Z_{\rm Double~log}^{\rm 1-particle} = \prod_{n=2}^\infty\frac{1}{|1-q^n|^2} \,
      \Big(1+\frac{2q^2+2\bar{q}^2}{|1-q|^2}\Big) \,.$$ We can thus interpret the result (\[Zpet\]) as $$\label{cftl2PET}\hskip 1cm
 Z_{\rm PET}^{\rm log^2}=Z_{\rm Double~log}^{\rm 1-particle}+{\rm multiparticle}\,,$$ where the multi-particle contribution is given by $$\label{Zmultiparticle}\hskip 1cm
 Z_{\rm multiparticle}=\sum_{h,\bar{h}}N_{h,\bar{h}}q^h\bar{q}^{\bar{h}}\prod_{n=1}^\infty\frac{1}{|1-q^n|^2} \,.$$ By explicit calculation of $N_{h,\bar{h}}$ for low values of $h$ and $\bar{h}$, and using the combinatorial counting argument from [@Gaberdiel:2010xv; @Bertin:2011jk], one can show that all $N_{h,\bar{h}}$’s are non-negative integers. Thus, we can interpret (\[Zmultiparticle\]) as the contribution of physical states to the partition function.\
*Single log* For $\beta=-3b_2/\ell^2-2\sigma\ell^2$ and $b_2\neq\pm2\sigma\ell^4$ PET gravity is conjectured to be dual to a parity even rank-two LCFT, plus an additional massive mode. The single-particle partition function is $$\label{log1mCFT}\hskip 1cm
   Z_{\rm Single~log}^{\rm 1-particle} = \prod_{n=2}^\infty\frac{1}{|1-q^n|^2} \,
      \Big(1+\frac{q^2+\bar{q}^2}{|1-q|^2}+\frac{q^{|\mathcal{M}|+1}\bar{q}^{|\mathcal{M}|-1}+q^{|\mathcal{M}|-1}\bar{q}^{|\mathcal{M}|+1}}{|1-q|^2}\Big) \,.$$ Again, we can write $$\label{cftlPET}\hskip 1cm
 Z_{\rm PET}^{\rm log}=Z_{\rm Single~log}^{\rm 1-particle}+Z_{\rm multiparticle} \,,$$ with $Z_{\rm multiparticle}$ given in (\[Zmultiparticle\]), and for given $\mathcal{M}$ show that all $N_{h,\bar{h}}$’s are non-negative integers.\
*Massive log* In the parameter range where $M_+=M_-=M$, PET gravity is conjectured to be dual to a parity even rank-two LCFT with non-vanishing central charge given by $$\hskip 1cm
 c_{R/L}=\frac{3\ell\sigma}{G}\frac{\ell^4M^4}{1+2\ell^2M^2+2\ell^4M^4}
        =\frac{3\ell\sigma}{G}\frac{(\ell^2\mathcal{M}^2-1)^2}{1-2\ell^2\mathcal{M}^2+2\ell^4\mathcal{M}^4}\,.$$ The single-particle CFT partition function for such a theory takes the form $$\label{mlogCFT}\hskip 1cm
   Z_{\rm Massive~log}^{\rm 1-particle} = \prod_{n=2}^\infty\frac{1}{|1-q^n|^2} 