Can a $CFT_2$ which can't be factorized into chiral and antichiral parts and/or have a central charge not a multiple of 24 have AdS duals?

In the article Three dimensional gravity reconsidered by Ed Witten, he remarked that the CFT dual to three dimensional quantum gravity has to admit a holomorphic factorization and have a central charge of $c=24k$ for some integer $k$. He argues this by some handwaving from a Chern-Simons crutch even though he admits the nonperturbative theory can't be Chern-Simons.

Anyway, does this mean if we have a two dimensional conformal field theory which does not factorize into chiral and antichiral parts and/or has a central charge which is not an integer multiple of 24, it does not have an AdS quantum gravity dual?

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What is the meaning of "chiral" (or "anti"chiral) in particle physics? –  Georg Mar 5 '11 at 17:14
Georg, "chiral" is derived from χείρ (cheir), the Greek word for the hand, and it's any theory or object that is inequivalent to its left-right mirror - much like the left hand differs from the right hand. So for spinors, it's a Weyl spinor- that includes supersymmetry, chiral SUSY is left-handed SUSY etc. For theories, it's a theory whose left-handed spinors couple differently than the right-handed ones. In 2D CFT, the solutions to wave and other equations are composed of left-moving and right-moving waves which are the two chiral sectors of the CFT (also equivalent to holom. and anti.), etc. –  Luboš Motl Mar 5 '11 at 17:50
The way I read the paper, the words "has to" in your description should be replaced by "are likely to", for the reasons he articulates in the article. The statement is then that CFT without those extra assumptions, if you can write it down, is not likely to correspond to non-perturbative formulation of pure gravity. Of course, the discussion in the abstract is only interesting to a point, he has some specific set of CFTs in mind, maybe you do also? –  user566 Mar 5 '11 at 18:01
@Lubos, I'm a chemist, and I know a little bit about chiralality. The problem is that "antichiral". There are achiral objects, but antichiral is a somewhat strange word to me. –  Georg Mar 5 '11 at 20:09

This actual reasons behind many statements you mentioned may be a bit different.

First, one doesn't need too much Chern-Simons handwaving to say that $c_L=24 k_L$ etc. The three-dimensional action of general relativity may be rewritten so that it only depends on $l/G$ where $G$ is Newton's constant (of dimension length) and $l$ is the curvature radius, so that $2/l^2$ is the cosmological constant term added to $R$ in the Einstein-Hilbert action.

Brown and Henneaux have studied the asymptotic action of the Virasoro group on the three-dimensional spacetime and from the commutators, they could simply determine that $c_L=c_R=3l/2G$. Now, $l/G$ is a multiple of 16 - it's plausible that one needs the Chern-Simons variables to argue this much (one uses the integer-valuedness of the levels). But the isomorphism between CS and GR works perturbatively.

Also, if $c_L$ and $c_R$ differ, it's still true that $c_L-c_R$ must be a multiple of 24 for any CFT - not just CFTs that are the duals of some AdS space. This follows from the $\tau\to\tau+1$ modular transformations of the torus - or, equivalently, from the requirement that $L_0-\tilde L_0$ has to be integer-valued for physical states (the implication holds because the ground states have energies $-c_{L,R}/24$).

It just happens that when you translate the levels of the Chern-Simons theory to the ground state energies of the boundary CFT, they're equal up to the sign. A necessary condition for holomorphic factorization of a CFT is that the ground state energies for the left-movers and right-movers are two integers - which naturally translates to the integrality of the levels.

Witten kind of assumed it was also a sufficient condition - i.e. that one can find a dual description with a monster symmetry not only for $k=1$ which is what he did but also for $k=2,3,\dots$. This naturally sounding conjecture was disproved, at least for $k=2$, by Davide Gaiotto by some representation theory of the monster group:

http://arxiv.org/abs/arXiv:0801.0988

It seems pretty likely that the theories with the monster group symmetry don't exist for any higher $k$, either. It may still be the case that the higher $k$ CFT duals exist and have a smaller sporadic symmetry - much like the ABJM at a general level generalizes the BLG construction but reduces its symmetry.

Various 2D CFTs may have AdS duals but they won't be pure gravity: they will include some massless bulk states i.e. marginal deformations. The Leech-lattice-based models that Witten studied were special because the Leech lattice doesn't produce any massless states - it only produces massive states which may be interpreted as black hole microstates. Witten hasn't discussed those in his paper. So he was not saying that non-factorizable CFTs can't have an AdS dual: they just can't have a pure gravity AdS dual in three dimensions.

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Excellent Lubos. I was wondering also what is the impact of this paper on the community of people playing with gauge theory formulations of pure gravity (in 3 and 4 dimensions). This should really give them a pause, since 3dim pure gravity is the most common toy model for them. –  user566 Mar 5 '11 at 18:05
Dear Moshe, good points. I was trying to focus on the positive insights. LQG assumes that the 2+1D gravity's equivalence to 2+1D Chern-Simons is the "trivial toy model" of the equivalence of 3+1D gravity to the SU(2) spin networks. Of course, even the first equivalence doesn't work at the non-perturbative level, as even Webb mentioned, but it works at some level. However, with all the quantum details about the BH microstates, there's a hidden monster symmetry that is totally invisible from a CS/LQG viewpoint. To summarize, I am not aware of any impact of Witten's paper on LQG. ;-) –  Luboš Motl Mar 5 '11 at 20:15
But the non-working hypotheses have bigger names under attack here, right? ;-) In particular, Witten's original expectation that there is a monster-symmetric "extremal" CFT for any level "k" - any radius of the AdS3 - has been falsified. I don't know what's the most qualified opinion about the existence of exact AdS/CFT duals for higher radii pure gravity as of 2011, do you? However, I think that Witten's expectation turned out to be wrong - there were a couple of similar cases I know. Of course, it's not the same kind of wrongness that one may see after the same time scale as in the LQG case. –  Luboš Motl Mar 5 '11 at 20:18
Hoehn independently arrived at the same result as Gaiotto (but in more mathy language): arxiv.org/abs/0801.1822 –  Scott Carnahan Mar 6 '11 at 20:06
Wow, very interesting, Scott. Quite an overlap. –  Luboš Motl Mar 12 '11 at 17:55