Hyperbolic 3-manifolds and zero energy ontology
The latest finding was the article Volumes of hyperbolic manifolds and mixed Tate motives by Goncharov- one of the great Russian mathematicians involved with the drama. The article is about polylogarithms emerging in twistor calculations and their relationship to the volumes of hyperbolic n-manifolds. I do not of course understand anything about the jargon of the article: it is written by a specialist for specialists and I can only try to understand the general notions and the possible meaning of the results from TGD point of view.
Hyperbolic n-manifolds are n-manifolds equipped with complete Riemann metric having constant sectional curvature equal to -1 (with a suitable choice of length unit) and therefore obeying Einstein's equations with cosmological constant. They are obtained as coset spaces on proper-time constant hyperboloids of n+1-dimensional Minkowski space by dividing by the action of discrete subgroup of SO(n,1), whose action defines a lattice like structure on the hyperboloid. What is remarkable is that the volumes of these closed spaces are homotopy invariants in a well-defined sense.
What is even more remarkable that hyperbolic 3-manifolds are completely exceptional in that there are very many of them. The complements of knots and links in 3-sphere are often cusped hyperbolic 3-manifolds (having therefore tori as boundaries). Also Haken manifolds are hyperbolic. Says Wikipedia:
According to Thurston's geometrization conjecture, proved by Perelman, any closed, irreducible, atoroidal 3-manifold with infinite fundamental group is hyperbolic. There is an analogous statement for 3-manifolds with boundary.
Therefore there are very many hyperbolic 3-manifolds.
The geometrization conjecture of Thurston allows to see hyperbolic 3-manifolds in a wider framework. The theorem states that compact 3-manifolds can be decomposed canonically into sub-manifolds that have geometric structures. It was Perelman who sketched the proof of the conjecture. The prime decomposition with respect to connected sum reduces the problem to the classification of prime 3-manifolds and geometrization conjecture states that closed 3-manifold can be cut along tori such that the interior of each piece has a geometric structure with finite volume serving as a topological invariant. There are 8 possible geometric structures in dimension three and they are characterized by the isometry group of the geometry and the isotropy group of point.
Important is also the behavior under Ricci flow ∂tgij= -2Rij: here t is not space-time coordinate but a parameter of homotopy. If I have understood correctly, Ricci flow is a dissipative flow gradually polishing the metric for a particular region of 3-manifold to one of the 8 highly symmetric local metrics defining topological invariants. This conforms with the general vision about dissipation as source of maximal symmetries. For compact n-manifolds the normalized Ricci flow ∂tgij= -2Rij +(2/n)Rgij preserving the volume makes sense. Interestingly, for n=4 the right hand side is Einstein tensor so that the solutions of vacuum Einstein's equations in dimension four are fixed points of normalized Ricci flow. Ricci flow expands the negatively curved regions and contracts the positively curved regions of space-time time. Hyperbolic geometries represent one these 8 geometries and for the Ricci flow is expanding. The outcome is amazingly simple and gives also support for the idea that the preferred extremals of Kähler action could represent maximally symmetries 4-geometries defining topological invariants: the preferred extremals would be maximally symmetric representatives with a given topology or algebraic geometry.
The volume spectrum for hyperbolic 3-manifolds forms a countable set which is however not discrete: the statement that one can assign to them ordinal ωω does not have any obvious meaning for the man of the street;-). What comes into my simple mind is that p-adic integers and more generally, profinite spaces with infinite number of points, might be something similar: one can enumerate them by infinitely long sequences of pinary digits so that they are countable (I do not know whether also infinite p-adic primes must be allowed and whether they could somehow correspond the hierarchy of infinite ordinals). They are totally disconnected in real sense but do not form a discrete set since since can connect any two points by a p-adically continuous curve.
What makes twistor people excited is that the polylogarithms emerging from twistor integrals (see this and this) seem to be expressible in terms of the volumes of hyperbolic manifolds. What fascinates me is that the polylogarithms in question make sense also p-adically and that the moduli spaces for causal diamonds -or rather, for the double light-cones associated with their M4 projections with second tip fixed - are naturally lattices of the 3-dimensional hyperbolic space defined by all positions of the second tip and 3-dimensional hyperbolic spaces are the most interesting ones! In the intersection of the real and p-adic worlds both algebraic universality and finite measurement resolution require number theoretic discretization so that the 3-volume volume could be quantized in discrete manner.
For n=3 the group defining the lattice is a discrete subgroup of the group of SO(3,1) which equals to PSL(2,C) obtained by identifying SL(2,C) matrices with opposite sign. The divisor group defining the lattice and hyperbolic spaces as its lattice cell is therefore a subgroup of PSL(2,Zc), where Zc denotes complex integers. Recall that PSL(2,Zc) acts also in complex plane (and therefore on partonic 2-surfaces) as discrete Möbius transformations whereas PSL(2,Z) correspond to 3-braid group. Reader is perhaps familiar with fractal like orbits of points of plane under iterated Möbius transformations. The lattice cell of this lattice obtained by identifying symmetry related points defines hyperbolic 3-manifolds. Therefore zero energy ontology realizes directly the hyperboliic manifolds whose volumes should somehow represent the poly-logarithms.
The volumes are topological invariants in the sense that homeomorphism does not affect the volume of the space in question if it is given hyperbolic metric. The spectrum of volumes is said to be highly transcendental. In the intersection of real and p-adic worlds only algebraic volumes are possible unless one allows extension by say finite number of roots of e (ep is p-adic number). The p-adic existence of polylogarithms suggests that also p-adic variants of hyperbolic spaces make sense and that one can assign to them volume as topological invariant although the notion of ordinary volume integral is problematic. In fact, hyperbolic spaces are symmetric spaces and the general arguments that I have developed earlier allow to imagine what the p-adic variants of real symmetric spaces could be.
What about AdS counterparts of hyperbolic manifolds?
Not surprisingly, also AdS-CFT enthusiasts would like to have similar invariants for for AdS (Minkowskian analog of hyperbolic space) and even dS (Minkowskian analog of sphere). Mitchell Porter gives a link to the talk of Maldacena. The expected non-compactness of these spaces implies infinite volume and this problem should be circumvented somehow.
Maybe the preferred role of hyperbolic spaces over AdS and dS might finally select between TGD and M-theory like approach. This would simplify matters enormously since 10-dimensional holography would reduce to 4-dimensional one and would have a direct connection with physics as we have used to know it. For condensed matter physicists expected to say something interesting about this real world already the complexities of 3-D world represent a tough enough challenge and the formulation of the problems in terms of 10-dimensional blackholes migh be too much;-).