Definition: Kinematic diffeomorphism rule, XI-B normalization

Let $c$ be the odd vector field $c^{\mu }{\partial }_{\mu }$ and write, with odd coefficients on the left,

Define the odd, ghost-number-one derivation $s$ of the field algebra by

Equivalently $sq=\frac{\kappa }{2}{\mathcal{L}}_{c}q$ for $q=\eta +\kappa \gamma$: the ghost generates the vector field $\frac{1}{2}c$, so that $c=2\xi$ recovers ${\delta }_{\xi }\gamma =K\xi +\kappa {\mathcal{L}}_{\xi }\gamma$ and ${\delta }_{\xi }\varphi =\kappa {\xi }^{\mu }{\partial }_{\mu }\varphi$. The backgrounds $g$, ${\sigma }_{\epsilon }$ are $s$-inert. At $\kappa =0$ the rule is XII-A's frozen $s_0$.