## Twistors, hyperbolic 3-manifolds, and zero energy ontologyWhile performing web searches for twistors and motives I have begun to realize that Russian mathematicians have been building the mathematics needed by quantum TGD for decades while realizing the great visions of Grothendieck. Maybe I am also beginning to vaguely grasp something about the connection of Grassmannian twistor approach to the motivic integrals. In the following I make comments about three articles that I found from web. 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.
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 ∂
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 ω
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 M
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,Z
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 (e 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;-). For more details see the new chapter Infinite Primes and Motives or the article with same title. |