and let $\omega\in\Omega^1(P;\mathfrak{g})$ be a principal connection with curvature $\Omega\in\Omega^2(P;\mathfrak{g})$ (see curvature 2-form of a principal connection ).

Let $P$ (unfortunately the same letter is standard) also denote an $\mathrm{Ad}$-invariant symmetric multilinear polynomial of degree $k$ on the Lie algebra $\mathfrak{g}$, i.e. an element of $(\mathrm{Sym}^k\mathfrak{g}^*)^G$.

The Chern–Weil form associated to the invariant polynomial $P$ and the connection $\omega$ is the $(2k)$-form on $P$

where the wedge product of the $\mathfrak{g}$-valued 2-forms is understood in the standard multilinear way.

A fundamental point is that $P(\Omega)$ is a basic form on $P$ (see the lemma that Chern–Weil forms are basic ), hence there exists a unique form $\mathrm{cw}_P(\omega)\in\Omega^{2k}(M)$ such that

By abuse of language, $\mathrm{cw}_P(\omega)$ is also called the Chern–Weil form on $M$.

What Chern–Weil theory guarantees

• The form $\mathrm{cw}_P(\omega)$ is closed, and its de Rham cohomology class does not depend on the choice of connection; this is the content of the Chern–Weil theorem .
• Consequently the class $[\mathrm{cw}_P(\omega)]\in H^{2k}_{\mathrm{dR}}(M)$ is an invariant of the underlying principal bundle (see Chern–Weil characteristic classes are bundle invariants ).

Examples

1. First Chern form for a unitary bundle Let $E\to M$ be a complex vector bundle with a Hermitian metric (see Hermitian metric ) and a unitary connection. The associated principal $U(n)$-bundle of unitary frames yields a curvature matrix $F\in\Omega^2(M;\mathfrak{u}(n))$. Taking the invariant polynomial $P(X)=\frac{i}{2\pi}\mathrm{tr}(X)$ gives the 2-form

   $$c_1(\nabla)=\frac{i}{2\pi}\,\mathrm{tr}(F),$$

   representing the first Chern class in de Rham cohomology (see Chern class and integrality of Chern classes ).

2. First Pontryagin form For a real vector bundle with structure group reduced to $SO(n)$ and a compatible connection, the curvature $F\in\Omega^2(M;\mathfrak{so}(n))$ defines the 4-form

   $$p_1(\nabla)= -\frac{1}{8\pi^2}\,\mathrm{tr}(F\wedge F),$$

   which represents the first Pontryagin class (see Pontryagin class ).

3. Euler form via the Pfaffian For an oriented rank-$2m$ real bundle with an $SO(2m)$-connection, the invariant polynomial given by the Pfaffian produces a $2m$-form representative of the Euler class (see Euler class ).