Chern–Simons form

Summary

A Chern–Simons form is an odd-degree differential form built from a connection whose exterior derivative is a characteristic form of Chern–Weil theory. The best-known one is the $3$-form $\mathrm{tr}(A \wedge dA + \frac23 A \wedge A \wedge A)$, whose derivative is $\mathrm{tr}(F \wedge F)$. Chern–Simons forms give secondary invariants, defined when the primary characteristic forms vanish. Their integrals over $3$-manifolds are the action of Chern–Simons gauge theory, which is central in low-dimensional topology and physics.