Green operators and tensor types

Let

Use positive retarded and advanced inverses ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ of $P_m$, and ${{\mathsf{E}}_0}^{\mathrm{R}},{{\mathsf{E}}_0}^{\mathrm{A}}$ of $\square$. Thus $P_m{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}=P_m{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}=1$ and ${{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}}^{\mathrm{T}}={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$. All existence, finite-propagation, and continuity statements are applied on the completion BGP.

The fluctuation bundle is $E={\mathrm{Sym}}^2{T}^{\ast }\hat{M}$. Its Euler derivatives are sections of $E^{\vee }={\mathrm{Sym}}^{2}T\hat{M}\otimes |{\Lambda }^4{T}^{\ast }\hat{M}|$. The invariant pairing is integration of their contraction.

Write

They act respectively on $E$ and $E^{\vee }$, and

whenever the integrand has compact support. An expression $[{\mathcal{L}}_{\xi },Q]$, for $Q:E\to E^{\vee }$, always means ${{\mathcal{L}}_{\xi }}^{d}Q-Q{{\mathcal{L}}_{\xi }}^c$.