paragraph: either we keep all modes and the theory is dual to a non-unitary LCFT, or it is not logarithmic but an “ordinary” CFT that propagates only null modes.

This work is organized as follows. In section \[sec2\] we calculate the partition function of PET gravity and discuss its similarity to LCFT single-particle partition functions. In section \[sec3\] we define the truncation by translating from black hole determinants to quasi-normal mode spectra and discriminating against certain modes. Finally we apply this truncation to critical NMG and PET gravity.

Partition function of PET gravity {#sec2}
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In this first part of the work we calculate the gravity one-loop partition function of PET gravity. Such a calculation is conveniently carried out by splitting the metric into a background metric $g_{\mu\nu}$ and a perturbation $h_{\mu\nu}$. Our background is an AdS space-time, but, as we will argue later, this is equivalent to taking the BTZ black hole [@Banados:1992wn] as background. The partition function then consist at least of a classical part, $Z_c$, corresponding to the background $g_{\mu\nu}$, and a one-loop contribution coming from the perturbation $h_{\mu\nu}$, $Z^{\rm 1-loop}$. This one-loop contribution is precisely what we are interested in.

We start by covering the whole 