\big[\det(-\nabla^2+2)_1^T\det(-\nabla^2+3)_0\big]^{1/2} \quad{\rm and}\quad
        J_1= \big[\det(-\nabla^2)_0\big]^{1/2}\,.$$

Path-integral for PET gravity
-----------------------------

We now consider the action (\[linaction\]) under the decompositions (\[tensorsplit\]) and (\[vectorsplit\]). All scalar, ’vector’ (T) and ’tensor’ (TT) modes decouple and we can write the action (\[linaction\]) as a sum of terms which are quadratic in the respective fluctuations. For the transverse-traceless part we find $$\begin{split}\label{actionTT}\hskip 1cm
 \delta^{(2)}S_{\rm PET}^{TT}= \int {\rm d} ^3x\sqrt{g} \,\Big\{
   &  -\frac{\bar{\sigma}}{4} h_{TT}^{\mu\nu}\big(-\nabla^2-\frac{2}{\ell^2}\big)h_{\mu\nu}^{TT}
      + \frac12 k_{TT}^{\mu\nu}\big(-\nabla^2-\frac{2}{\ell^2}\big)h_{\mu\nu}^{TT} + \\ 
  & + b_2 f_{TT}^{\mu\nu}\big(-\nabla^2+\frac{\beta}{b_2}\big)f_{\mu\nu}^{TT}  -f^{\mu\nu}_{TT}k_{\mu\nu}^{TT}\Big\} \,.
\end{split}$$ The transverse vector part is given by $$\label{actionT}\hskip 1cm
 \delta^{(2)}S_{\rm PET}^T = 2\int{\rm d} ^3x\sqrt{g} \,\Big\{
       \big(\frac{2b_2}{\ell^2}+\beta\big)u_T^\mu\big(-\nabla^2+\frac{2}{\ell^2}\big)u_\mu^T
       -u_T^\mu\big(-\nabla^2+\frac{2}{\ell^2}\big)v_\mu^T\Big\} \,,$$ and the scalar contribution is $$\begin{aligned}
\hskip -.8cm
 \delta^{(2)}S_{\rm PET}^{\rm scalar}=\int{\rm d} ^3x\sqrt{g}\,\Big\{ &
     \frac{\bar{\sigma}}{18}h\big(-\nabla^2+\frac{3}{\ell^2}\big)h
       -\frac19\bar{K}\big(-\nabla^2+\frac{3}{\ell^2}\big)h
     -\frac{2b_2}{9}\bar{F}\big(-\nabla^2+\frac{3}{\ell^2}\big)\bar{F} \nonumber\\
    &+ \big(\frac{2b_2}{\ell^2}+\beta\big)\big[
             -\frac23\bar{F}^2-\frac83\bar{F}\nabla^2\delta-\frac{8}{\ell^2}\delta\nabla^2\delta\big] \label{actionS}\\
    &+ \frac23\bar{F}\bar{K}+\frac43\bar{F}\nabla^2\varepsilon+\frac43\bar{K}\nabla^2\delta
             +\frac{8}{\ell^2}\delta\nabla^2\varepsilon  \Big\} \,, \nonumber\end{aligned}$$ where $\varepsilon$ is the scalar part coming from the decomposition $v^\mu=v^\mu_T + \nabla^\mu\varepsilon$.

### The critical/logarithmic loci {#critloc}

We now evaluate the path-integrals (\[actionTT\])–(\[actionS\]) for the critical values of the parameters $\beta$ and $b_2$ that lead to dual LCFTs of rank two and three.\
*Tricritical point* Let us now consider in particular the tricritical point. At this critical locus the combinations $\bar{\sigma}$ and $(2b_2/\ell^2+\beta)$ vanish, thus the formulas (\[actionTT\])–(\[actionS\]) simplify considerably. The path-integral 