We consider the problem of identifying the CFT's that may be dual to pure gravity in three dimensions with negative cosmological constant. The c-theorem indicates that three-dimensional pure gravity is consistent only at certain values of the coupling constant, and the relation to Chern-Simons gauge theory hints that these may be the values at which the dual CFT can be holomorphically factorized. If so, and one takes at face value the minimum mass of a BTZ black hole, then the energy spectrum of three-dimensional gravity with negative cosmological constant can be determined exactly. At the most negative possible value of the cosmological constant, the dual CFT is very likely the monster theory of Frenkel, Lepowsky, and Meurman. The monster theory may be the first in a discrete series of CFT's that are dual to three-dimensional gravity. The partition function of the second theory in the sequence can be determined on a hyperelliptic Riemann surface of any genus. We also make a similar analysis of supergravity.
1. Key ideas
The key observations are following.
- In 3-D case Einstein-Hilbert action can be written as Chern-Simons action for gauge group SO(2,2) by combining vielbein connection form with vielbein (this for negative cosmological constant).
- Although 3-D gravity is trivial dynamically, it allows so called BTZ blackholes for negative cosmological constant. These blackholes have a huge degeneracy of states and the idea is that this degeneracy could correspond to primary fields of QFT defined by Chern-Simons action.
- The Virasoro algebra decomposes into direct sum of left and right algebras corresponding to left and right movers. If gravitational Chern-Simons action is assumed to vanish, one has left-right symmetry and k=kL=kR=k and c=24k. Holomorphic factorization of the partition function into a product occurring for cL= cL=24k makes the model calculable and the partition function can be expressed as power series using the well known modular invariant J-function j(q)= 1/q+ 19688q+... appearing in number theory and associated with the Monster group. This allows to identify blackhole states in terms of primary fields belonging to the representations of the Monster group.
Witten considers also some objections against the equivalence of 3-D quantum gravity with Chern-Simons gauge theory.
- Witten makes a comment about the invertibility of the vielbein as a condition for perturbative well-definedness since in the gauge theory formulation the gauge potential (ω,e/l)=(0,0) (l is the length scale defining cosmological constant) is legitimate whereas the perturbation theory is performed around a solution for which e is invertible (3-metric is non-degenerate). Exactly this would occur in TGD by effective metric 2-dimensionality of lightlike 3-surfaces.
- Witten also notices that in quantum gravity description using path integral one expects sum over all topologies whereas in gauge theory description no such sum is necessary. In TGD framework where the generalization of S-matrix defines time like entanglement coefficients between positive and negative energy parts of zero energy state there is no need to sum over intermediate topologies.
The formula for Kac Moody central charge in absence of gravitational Chern-Simons term is of form k = l/16G, l =sqrt(-1/λ). For λ→0 l and thus k infinite and the theory would become classical. Also c= 24k would diverge at this limit meaning classicality in Virasoro degrees of freedom. The number of "blackhole states" would become infinite. Note however that both curvature scalar and gravitational Chern-Simons term vanish at this limit if the metric corresponds to that of a lightlike surface.
3. Comparison with TGD
It is interesting to look the situation from the TGD point of view since usually this kind of comparisons bring new insights to TGD.
- I have already told about strong analogies between this theory and quantum TGD as an almost topological QFT for lightlike 3-surfaces as basic objects with extended conformal invariance (see this and this). There are of course also deep differences. In Witten's theory 3-D space-time is not a surface. In TGD one has light-like 3-surfaces which correspond to solutions of Einstein's equations with degenerate metric for vanishing cosmological constant. Thus light-likeness characterizing a property as a 3-surface forces vacuum Einstein's equations characterizing purely geometric property. Dynamical variables in TGD framework correspond to second quantized free induced spinor field and deformations of lightlike 3-surface.
- In the case of a degenerate metric with vanishing cosmological constant, the vielbein group would reduce to SO(2) and extension by vielbein would reduce to trivial extension. What makes this interesting is that induced Kähler form corresponds to U(1), which strictly speaking does not define a gauge symmetry but spin glass type dynamical symmetry acting as gauge transformations only for vacuum extremals. It is just the classical gravitation which breaks the gauge symmetry character of U(1). This picture conforms nicely with quantum criticality of TGD Universe.
- Partition function can be determined for hyperelliptic 2-surfaces of any genus (all g<3 surfaces are hyperelliptic). In TGD framework hyper-elliptic surfaces should correspond to fermions and gauge bosons having identification as fermion-antifermion bound states and hyperellipticity implied by the generalization of the imbedding space concept and quantum classical correspondence implies the vanishing of the elementary particle vacuum functionals for g>2 so that only three fermion and gauge boson families are possible.
- An interesting question is whether one should add to TGD 3-D gravitational action and corresponding Chern-Simons term. The curvature scalar part vanishes identically by the metric 2-dimensionality. For the same reason also Chern-Simons term should vanish, at least in a suitable gauge. If not, gravitational Chern-Simons would remove the vacuum degeneracy crucial for TGD and spoil also the almost topological QFT property of quantum TGD.
Usually this kind of comparisons have yielded new insights also into TGD. This was the case also now. The conjectured quantization of the coupling constant guaranteing so called holomorphic factorization is implied by the integer valuedness of the Chern-Simons coupling strength k. As Witten explains, this follows from the quantization of the first Chern-Simons class for closed 4-manifolds plus the requirement that the phase defined by Chern-Simons action equals to 1 for a boundaryless 4-manifold obtained by gluieing together to 4-manifolds along their boundaries.
The quantization argument seems to generalize to the case of TGD. What is clear that this quantization should closely relate to the quantization of Kähler coupling strength appearing in the 4-D Kähler action defining Kähler function for the world of classical worlds and conjectured to result as a Dirac determinant. The argument leading leading to an extremely simple formula for Kähler coupling strength as αK =1/4k and allowing to identify p-adic temperature as Tp=1/k with k =26 for bosons and k=1 for fermions, is discussed in separate posting.