BRST from finite geometric transformations

Builder/output: brst_from_geometry.py.

From $\delta g={\mathcal{L}}_{\xi }g$ with $\xi =\kappa ac$ and the affine metric:

The ghost law $s{\xi }^{\mu }=\lambda {\xi }^{\rho }{\partial }_{\rho }{\xi }^{\mu }$ is an ansatz whose coefficient is derived: $s^{2}h=s^2\varphi =0$ on all jets (115 independent conditions) has the unique solution $\lambda =1$. {Derived $s_2$ per sector:} $s_{2}h=0$, $s_{2}c=0$, $s_2\varphi =0$, $s_2\bar{c}=0$, $s_{2}b=0$. Reason (a derivation, not the p09 policy “no invented $s_2$”): $g$ is affine in $h$ and ${\mathcal{L}}_{\xi }$ is linear in $g$, so ${\kappa }^{-1}{\mathcal{L}}_{\kappa ac}g$ has $\kappa$-degree exactly 1; $s\xi$ is quadratic in $\xi$, so $sc$ has degree exactly 1; $s\varphi$ is linear in $\xi$; the pair $(\bar{c},b)$ is $\kappa$-independent. $s^2=0$ holds exactly to all orders. Discriminating control: for $g=\eta +\kappa h+\frac{{\kappa }^2}{2}h{\eta }^{-1}h$ the same construction gives $s_{2}h\ne 0$, and dropping it breaks $s^2=0$ at $O({\kappa }^2)$.