Spacetime, formal ring, configurations

Indices are raised and lowered with $\eta$; parentheses on index pairs include the factor $\frac{1}{2}$. Put

Equality modulo ${\kappa }^3$ means coefficient equality; no numerical remainder estimate is implied. For a covector field ${\xi }_{\mu }$ and vector field ${\xi }^{\mu }={\eta }^{\mu \nu }{\xi }_{\nu }$,

Here $D_{\xi }$ is the Lie derivative of a scalar density represented relative to ${\mathrm{d}}^{4}x$.

The configuration space is

All fields are smooth on a neighbourhood of $M_T$ and are used on $M_T$. The case of a $\kappa$-independent scalar is ${\varphi }_1={\varphi }_2=0$.