Chern-Simons 5-form
= Chern-Simons 5-form
{c}
{title2=$Q_5(A)=\operatorname{Tr}(AF^2-\tfrac12FA^3+\tfrac1{10}A^5)$}
For a matrix-valued <gauge field> $A$, let $F=dA+A\wedge A$ be its <gauge field strength>; products below combine matrix multiplication with the <wedge product of differential forms>. The five-form can be written $Q_5=\operatorname{Tr}(A(dA)^2+\tfrac32A^3dA+\tfrac35A^5)$. Applying the <exterior derivative>, using its graded product rule and the graded cyclic property of the <matrix trace>, gives $dQ_5=\operatorname{Tr}(F^3)$. The six-form is an invariant polynomial in the <gauge curvature>, while its local potential depends on the connection.