## Tuesday, July 31, 2007

### Monstrous developments

Off-topic: a blog of Scientific American: The Simpsons embiggen cromulant papers on string theory
This text is about Witten's proposal that
pure AdS3 black holes carry monstrous symmetry.

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.

#### 1 comment:

1. Dear Lubos

I would extremely appreciate your opinion on definition of the total black
hole partition function via the Feynmanian integral over moduli space M

Z_BH = int_M DM Z_TOP (M)

Define graded Hopf/Grothendieck-Teichmuller group manifold M, whose
submoduli spaces of graded moduli space generate the modular union by
the inclusion system

qE_8 -> qE_9 -> qE_10 -> ... -> qE_n

Moduli space of graded hyperbolic quantum deformed group manifold M leads to
exponential growth of rank of holographically dual group of cascading throat
of generalized conifold and in the process whose Weyl group of root space
makes rank of vacuum torsion variable. Total number of generators grows
expo-exponentially with exponentially growing rank of M. Outgrowth is that
the roots of M have uncountable degeneration. Ireducible representation of
U-dual modular group M are K-theoretic knots of dilaton/tachyon which are
just condensates of one unstable graded quantum deformed octonionic black
hole throat. We identify T-dual modular group which makes the group rank
variable, S-duality cascade of generalized conifold throat and U-duality
chain which permutes NS-NS/R-R potentials and arbitrarily sets dilaton value
on the one side, with affine Weyl group of quantum deformed root system of M
on the other side.

There exists isomorphism between U-dual modular group M and moduli space of
U-dual instanton. The interchange of moduli of U-dual instanton is just
Weyl transformation in M. Isomorphism between topological amplitudes of
U-dual instanton and root lattice of M provides information about algebraic
structure of nonperturbative contributions to vacuum potential. U-dual flux
through U-dual cycle is

Ng = int psi = V_vaccum = 0

Monodromy of M makes volume of U-dual cycle and dimension of wrapping U-dual
brane variable. (Group manifold M permutes charges of the U-dual black
brane/U-dual vacuum torsion.)

There exists module homomorphism between affine Weyl group of root space M
and nonassociative deformed fractal attractor of condensation of Hagedornian
tachyon/U-dual instanton inside deformed throat of generalized conifold
(throat which fills group manifold M). Hagedornian tachyon orbits of affine
Weyl group of U-dual root system is fundamental observation. U-dual black
hole throat degenerations are determined by the U-dual automorphic forms
(degenerations are generalized Fourier coefficients of modular forms of M).
Outgrowth is that nilpotent orbits of M defines topological string
amplitudes. Thus we're Feynmanian integrating over U-dual modular space M
because of U-dual instanton tunneling amplitude we require observe. Notable
consequence is

uncountably degenerate U-dual root space = uncountably degenerate U-dual
black hole throat = int_M DM Z_TOP (M) = 0

Thank you very much for your opinion on this consequence!
With great respect
Marcel