I am bit confused as to if both the things are the same since it seems that people refer to both as being given by the ``Liouville action",

\(\frac{1}{4\pi}\int d^2z ( \vert \partial \phi \vert ^2 + \mu e^{\phi } ) + bounday-terms \)

- Is the above action (which I would have thought is for the Liouville CFT) the same as $S_{ZT}$ which is said to satisfy the identity $S_{E}(M_3) = \frac{c}{3}S_{ZT}(M_2,\Gamma_{i=1,...,g})$?

In the above equality $S_E$ is the Einstein gravity action evaluated on-shell on a 3-manifold which is formed by "filling" in the interiors of a choice $\Gamma_{i=1,..,g}$ of $g$ non-contractible cycles on a genus-g Riemann surface $M_2$. (...in the context of $AdS/CFT$ one would want to interprete $M_2$ as being the conformal boundary of the space-time $M_3$...)

If $S_{ZT}$ is indeed the same as the Liouville CFT action then how does one understand the need to choose these non-contractible cycles to define that?

- In the same strain I would want to know what exactly is the meaning of the conjecture made in equation 5.1 in this paper, http://arxiv.org/abs/1303.6955