Witten's new ideas about 1+2 dimensional quantum gravity
Witten has been talking at Friday in string cosmology workshop in New York about his new ideas relating to 1+2 D quantum gravity. Peter Woit has been listening the talk and represents his understanding in NotEvenWrong. Lubos Motl gives a nice summary about the conformal field theoretic ideas involved. Why I got interested (even with this miserable technical background) is that Witten's talk gives an interesting perspective to quantum TGD, which reduces to almost topological QFT for lightlike partonic surfaces defined by ChernSimons action and its fermionic super counterpart.
1. Brief summary of main points
Very concisely, the message seems to be following.
 The motivation of Witten is to find an exact quantum theory for blackholes in 3D case. Witten proposes that the quantum theory for 3D AdS_{3} blackhole with a negative cosmological constant can be reduced by AdS_{3}/CFT_{2} correspondence to a 2D conformal field theory at the 2D boundary of AdS_{3} analogous to blackhole horizon. This conformal field theory would be a ChernSimons theory associated with the isometry group SO(1,2)×SO(1,2) of AdS_{3}.
 This conformal theory would have so called monster group (see also the posting of Lubos) as the group of its discrete hidden symmetries. The primary fields of the corresponding conformal field theory would form representations of this group.
2. Questions
 Why negative cosmological constant? The answer is that Λ=0 does not allow 1+2 D blackholes and Witten believes that for Λ>0 is nonperturbatively unstable.
Remark: The situation changes in TGD framework, where AdS_{3} is replaced by a generic lightlike 3surface so that only Λ=0 situation is encountered.
 Why monster group should act as symmetry group?
 The existence of this kind of conformal theory has been demonstrated already earlier. Lubos Motl gives a nice description of how this symmetry results when one compactifies chiral bosonic fields in 24dimensional space to a torus defined by Leech lattice.
 The crucial observation is that the partition function of this conformal field theory contains no term coming from massless particles. Hence one can hope that it could correspond via AdS correspondendence to 1+2 D quantum gravitational theory which does not allow any gravitons since empty space Einstein equations stating the vanishing of Ricci tensor also imply the vanishing of curvature tensor.
 Could these results generalize to 1+3 D case? According to Witten this is not the case.
Remark: In TGD framework the lightlike partonic 3surfaces are boundaries of 4D spacetime sheets providing classical physics representations for the partonic quantum dynamics required by quantum measurement theory. Kind of inverse of holography. The generalization is therefore trivial in TGD framework.
 Does one obtain this 3D quantum gravity from string or Mtheory in the sense that this 3D gravity would emerge by compactication. Lubos is pessimistic about this.
3. How could this relate to Quantum TGD?
There are very strong resemblances between Witten's model and the formulation quantum TGD at parton level.
 In quantum TGD holography would be realized in 4D case by identifying lightlike 3surfaces as basic dynamical objects and 4D spacetime sheets as surfaces containing partonic lightlike surface (of arbitrarily large size) as analogs of black hole horizons. It should be emphasized that general coordinate invariance in 4D sense allows the assumption about lightlikeness. The additional conformal symmetries correspond to the symmetries predicted already much before the realization of fundamental role of lightlikeness from the fact that spacetime surfaces correspond to preferred extremals of Kähler action analogous to Bohr orbits.
Partonic 3surfaces would be something much more general than blackhole horizons. They define the "world of classical worlds", arena of quantum dynamics. In Witten's theory these degrees of freedom are frozen. The absolutely essential point would be a huge extension of 2D conformal invariance to 3D case made possible by lightlikeness implying metric 2dimensionality. Obviously this picture provides a different approach to 4D gravitational holography using lightlike 3surfaces as basic dynamical objects.
 AdS_{3} is replaced with lightlike partonic 3surface. Euclidian partonic 2surface corresponds to the boundary of AdS_{3}. This surface is determined as the intersection of the lightlike 3surface and future or past lightcone of M^{4}× CP_{2} and is thus unique. There is thus no need for the analog of black hole horizon to result as a singularity of Einstein equations.
These lightcones are an essential element in the definition of Smatrix and are associated with each argument of Npoint function. They are also essential for the TGD inspired manysheeted Russian doll cosmology. Jones inclusions and related quantization of Planck constant make them also necessary and they relate very closely to the representation of choice of quantization axes at the level of spacetime, imbedding space, and "world of classical worlds" (everything quantal must have geometric correlates).
 Vacuum Einstein's equations are satisfied in the following sense. Due to the effective 2dimensionality of the induced metric the situation is effectively 2dimensional so that Einstein tensor vanishes identically and vacuum Einstein equations are satisfied for Λ=0. Gravitation would become purely topological in absence of the overall important attribute "lightlike". Note that curvature tensor is nonvanishing but since the time direction disappears from metric there can be no propagating waves.
 ChernSimons action for the induced Kähler form  or equivalently, for the induced classical color gauge field proportional to Kähler form and having Abelian holonomy  corresponds to the ChernSimons action in Witten's theory. Also the fermionic counterpart of this action for induced spinors and dictated by superconformal symmetry is present.
The very notion of lightlikeness involves induced metric implying that the theory is almost topological but not quite. This small but important distinction guarantees that the theory is physically interesting.
 In Witten's theory the gauge group corresponds to the isometry group SO(1,2)× SO(1,2) of AdS_{3}. The group of isometries of lightlike 3surface is something much much mightier. It corresponds to the conformal transformations of 2dimensional section of the 3surfaces made local with respect to the radial lightlike coordinate in such a manner that radial scaling compensates the conformal scaling of the metric produced by the conformal transformation.
The direct TGD counterpart of the Witten's gauge group is thus infinitedimensional and essentially same as the group of 2D conformal transformations! Presumably this can be interpreted in terms of the extension of conformal invariance implied by the presence of ordinary conformal symmetries associated with 2D cross section plus "conformal" symmetries with respect to the radial lightlike coordinate.
 Monster group does not have any special role in TGD framework. However, all finite groups and  as it seems  also compact groups can appear as groups of dynamical symmetries at the partonic level in the general framework provided by the inclusions of hyperfinite factors of type II_{1}(see this). Compact groups and their quantum counterparts would closely relate to a hierarchy of Jones inclusions associated with the TGD based quantum measurement theory with finite measurement resolution defined by inclusion as well as to the generalization of the imbedding space related to the hierarchy of Planck constants. Discrete groups would correspond to the number theoretical braids providing representations of Galois groups for extensions of rationals realized as braidings (see this).
 To make it clear, I am not suggesting that AdS_{3}/CFT_{2} correspondence should have a TGD counterpart. If it had, a reduction of TGD to a closed string theory would take place. The almosttopological QFT character of TGD excludes this on general grounds.
More concretely, the dynamics would be effectively 2dimensional if the radial superconformal algebras associated with the lightlike coordinate would act as pure gauge symmetries. Concrete manifestations of the genuine 3D character are following.
 Generalized superconformal representations decompose into infinite direct sums of stringy superconformal representations.
 In padic thermodynamics explaining successfully particle massivation radial conformal symmetries act as dynamical symmetries crucial for the particle massivation interpreted as a generation of a thermal conformal weight.
 The maxima of Kähler function defining Kähler geometry in the world oc classical worlds correspond to special lightlike 3surfaces analogous to bottoms of valleys in spin glass energy landscape meaning that there is infinite number of different 3D lightlike surfaces associated with given 2D partonic configuration each giving rise to different background affecting the dynamics in quantum fluctuating degrees of freedom (see this ). This is the analogy of landscape in TGD framework but with a direct physical interpretation in say living matter.
For more details see the chapter Construction of Configuration Space Kähler Geometry
from Symmetry Principles: part II .
