Fields, operators, and the inverse

The independent fields are $({\gamma }_{\mu \nu },\varphi ,c^{\mu },{\bar{c}}_{\mu },b_{\mu })$ with parities $(0,0,1,1,0)$ and ghost numbers $(0,0,1,-1,0)$. Odd identities are coefficientwise in finite Grassmann algebras, with odd coefficients placed on the left. Put

The inverse frozen on this sheet is

X-G's $-4\mathcal{P}{E}_0^{\mathrm{R}}$ inverts the differently normalized Hessian $-\frac{1}{4}\mathcal{P}\square$; it is not converted into (eq:inverse) by vocabulary and is not used.

The classical action is XII-C's

with $S_{\mathrm{grav},\mathrm{D}}$ from (eq:action) and ${\mathfrak{L}}_q=\frac{1}{2}\sqrt{-q}(q^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{\varphi }^2)$. It is a formal bulk variational functional tested with compact interior variations; its Euler derivatives ${e_{\mathrm{D}}}^{\mu \nu }$, $E_{\varphi }$ are the exact expressions of XII-C. The gauge-fixing term $S_{\mathrm{gf}}=\int D_{\nu }\gamma D^{\nu }\gamma$ is generated below by XI-B's fermion; it is not added by hand.