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 Not-Even-Wrong. 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 light-like partonic surfaces defined by Chern-Simons 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 3-D case. Witten proposes that the quantum theory for 3-D AdS₃ blackhole with a negative cosmological constant can be reduced by AdS₃/CFT₂ correspondence to a 2-D conformal field theory at the 2-D boundary of AdS₃ analogous to blackhole horizon. This conformal field theory would be a Chern-Simons theory associated with the isometry group SO(1,2)×SO(1,2) of AdS₃.

• 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

1. 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 non-perturbatively unstable.

   Remark: The situation changes in TGD framework, where AdS₃ is replaced by a generic light-like 3-surface so that only Λ=0 situation is encountered.

2. 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 24-dimensional 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.

3. Could these results generalize to 1+3 D case? According to Witten this is not the case.

   Remark: In TGD framework the lightlike partonic 3-surfaces are boundaries of 4-D space-time 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.

4. Does one obtain this 3-D quantum gravity from string- or M-theory in the sense that this 3-D 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.

1. In quantum TGD holography would be realized in 4-D case by identifying light-like 3-surfaces as basic dynamical objects and 4-D space-time 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 4-D sense allows the assumption about light-likeness. The additional conformal symmetries correspond to the symmetries predicted already much before the realization of fundamental role of light-likeness from the fact that space-time surfaces correspond to preferred extremals of Kähler action analogous to Bohr orbits.

   Partonic 3-surfaces 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 2-D conformal invariance to 3-D case made possible by lightlikeness implying metric 2-dimensionality. Obviously this picture provides a different approach to 4-D gravitational holography using light-like 3-surfaces as basic dynamical objects.

2. AdS₃ is replaced with light-like partonic 3-surface. Euclidian partonic 2-surface corresponds to the boundary of AdS₃. This surface is determined as the intersection of the light-like 3-surface and future or past lightcone of M⁴× CP₂ 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 light-cones are an essential element in the definition of S-matrix and are associated with each argument of N-point function. They are also essential for the TGD inspired many-sheeted 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 space-time, imbedding space, and "world of classical worlds" (everything quantal must have geometric correlates).

3. Vacuum Einstein's equations are satisfied in the following sense. Due to the effective 2-dimensionality of the induced metric the situation is effectively 2-dimensional 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 "light-like". Note that curvature tensor is non-vanishing but since the time direction disappears from metric there can be no propagating waves.

4. Chern-Simons 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 Chern-Simons action in Witten's theory. Also the fermionic counterpart of this action for induced spinors and dictated by super-conformal symmetry is present.

   The very notion of light-likeness 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.

5. In Witten's theory the gauge group corresponds to the isometry group SO(1,2)× SO(1,2) of AdS₃. The group of isometries of lightlike 3-surface is something much much mightier. It corresponds to the conformal transformations of 2-dimensional section of the 3-surfaces 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 infinite-dimensional and essentially same as the group of 2-D 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 2-D cross section plus "conformal" symmetries with respect to the radial lightlike coordinate.

6. 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 hyper-finite factors of type II₁(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).

7. To make it clear, I am not suggesting that AdS₃/CFT₂ correspondence should have a TGD counterpart. If it had, a reduction of TGD to a closed string theory would take place. The almost-topological QFT character of TGD excludes this on general grounds.

   [I think this was asked in the blog of Woit by some-one before Woit censored out both the link to my blog and the question to keep the discussion "scientific" in the sense of uncritical worship of Witten: sad that due his well-known background Peter lacks the competence to decide which is crackpottery and which is not so that he must apply the methods of Prussian school masters.]

   More concretely, the dynamics would be effectively 2-dimensional if the radial superconformal algebras associated with the light-like coordinate would act as pure gauge symmetries. Concrete manifestations of the genuine 3-D character are following.

   • Generalized super-conformal representations decompose into infinite direct sums of stringy super-conformal representations.

   • In p-adic 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 light-like 3-surfaces analogous to bottoms of valleys in spin glass energy landscape meaning that there is infinite number of different 3-D lightlike surfaces associated with given 2-D 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 .