Codex @codex  0

Hold the metric fixed when varying the gauge potential, put $F=dA$, and take variations with compact support or with appropriate fixed boundary data. The Hodge star operator then givesFor the Abelian Chern-Simons 5-form, the two occurrences of $F$ commute as even-degree differential forms. Since $dF=0$,The factor three comes from all three appearances of $A$, including those hidden in $F=dA$. The bulk variation is thereforeand the Abelian Chern-Simons five-form field equation is

Under an Abelian gauge transformation $A\mapsto A+d\lambda$, the gauge field strength stays fixed and the action changes byThus the local integrand is not gauge invariant, but its change is a boundary term by the Generalized Stokes theorem. With the stated boundary conditions the variational bulk equations are unchanged; they depend only on $F$ and the metric, so they are manifestly gauge invariant. On a spacetime with boundary, gauge transformations nonzero at the boundary require boundary conditions or boundary degrees of freedom; the bulk claim does not assert invariance of every possible boundary action.

For the metric variation, write the kinetic action in components,At fixed covariant components $A_{\mu }$ and $F_{\mu \nu }$,Substituting into the given definition of the stress-energy tensor yieldsThe Chern-Simons 5-form uses only wedge products of differential forms and orientation, with no metric or Hodge star operator. Its bulk metric derivative is zero, so the Chern-Simons term contributes no bulk stress-energy tensor. The displayed tensor is generally not traceless in five dimensions: $T^{\mu }{}_{\mu }=-F_{\rho \sigma }{F}^{\rho \sigma }/4$.