Abstract
Three-dimensional Topologically Massive Gravity at its critical point has been conjectured to be holographically dual to a Logarithmic CFT. However, many details of this correspondence are still lacking. In this work, we study the 1-loop partition function of Critical Cosmological Topologically Massive Gravity, previously derived by Gaberdiel, Grumiller and Vassilevich, and show that it can be usefully rewritten as a Bell polynomial expansion. We also show that there is a relationship between this Bell polynomial expansion and the plethystic exponential. Our reformulation allows us to match the TMG partition function to states on the CFT side, including the multi-particle states of t (the logarithmic partner of the CFT stress tensor) which had previously been elusive. We also discuss the appearance of a ladder action between the different multi-particle sectors in the partition function, which induces an interesting sl(2) structure on the n-particle components of the partition function.
Highlights
One can deform pure 3d gravity by adding a gravitational Chern-Simons term
We study the 1-loop partition function of Critical Cosmological Topologically Massive Gravity, previously derived by Gaberdiel, Grumiller and Vassilevich, and show that it can be usefully rewritten as a Bell polynomial expansion
Li, Song and Strominger [8] showed that the situation can be improved if one replaces Einstein gravity by chiral gravity, which can be viewed as a special case of topologically massive gravity [4, 5] at a specific tuning of the couplings, and is asymptotically defined with Brown-Henneaux boundary conditions [2]
Summary
As shown in [19], the partition function (1.2) has the general structure expected from a dual logarithmic conformal field theory. The coefficients Nh,hare higher-order in h, hand should correspond to multi-particle states in t. A full understanding of the combinatorics leading to the multi-particle part of (2.1) was not attempted in [19], the coefficients Nh,hwere shown to be all positive, as should be the case if they are counting states in a CFT. The bulk mode dual to t results in the single double product appearing in (1.2), while the bulk modes dual to t and t (the logarithmic partner of the right-moving CFT stress tensor) lead to the two double products appearing in (2.3). We will be interested in better understanding the structure of such double products, focusing mainly on the TMG case
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