Tuesday, January 08, 2008 ... /////

Davide Gaiotto: Killing the two-headed monster

Edward Witten has proposed his AdS/CFT dual of pure gravity in three-dimensional anti de Sitter space.

In the bulk, pure gravity looks as simple as you can get. However, the boundary CFT was conjectured to incorporate what is arguably the most complex symmetry group in mathematics, namely the monster group.

In the usual AdS/CFT fashion, the central charge of this CFT increases with the curvature radius of the AdS3 space. The central charge can be written as c=24k. A known theory exists for k=1 but for higher values of k, it was not known whether a CFT existed and whether it was unique. More precisely, we are looking for an extremal CFT which is a CFT whose lowest dimension of a non-identity primary field is k+1, the highest value of the lower bound that general rules of CFTs can allow. All these non-identity primary fields are then good enough to be identified with BTZ black holes.

Matthias Gaberdiel proposed a conjecture based on an analysis of some data about the CFTs. One of the key implications of his conjecture was that the CFTs with k=42 and higher do not exist. It would look like the size of the AdS3 space could not exceed 42 units if you required pure gravity in the bulk.

The new paper

Davide Gaiotto
argues that the Gaberdiel conjecture banning high values of k is wrong because a simple power of the monster module is a counterexample. (A refined version of the Gaberdiel conjecture involving "irreducible" CFTs has not yet been falsified, I think.) But he offers a much more fascinating claim about the extremal CFTs with monster symmetry:
The k=2 i.e. c=48 theory already doesn't exist.
While the monster group CFT exists for k=1, already the doubled "size" of the AdS3 leads to contradictions with axioms of CFTs and the structure of the conjectured monster symmetry.

Gaiotto's strategy

Which contradictions? Davide doesn't use any other methods than the axioms of CFTs themselves: his approach is based on conformal bootstrap. The monster group has nearly 1054 elements so Davide is essentially going to build the twist fields for each element in this rather large set. It's not an impossible task because the answer only depends on the conjugacy class and there are 194 conjugacy classes of the monster.

He starts with the 2A conjugacy class and determines the lowest dimension of the twist field in the appropriate sector. The twisted partition sums are certain hauptmodules and the coefficients in them are incompatible with the positivity and integrality properties of the CFT combined with the information from the OPE of the two 2A twist fields.

Message

Unless there is a mistake in Gaiotto's analysis, the main lesson is clear. You must allow the proper laws of mathematics to determine what theories can exist and what theories can't. And the proper laws of mathematics are the full microscopic laws, in this case the stringy laws describing the boundary CFT.

The existence of the k=2 theory may be motivated by heuristic albeit controversial, loop-quantum-gravity-like "gravity as a gauge theory" arguments, the picture looks complicated enough to avoid obvious contradictions, the CFT has Edward Witten's signature below it - but one (or at least Davide) may still prove the theory doesn't exist if he looks carefully enough.

What is the actual structure of the AdS/CFT for pure AdS3 gravity? Well, it might be that the theory exists for all k but most of them don't have the monster symmetry; they can still be extremal. However, it looks somewhat strange that a bigger AdS space should break the symmetries of the smallest one. Alternatively, the pure gravity in AdS3 with k=2 or higher k (and its dual CFT) might simply be non-existent.