M Theory Lesson 192
Are the Foaming Loopies secretly doing String Theory at last? The latest paper by Markopoulou et al looks at ribbon graphs in two dimensions and three stranded diagrams in three dimensions. 
The three strands may be considered as tubes, much as in closed string diagrams. Then the open-closed string duality becomes a duality between 2d simplices and 3d ones. But how can this be? As a Poincare duality one exchanges 2d and 3d objects only in dimension 5, whereas this stringy duality is usually associated with 2-categorical structures. Fortunately, a moduli space perspective solves the mystery. The tube diagram for the tetrahedron is a 4 punctured sphere, the moduli of which is indeed two dimensional. The other two dimensional moduli space is the space of elliptic curves. These two moduli describe duality as envisaged by Grothendieck in his work on ribbon graphs for surfaces.
The three strands may be considered as tubes, much as in closed string diagrams. Then the open-closed string duality becomes a duality between 2d simplices and 3d ones. But how can this be? As a Poincare duality one exchanges 2d and 3d objects only in dimension 5, whereas this stringy duality is usually associated with 2-categorical structures. Fortunately, a moduli space perspective solves the mystery. The tube diagram for the tetrahedron is a 4 punctured sphere, the moduli of which is indeed two dimensional. The other two dimensional moduli space is the space of elliptic curves. These two moduli describe duality as envisaged by Grothendieck in his work on ribbon graphs for surfaces.
posted by Kea | 3:54 PM
3 Comments:
It's a shame if loop quantum gravity is now describing things which can't ever be experimentally refuted if incorrect.
Smolin put some Perimeter Institutes of his on quantum gravity online a few years ago, and I was impressed with the basic concept: to quantize gravity with a minimal amount of speculation.
In particular, I liked the idea of arriving at a gravitational force path integral by summing all of the interaction graphs for gravitational interactions in spacetime.
This seems a physically sensible approach. Smolin showed (in outline) in the first lectures that you can sum interaction graphs to get general relativity without a metric, which is what he calls background independence.
A metric is an output of general relativity for a particular set of input assumptions. So it's interesting that you can get the basic field equation without a metric from summing spin-foam interaction graphs over spacetime.
However, all of this is very abstract and it doesn't predict anything comparable to observation such as the relatively weak force of the gravitational interaction (relative to other fundamental forces).
One thing that I don't see any evidence for is the assumption in loop quantum gravity that the penrose "spin network" model for gravitational interactions in spacetime is physically the best model to use! Smolin's summation of interaction graphs is basically a summation of the Penrose spin network graphs, which is a very abstract ans questionable model of spacetime.
Penrose's own papers on spin networks, http://math.ucr.edu/home/baez/penrose/Penrose-AngularMomentum.pdf and http://math.ucr.edu/home/baez/penrose/Penrose-OnTheNatureOfQuantumGeometry.pdf, are entirely abstract models with no checkable predictions or even solid foundations in physical facts.
On page 18 of the first paper Penrose states:
"When the vertex connections have been completed at every vertex of a closed spin-network, then we shall have a number of closed loops, with no open-ended strands remaining."
This could physically be a model for the closed loops of graviton radiation being exchanged from gravitational charge A to charge B and then back again to charge A, in a endless cycle (closed loop).
However, the linkage between mathematical or geometric model and physical fact is so indirect and vague that it's just not very helpful physically.
On page 4 of the first paper, Penrose writes:
"I have referred to these line segments [in the Penrose spin network spacetime illustration] as representing, in some way, the world-lines of particles. But I don't want to imply that these lines stand just for elementary particles (say). Each line could represent some compound system which separates itself from other such systems for long enough that (in some sense) it can be regarded as isolated and stationary, with a well-defined total angular momentum n*(1/2)*h-bar. Let us call such a system or particle an n-unit. (We allow n = 0, 1, 2, ...) For the precise model I am describing, we must also imagine that the particles or systems are not moving relative to one another. They just transfer angular momentum around, regrouping themselves into different subsystems, perhaps annihilating one another, perhaps producing new units."
This is needlessly very vague, which is a pity. Why not physically describe something specific, such as gauge boson exchange between gravitational charges, and see where it leads? Why instead do they just pick one very vague spacetime model and work on that (the Penrose spin network)?
Sorry, the second paragraph of the first comment above is missing the word "lectures", and should begin:
"Smolin put some Perimeter Institute lectures of his..."
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