Characteristic Classes

Characteristic classes are the cohomology classes produced by Chern–Weil theory: topological invariants of a bundle, built from curvature but independent of the connection. They are the mathematical home of the instanton number, the monopole charge, and the $\theta$-term. This page catalogues the standard classes and their physics incarnations, building on chern-weil.md and de-rham.md.

Chern classes (complex / gauge bundles)

For a complex vector bundle (or a gauge bundle with structure group $U(n)$/$SU(n)$) with curvature $F$, the total Chern class is $$c(F)=\det (1+\frac{iF}{2\pi })=1+c_1+c_2+\cdots ,$$ with $c_k\in H^{2k}(M;\mathbb{Z})$. The low ones:

• First Chern class $c_1=\frac{i}{2\pi }\mathrm{tr}F$. For a $U(1)$ bundle its integral over a closed surface is an integer — the magnetic flux / monopole charge / Dirac quantization: $$\frac{1}{2\pi }{\int }_{S^2}F=c_1\in \mathbb{Z}={\pi }_1(U(1)).$$
• Second Chern class $c_2\sim \frac{1}{8{\pi }^2}\mathrm{tr}(F\wedge F)$. For an $SU(2)$ bundle over a 4-manifold its integral is the instanton number $$\nu =\frac{1}{8{\pi }^2}\int \mathrm{tr}(F\wedge F)\in \mathbb{Z}={\pi }_3(SU(2)),$$ the topological charge of advanced/topological.md.

Chern numbers (integrals of top-degree products of $c_k$) are integers by the integrality of the underlying integer cohomology class — the deep reason topological charges are quantized.

Pontryagin and Euler classes (real / tangent bundles)

For a real vector bundle (e.g. the tangent bundle), curvature is $\mathfrak{so}(n)$- valued and the relevant invariants are:

• Pontryagin classes $p_k\in H^{4k}(M;\mathbb{Z})$, from $\det (1+\frac{F}{2\pi })$ keeping even powers. The first Pontryagin number governs gravitational instantons and the gravitational anomaly.
• Euler class $e\in H^n(M;\mathbb{Z})$ (oriented, even rank $n$), the Pfaffian of the curvature. Its integral over a closed manifold is the Euler characteristic: $$\chi (M)={\int }_{M}e=\frac{1}{2\pi }{\int }_{M}KdA(\text{2D, Gauss–Bonnet}),$$ tying curvature to the topology counted by $\chi =\sum (-1{)}^p{b}_p$ (see riemannian-bridge.md and geometry/curvature.md).

The $\theta$-term

Because $\mathrm{tr}(F\wedge F)=d{\omega }_{CS}$ is locally exact (chern-weil.md), adding $$S_{\theta }=\frac{\theta }{8{\pi }^2}\int \mathrm{tr}(F\wedge F)=\theta \nu$$ to the action changes it only by a topological integer $\times \theta$. It does not affect the classical equations of motion (a total derivative), but it weights different instanton sectors in the quantum path integral by $e^{i\theta \nu }$ — the origin of the strong-CP problem and the physics in advanced/topological.md § the θ-vacuum.

The classification connection

Characteristic classes are the computable image of the abstract bundle classification $[M,BG]$ of principal-bundles.md: they are pullbacks of universal classes on the classifying space $BG$. For the cases physics uses most,

The homotopy groups in the third column come from homotopy.md; the translation to zero modes of a Dirac operator is index-theorem.md.

References

• Milnor & Stasheff, Characteristic Classes — the definitive text.
• Bott & Tu, Differential Forms in Algebraic Topology, Ch. 4.
• Nakahara, Geometry, Topology and Physics, Ch. 11.