Graviton 1-loop partition function for 3-dimensional massive gravity

TL;DR

The paper computes the 1-loop partition function $Z_{\textrm{TMG}}$ for Euclidean AdS$_3$ topologically massive gravity using heat-kernel and zeta-regularization techniques, establishing a gauge-fixed path integral framework that yields $Z_{\textrm{TMG}}=Z_{\textrm{Ein}}\cdot Z_{M}$. At the critical point $\mu\ell=1$, the result is not holomorphically factorized and matches the expectations for a logarithmic CFT (LCFT) dual with $c_L=0$ and $c_R=3\ell/G_N$, including a non-chiral massive sector associated with the logarithmic partner of the stress tensor. The explicit form $\ln Z_M=\sum_{n=1}^{\infty}\frac{1}{n}\frac{q^{2n}}{(1-q^{n})(1-\bar{q}^{n})}$ leads to the gravity partition function $Z_{\textrm{TMG}}=\prod_{n=2}^{\infty}\frac{1}{|1-q^{n}|^{2}}\prod_{m=2}^{\infty}\prod_{\bar{m}=0}^{\infty}\frac{1}{1-q^{m}\bar{q}^{\bar{m}}}$. The results reinforce the LCFT holographic conjecture for TMG and extend naturally to New Massive Gravity, with the LCFT structure confirmed by positive multiplicities in the extended partition functions. The work thus clarifies the holographic content and the role of non-chiral sectors in 3D gravity at criticality.

Abstract

The graviton 1-loop partition function in Euclidean topologically massive gravity (TMG) is calculated using heat kernel techniques. The partition function does not factorize holomorphically, and at the chiral point it has the structure expected from a logarithmic conformal field theory. This gives strong evidence for the proposal that the dual conformal field theory to TMG at the chiral point is indeed logarithmic. We also generalize our results to new massive gravity.