include them in the quasi-normal mode spectrum or not. Here we do include them because they are related to non-trivial excitations in the dual conformal field theory with non-zero central charge ($c=3\ell/2G$). In the same vein, for a critical theory we can choose not to — or, as spelled out in [@Castro:2011ui], shall not — include them in the spectrum if they correspond to null states that lead to zero central charge in the dual CFT.

It was pointed out too in [@Castro:2011ui] that even formula (\[ZCFT\]) is over-counting states if the central charge is too small. Then we find null states that are multi-particle states, combinations of states which have positive norm when considered as single-particle excitations. This lead to the restriction from (\[ZCFT\]) to a (Virasoro) minimal model character. Since in the following we are dealing with critical theories the central charge is always zero in our case. Therefore all non-log-modes are null modes.

We will now go on to our main objective, the partition functions of critical gravities.

Partition function of critical NMG and PET gravity
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Let us consider the partition function of critical NMG, which is given by [@Gaberdiel:2010xv] $$\label{ZcNMG}\hskip 1cm
  Z_{\rm cNMG}=Z_{\rm Ein}\cdot\frac{1}{\sqrt{\det(-\nabla^2-2)_2^{TT}}} \,.$$ 