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",
14π∫d2z(|∂ϕ|2+μeϕ)+bounday−terms
- Is the above action (which I would have thought is for the Liouville CFT) the same as SZT which is said to satisfy the identity SE(M3)=c3SZT(M2,Γi=1,...,g)?
In the above equality SE is the Einstein gravity action evaluated on-shell on a 3-manifold which is formed by "filling" in the interiors of a choice Γi=1,..,g of g non-contractible cycles on a genus-g Riemann surface M2. (...in the context of AdS/CFT one would want to interprete M2 as being the conformal boundary of the space-time M3...)
If SZT 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