Chern–Weil Theory

Chern–Weil theory manufactures topological invariants out of curvature: from any invariant polynomial of the curvature 2-form it builds a closed differential form whose de Rham class is independent of the connection. This is the machine that turns the local, connection-dependent field strength into global, quantized characteristic numbers — the instanton number, the monopole charge, the Euler characteristic. It builds on curvature.md and de-rham.md; the resulting classes are catalogued in characteristic-classes.md.

Invariant polynomials

Let $P(X)$ be a symmetric, $\mathrm{Ad}$-invariant polynomial on the Lie algebra $\mathfrak{g}$ — $P(g^{-1}Xg)=P(X)$ — for example $P(X)=\mathrm{tr}(X^k)$ or $\det (1+X)$. Evaluated on the curvature 2-form $F$ (with wedge products of the form-part), $P(F)$ is an ordinary differential form of degree $2k$ on the base. Invariance is what makes $P(F)$ gauge-independent: since $F\mapsto g^{-1}Fg$ under a gauge change (curvature.md), an $\mathrm{Ad}$-invariant $P$ sees no change.

The Chern–Weil theorem

Chern–Weil theorem. For an $\mathrm{Ad}$-invariant polynomial $P$ and the curvature $F$ of any connection:

1. $P(F)$ is closed: $dP(F)=0$ (using the Bianchi identity).
2. Its de Rham class $[P(F)]\in H_{\mathrm{dR}}^{2k}(M)$ is independent of the connection — changing $A$ shifts $P(F)$ by an exact form.

So while $F$ itself depends on the choice of connection (a physical field), the cohomology class $[P(F)]$ depends only on the bundle. That class is a characteristic class; its integral over a cycle is a characteristic number, a topological invariant.

Sketch. Closedness: $dP(F)=kP(d_{\nabla }F,F,\dots )=0$ by Bianchi $d_{\nabla }F=0$ and invariance. Connection-independence: interpolate $A_t=A_0+ta$ between two connections; $\frac{d}{dt}P(F_t)=d(\text{something})$ is exact — the transgression / Chern–Simons form is that "something".

Chern–Simons forms

The exact form witnessing connection-independence is the Chern–Simons form. For the second Chern class, $$\mathrm{tr}(F\wedge F)=d{\omega }_{CS},{\omega }_{CS}=\mathrm{tr}(A\wedge dA+\frac{2}{3}A\wedge A\wedge A).$$ Locally $\mathrm{tr}(F\wedge F)$ is exact (it is $d{\omega }_{CS}$), so its integral over a closed manifold is topological — nonzero only because of global twisting. The Chern–Simons form ${\omega }_{CS}$ is itself the Lagrangian of Chern–Simons theory (a topological field theory) and appears in the descent equations for anomalies (index-theorem.md, anomalies/chiral-anomaly.md).

Why this is the key theorem

Chern–Weil is the bridge that makes "topological charge = curvature integral" rigorous:

• it explains why the instanton number $\frac{1}{8{\pi }^2}\int \mathrm{tr}(F\wedge F)$ is an integer independent of the field configuration within a topological sector (characteristic-classes.md);
• it is why the $\theta$-term $\theta \int \mathrm{tr}(F\wedge F)$ is topological (a total derivative locally) yet physically consequential globally (strong-CP);
• it underlies the Gauss–Bonnet theorem when applied to the tangent bundle's curvature (riemannian-bridge.md).

References

• Bott & Tu, Differential Forms in Algebraic Topology, Ch. 4.
• Nakahara, Geometry, Topology and Physics, Ch. 11.
• Chern, Complex Manifolds Without Potential Theory.