gravity with a negative cosmological constant. We discuss the contributions to the partition functions of Einstein gravity, NMG and PET gravity. Then we will define a truncation of the theory and apply it to critical NMG and tricritical PET gravity. Finally we summarize and comment on the results we have obtained.

Partition function of Einstein gravity
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The one-loop contribution to the partition function of Einstein gravity is (see e.g. [@David:2009xg]) $$\label{Zein}\hskip 1cm
 Z_{\rm Ein}=\sqrt{\frac{\det(-\nabla^2+2)_1^T}{\det(-\nabla^2-2)_2^{TT}}} \,.$$ Let us briefly comment on the two terms contributing to the partition function in (\[Zein\]). The determinant in the numerator is due to the gauge choice and corrects the path-integral measure (see e.g. [@Gaberdiel:2010xv]), while the determinant in the denominator stems from the gauge-fixed equations of motion. Thus, for a critical theory where we observe degeneration of the equations of motion, we expect to find a multiple of the contribution $\det(-\nabla^2-2)_2^{TT}$ in the denominator.

According to [@Denef:2009kn] and [@Datta:2011za] the determinants are evaluated using $$\label{zdelta}\hskip 1cm
 \frac12\ln Z_{\Delta_s}=\ln\prod_{\kappa\geq0}^\infty|1-q^{\kappa+\Delta_s}|^{-2(\kappa+1)} \,,$$ where $\Delta_s=1+|m_s|$ can be read off from the definitions $$\hskip 1cm
 \det(-\nabla^2+m^2_1-2)_1^T \quad{\rm and}\quad \det(-\nabla^2+m^2_2-3)_2^{TT} \,.$$ Comparison with (\[Zein\]) yields $\Delta_1=3$ and $\Delta_2=2$ to give $$\label{ZCFT}\hskip 1cm
 Z_{\rm Ein}=\prod_{n\geq2}^\infty\frac{1}{|1-q^n|^2}\,.$$

This has a nice interpretation as 