\,
      \Big(1+\frac{2q^{|\mathcal{M}|+1}\bar{q}^{|\mathcal{M}|-1}+2q^{|\mathcal{M}|-1}\bar{q}^{|\mathcal{M}|+1}}{|1-q|^2}\Big) \,.$$ Again, we can write $$\label{cftlmPET}\hskip 1cm
 Z_{\rm PET}^{\rm mlog}=Z_{\rm Massive~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.

Quasi-normal modes and partition functions of (truncated) critical gravities {#sec3}
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In this section we address the question if there are other methods to calculate the gravity one-loop partition function circumventing the (direct) use of heat kernel techniques. We will focus on one specific idea which relies on the work of Denef, Hartnoll and Sachdev [@Denef:2009kn], where they claim that the calculation of black hole determinants via heat kernel techniques is equal to summing over the quasi-normal mode spectrum of the theory. Calculations in higher spin gravity lend support to this conjecture, see e.g. [@Datta:2011za]. However, the issue of boundary conditions does not play such a prominent role there and the method of summing over quasi-normal mode spectra has not been applied to critical gravity theories yet.

One motivation to look for other-than heat kernel methods is the apparent difference of the results obtained for the gravity one-loop partition function of log-TMG [@Gaberdiel:2010xv] (TMG at the chiral point with log boundary conditions) and chiral gravity [@Maloney:2009ck] (TMG 