Off-topic: a blog of Scientific American: The Simpsons embiggen cromulant papers on string theoryThis text is about Witten's proposal that
There have been various CFT-centered developments here.
First, Jan Manschot (UVA) rewrote the partition sum as an actual sum instead of product, claiming that this form makes it clear that it is a Feynman sum over geometries even though it is not explained too clearly why the individual terms should be interpreted as "geometries".
A week ago, Davide Gaiotto and Xi Yin (HU) have calculated genus two partition sums of these "extremal conformal field theories" or "ECFTs" of Witten. It's an impressive calculation that uses very much the same methods that they know from more conventional compactifications of string theory, further suggesting that neither they nor Witten work on LQG as argued by a supreme crackpot. You should see the formula pi (3.14) on page 9 - the very existence of this solution is a surprise but it exists, in analogy with other stringy miracles. Their methods are effective up to k=10.
A few days ago, Matthias Gaberdiel (ETH) has proposed a general rule for CFTs, especially their modular differential equations whose order should be linked to the vanishing of a certain vector in Zhu's C2 quotient space. This link works in all examples he has checked and if this rule is true in general, Witten's monstrous ECFTs shouldn't exist for k=42 and higher! Recall that the central charge is c=24k. The AdS space wouldn't be allowed to get too large.
If Wolfgang Beirl is still in search of 42, he may find it in Gaberdiel's paper in the role of a killer. ;-)
Witten doesn't want to believe that his theories could be non-existent for k=42 and higher so he proposes a loophole that sounds rather contrived but can't be easily ruled out: namely that there are new states with a high dimension (in the flat space limit) that grows like sqrt(k) that repair the partition sum. ;-)
Well, if I had to make a bet, I would bet that what breaks at k=42 and above is not the sequence of AdS/CFT dual theories by Witten but rather Gaberdiel's link and that Witten's loophole is not needed.
And that's the memo.