Backgrounds, ramp, plateau, and gauge action

Let

with the smooth compact-open topology, and let

The backgrounds are real, smooth, classical, and off shell. No Einstein equation, graviton propagator, or quantum metric variable is introduced.

Fix $m\ge 0$, $0<\epsilon <T_{\ast }$, and

The ramp is

It is smooth and flat at both endpoints.

Retain the same nonempty relatively compact open plateau $U$ as in X-A-ST:

The formal group ${\mathcal{G}}_p$ is generated through ${\kappa }^2$ by flows of $\kappa \xi +{\kappa }^2\zeta$, $\xi ,\zeta \in {\mathfrak{g}}_p$. The vector fields are extended by zero. They preserve $g,\sigma$, are initial-trivial, and do not meet the worldsheet.

Write $X_{\xi }={\xi }^{\mu }{\partial }_{\mu }$ on scalars and $D_{\xi }r={\partial }_{\mu }({\xi }^{\mu }r)$ on scalar densities. The formal metric transformation is

The last term is not omitted in a second-order diagnostic.