= Boundary variation of the Yang-Mills theta term
{title2=$\delta\int_M\operatorname{tr}(F\wedge F)=2\int_{\partial M}\operatorname{tr}(\delta A\wedge F)$}
For $F=dA+A\wedge A$, the <gauge-theory Bianchi identity> is $DF=0$ and the variation is $\delta F=D\delta A$. The graded product rule for the <covariant exterior derivative> gives the displayed boundary integral. Thus a constant-coefficient <Yang-Mills theta term> leaves the bulk <Yang-Mills equations> unchanged, but can change the boundary variational condition. Its four-form density is exactly <gauge-invariant> by conjugation of $F$ and cyclicity of the <matrix trace>, even if the gauge parameter is nonzero on the boundary. Gauge changes of a local <Chern-Simons 3-form> must not be confused with gauge changes of this globally defined curvature density.
Back to article page
