ChernSimons

Chern-Simons theory, named after Shiing-Shen Chern and James Simons, appears in differential geometry as a secondary characteristic class and in physics as a three-dimensional topological quantum field theory. It is formulated using a connection on a principal bundle and, in mathematics, provides a transgression between characteristic classes.

Let M be a 3-manifold and A a connection on a principal G-bundle with Lie algebra g and an invariant bilinear form Tr. The Chern-Simons 3-form is cs(A) = Tr(A ∧ dA + (2/3) A ∧ A ∧ A). The Chern-Simons functional is S_CS(A) = (k/4π) ∫_M cs(A). The exterior derivative satisfies d cs(A) = Tr(F ∧ F), where F = dA + A ∧ A is the curvature. The quantity S_CS is not strictly gauge-invariant; under a gauge transformation it shifts by an integer multiple of 2π k, so for the quantum theory to be well-defined the level k must be an integer for compact gauge groups.

In physics, Chern-Simons theory is a metric-independent topological quantum field theory in three dimensions. Observables include Wilson loops W_R(C) = Tr_R P exp(∮_C A). Their expectation values yield knot and link invariants; in particular, Witten showed that, at appropriate levels, these observables reproduce the Jones polynomial. The theory is closely related to three-dimensional gravity via gauge formulations with groups such as SL(2, R) or ISO(2,1), and it serves as a source of rich mathematical structures in low-dimensional topology.

In mathematics, the Chern-Simons form is a secondary characteristic class whose integrals define invariants of 3-manifolds and of flat connections, linking geometric data to topological properties.