We know that the AdS/CFT duality is a valid correspondence for large central charge (in the semi-classical limit). If this duality is valid for small central charge (in the quantum regime) is unclear.
It is often said that
a high energy CFT state on a torus is dual to a Euclidean BTZ black hole spacetime with a torus conformal structure at asymptotic infinity.
On the other hand, there has been an explicit check (see this link) that
the Ising model CFT on a torus is dual to the full AdS quantum gravity theory with a torus conformal structure at asymptotic infinity.
This check has been performed by computing the partition functions of the Ising model CFT and the quantum gravity theory, where the partition function of the quantum gravity theory is a sum over geometries globally diffeomorphic to thermal Anti-de Sitter spacetime ( this includes the thermal Anti-de Sitter space and the Euclidean BTZ black hole spacetime).
My question is the following:
How can a CFT be dual to the BTZ spacetime in one case, and the sum over all AdS-like spacetimes in the second case? Does this have something to do with the duality at small and large central charge?
