over the transverse-traceless tensor modes is[^2] $$\label{ZTT}\hskip 1cm
 Z^{TT}_{\rm crit} = \int\mathcal{D}h_{\mu\nu}^{TT}\mathcal{D}k_{\mu\nu}^{TT}\mathcal{D}f_{\mu\nu}^{TT}\,
     e^{-\delta^{(2)}S_{\rm PET}^{TT}} = \big[\det(-\nabla^2-\frac{2}{\ell^2})_2^{TT}\big]^{-3/2} \,.$$ To perform the integral over the transverse vector and scalar fluctuations we note that the kinetic terms have the wrong sign. This can be remedied by a Gibbons–Hawking–Perry [@Gibbons:1978ac] rotation of the fluctuations to imaginary values, which we shall also employ repeatedly in the remainder of this work. The results are $$\label{ZT}\hskip 1cm
 Z^T_{\rm crit}=\int\mathcal{D}v^T_\mu\mathcal{D}u^T_\mu\,e^{-\delta^{(2)}S_{\rm PET}^T}
     = \big[\det(-\nabla^2+\frac{2}{\ell^2})_1^T\big]^{-1} \,,$$ and after integration over $h$, $\bar{K}$, $\delta$, $\varepsilon$ and finally $\bar{F}$ in that order: $$\hskip -.8cm
 Z^{\rm scalar}_{\rm crit} = \int\mathcal{D}h\mathcal{D}\bar{K}\mathcal{D}\varepsilon\mathcal{D}\bar{F}\mathcal{D}\delta \,
     e^{-\delta^{(2)}S_{\rm PET}^{\rm scalar}}
     = \big[\det(-\nabla^2+\frac{3}{\ell^2})_0\big]^{-3/2}\big[\det(-\nabla^2)_0\big]^{-1} \,. \label{ZS}$$ The gravity one-loop partition function for PET gravity at the tricritical point is obtained by carefully collecting all ghost determinants, see (\[Zghost\]), and the determinants (\[ZTT\]), (\[ZT\]) and (\[ZS\]): $$\begin{split}\label{Zpetdet}\hskip 1cm
 Z_{\rm crit~PET}^{\rm 1-loop}=& \,Z_{\rm gh}^3\cdot J_1^2\cdot Z^{TT}_{\rm crit}\cdot Z^T_{\rm crit}\cdot Z^{\rm scalar}_{\rm crit} \\
     =&\,\frac{\big[\det(-\nabla^2+\frac{2}{\ell^2})_1^T\big]^{1/2}}{\big[\det(-\nabla^2-\frac{2}{\ell^2})_2^{TT}\big]^{3/2}}
     = Z_{\rm Ein}\cdot \Big(\big[\det(-\nabla^2-\frac{2}{\ell^2})_2^{TT}\big]^{-1/2}\Big)^2 \,,
\end{split}$$ with $Z_{\rm Ein}$ being the one-loop contribution to the partition function of Einstein gravity, see eq. (\[Zein\]). Using heat kernel techniques [@Giombi:2008vd; @David:2009xg] it is straightforward to obtain the result (for positive temperature \[$\tau_2>0$\]) $$\label{Zpet}\hskip 1cm
 Z_{\rm crit~PET}^{\rm 1-loop}= \prod_{n=2}^\infty\frac{1}{|1-q^n|^2} \,\bigg[
                           \prod_{m=2}^\infty\prod_{\bar{m}=0}^\infty\frac{1}{1-q^m\bar{q}^{\bar{m}}}
                           \prod_{l=0}^\infty\prod_{\bar{l}=2}^\infty\frac{1}{1-q^l\bar{q}^{\bar{l}}} 