= Chern-Simons 3-form
{c}
= Chern-Simons three-form
{c}
{synonym}
For a local matrix-valued <connection one-form> $A$, put
$$
Y(A)=\frac1{8\pi^2}\operatorname{tr}\left(A\wedge dA+\frac23A\wedge A\wedge A\right).
$$
Its <exterior derivative> is
$$
dY=\frac1{8\pi^2}\operatorname{tr}(F\wedge F),\qquad F=dA+A\wedge A.
$$
Thus the <Chern-Simons 3-form> is a local primitive of the trace-square <curvature form>. For traceless <unitary connection> <curvature>, this equals the <Second Chern form> in the stated convention; a general unitary <principal bundle> needs the additional trace-product correction. This is an instance of transgression from a characteristic four-form to a three-form.<Characteristic forms and geometric invariants by Shiing-Shen Chern and James Simons>[Section 3, equation (3.1) and Proposition 3.2, page 52]{ref}
The expression uses a chosen local frame and is not itself a globally gauge-invariant three-form. Its precise change under <gauge transformations> is recorded in <gauge change of the Chern-Simons three-form>.
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