I've tried to give a general sketchy introduction to Witten's paper in week254
- you might look at that.
I don't really understand that term. It should be defined in Schelleken's paper
--- this paper speaks of "meromorphic conformal field theories" instead of "conformal field theories with holomorphic factorization", but they must be the same thing. However, I'm having a bit of trouble finding the precise definition! I just know a bunch of properties of these theories.
First, the central charge c is an integer multiple of 24.
Second, as a consequence, the partition function is really a well-defined number, not just defined up to (24/c)th root of unity. In other words, it's "modular invariant".
These two are very important in Witten's paper.
Third, as another consequence, the Schwinger functions
, otherwise known as "n-point functions" are all well-defined meromorphic functions
--- that is, holomorphic except for poles. This is not so important in Witten's paper, though.
Witten gives an argument that 3d quantum gravity has as its AdS/CFT dual a conformal field theory with c = 24k for some integer k = 1,2,3,... The main
nice thing is that - modulo a certain conjecture - Schellekens classified these conformal field theories for k = 1.
Yes, he argues this is true for any k. Then, around equation (3.13), he shows that this property, together with modular invariance of the exact partition function, completely determines the exact partition function! It's a certain explicit polynomial in the J function.
For k=1 he goes through Schelleken's list of 71 conformal field theories with c = 24 and picks the one that has the Monster group as its symmetries. He gives an argument for why this one is the right one, but it's not airtight.
He doesn't actually find the relevant conformal field theories with c = 24k for
higher values of k. He just figures out their supposed partition functions. Since the coefficients of their partition functions are - just as in the k = 1 case - dimensions of representations of the Monster group, it seems awfully plausible that these theories (if they really exist!) have the Monster group as symmetries.
However, this is something one would want to check. Nobody seems to know a c = 48 theory with Monster group symmetries, for example.
I will copy your questions and my answers to the n-Category Cafe
, and hope some experts on conformal field theory (like Urs Schreiber and Jacques Distler) can help us out.