Proposition: Recovery of the frozen gauge term

The free auxiliary action

has

Its auxiliary equation is $b_{\mu }=D_{\mu }\gamma$. Eliminating $b$ gives precisely $S_{\mathrm{gf}}$ and the ghost term $i\int {\bar{c}}_{\mu }\square c^{\mu }$.

Proof. Equation (eq:basic) gives $s_0{D}^{\mu }\gamma =\frac{1}{2}\square c^{\mu }$. The minus sign from applying an odd derivation to the second factor of $\bar{c}(2D-b)$ yields (eq:sPsi). The auxiliary variation is $2(D^{\mu }\gamma -b^{\mu })$. Finally,