in the final expression (\[Zlogpetdet\]), we find that it reduces to (\[Zpetdet\]). Another cross-check is the limit going to critical NMG, $b_2\to0$. This is not apparent from formula (\[Zlogpetdet\]). We need to set $b_2=0$ in the action (\[actionTT\]) to see that there is no contribution from the integral over $\mathcal{D}f_{\mu\nu}^{TT}$, i.e. we do not get the last factor in (\[Zlogpetdet\]). Therefore we obtain exactly the same result that was obtained for critical NMG [@Gaberdiel:2010xv].

For arbitrary $b_2$ ($\neq-2\sigma\ell^4$) we find that the partition function (\[Zlogpetdet\]) consist of the contribution of one massive mode (third term) and the contribution of critical NMG. We thus find a parity even rank-two LCFT plus one massive mode.

The last term of eq. (\[Zlogpetdet\]) in terms of $q$’s is $$\label{ZM}\hskip 1cm
 Z_\mathcal{M}=\prod_{l=|\mathcal{M}|}^\infty\prod_{\bar{l}=|\mathcal{M}|}^\infty
          \frac{1}{(1-q^{l+1}\bar{q}^{\bar{l}-1})(1-q^{l-1}\bar{q}^{\bar{l}+1})}\,,$$ where we defined $\mathcal{M}^2=-2\sigma\ell^2/b_2$.[^3]\
*Massive log* In the massive log case the parameters $\beta$ and $b_2$ are restricted by the equation $$\label{Mp=Mm}\hskip 1cm
 \beta^2+4b_2\sigma+\frac{6b_2\beta}{\ell^2}+\frac{10b_2^2}{\ell^4}=0 \,.$$ Here a little more effort is needed to evaluate the ’TT’ and the scalar path-integrals. The vector components do not change; we find the result (\[ZT\]). To evaluate the ’TT’ part we first rescale $2k^{TT}_{\mu\nu}=\bar{\sigma}\bar{k}^{TT}_{\mu\nu}$ and shift $\bar{h}^{TT}_{\mu\nu}=h^{TT}_{\mu\nu}-1/2\bar{k}^{TT}_{\mu\nu}$. Then $\bar{h}^{TT}$ decouples and we can integrate over it. 