M Theory Lesson 30
A paper by D. Thurston introduces the idea of a knotted trivalent (ribbon) graph (KTG). It turns out that these gadgets may be generated from three simple graphs, namely the unknotted tetrahedron 
and two unknotted Mobius strips, one with a left twist and one with a right twist. The moves allowed on single graphs are the bubble move 
and the unzip move 
There is also a connected sum operation, which splices two graphs together along an edge. Any knot may be represented by a string of KTG operations. Bar-Natan's important paper on non-associative tangles includes a pentagon relation, which Thurston encodes via KTG moves. The sequence of three moves begins with three unknotted tetrahedra, which are connected via sum and then unzipped to obtain the final triangular prism 
This is of course reminiscent of the gluing of three sides of the parity cube 
on which the categorified Mac Lane pentagon appears.
and two unknotted Mobius strips, one with a left twist and one with a right twist. The moves allowed on single graphs are the bubble move 
and the unzip move 
There is also a connected sum operation, which splices two graphs together along an edge. Any knot may be represented by a string of KTG operations. Bar-Natan's important paper on non-associative tangles includes a pentagon relation, which Thurston encodes via KTG moves. The sequence of three moves begins with three unknotted tetrahedra, which are connected via sum and then unzipped to obtain the final triangular prism 
This is of course reminiscent of the gluing of three sides of the parity cube 
on which the categorified Mac Lane pentagon appears.
posted by Kea | 12:53 PM
4 Comments:
Dear Kea,
I wish I understood more about this. In any case, I have the feeling that (contrary to my own earlier beliefs) the generalizations of braid diagrams by allowing vertices between strands is not the solution to the construction of generalized Feynman graphs (the above-illustrated trivialization of a tree diagram describing two-strand scattering in s-channel supports this view).
Rather, I believe that one should give up stringy diagrams and generalize old-fashioned Feynman diagrams by glueing the 3-D light-like surfaces carrying braids along their 2-D partonic ends. For stringy option light-like 3-surfaces would be manifolds but vertices highly singular 1-manifolds (say "eye-glasses") and tensor product of incoming and outgoing state spaces would have have no natural representation which certainly relates to the difficulties in construction of vertex operators.
For generalized Feynman diagrams vertices would be 2-manifolds and tensor product would be extremely naturally represented. Light-like 3-surfaces would be singular (branched) manifolds. An interesting question is what this implies for 4-surfaces (lightlike 3-surfaces need not be boundaries of 4-surface but more like causal horizons).
In any case, entire braids would replicate. I would be happy if these brilliant mathematicians could categorize this view and could explain everything so that even I could understand!
You mentioned earlier planar algebras and operads, which relate very naturally to inclusions. Do you have a good not-too-technical reference about this?
Matti
Dear Kea,
it occurred to me that planar algebras might have interpretation in terms of planar projections of generalized Feynman diagrams (these structures are metrically 2-D by presence of one light-like direction so that 2-D representation is especially natural).
Some arguments in favor of this interpretation.
a) Planar diagrams describe sequences of inclusions of HFF:s and assign to them a multi-parameter algebra corresponding indices of inclusions. They describe also Connes tensor powers in the simplest situation corresponding to Jones inclusion sequence. Suppose that also general Connes tensor product has a description in terms of planar diagrams. This might be trivial.
b) Generalized vertices identified geometrically as partonic 2-surfaces indeed contain Connes tensor products. The smallest sub-factor N would play the role of complex numbers meaning that due to a finite measurement resolution one can speak only about N-rays of state space and the situation becomes effectively finite-dimensional but non-commutative.
c) The product of planar diagrams could be seen as a projection of 3-D Feynman diagram to plane or to one of the partonic vertices. It would contain a set of 2-D partonic 2-surfaces. Some of them would correspond vertices and the rest to partonic 2-surfaces at future and past directed light-cones corresponding to the incoming and outgoing particles.
The basic fact about planar algebras is that in the product of planar diagrams one glues two disks with identical boundary data together. One should understand the counterpart of this in detail.
a) The boundaries of disks would correspond to 1-D closed space-like stringy curves at partonic 2-surfaces along which fermionic anti-commutators vanish.
b) The lines connecting the boundaries of disks to each other would correspond to the strands of number theoretic braids and thus to braidy time evolutions. The intersection points of lines with disk boundaries would correspond to the intersection points of strands of number theoretic braids meeting at the generalized vertex.
[Number theoretic braid belongs to an algebraic intersection of a real parton 3-surface and its p-adic counterpart obeying same algebraic equations: of course, in time direction algebraicity allows only a sequence of snapshots about braid evolution].
c) Planar diagrams contain lines, which begin and return to the same disk boundary. Also "vacuum bubbles" are possible. Braid strands would disappear or appear in pairwise manner since they correspond to zeros of a polynomial and can transform from complex to real and vice versa under rather stringent algebraic conditions.
d) Planar diagrams contain also lines connecting any pair of disk boundaries. Stringy decay of partonic 2-surfaces with some strands of braid taken by the first and some strands by the second parton might bring in the lines connecting boundaries of any given pair of disks (if really possible!).
Best Regards,
Matti
Rather, I believe that one should give up stringy diagrams and generalize old-fashioned Feynman diagrams by glueing the 3-D light-like surfaces carrying braids along their 2-D partonic ends.
OK, Matti. We are not trying to do string diagrams, but we are turning Feynman diagrams into twistor ones, and then categorifying. If I try and translate to your language, that would mean the light-like surfaces are our tri-operad twistor diagrams for octonions (somehow) and the braids live in the 2-categorical boundaries. Hmmm. This is starting to sound similar. Good. Does that make sense?
Although I am still learning Max Plus Algebra, I do think that it may be possible to relate Feynman diagrams to Petri Nets [with transitions and representative token economy]. Petri Nets are said to be able to handle “thousands of variables” and I think, though have not seen, that 3D models are possible.
The tokens should be able to represent not only money but energy quanta of various scales or gauges.
positron \ ............... / electron
........... >~~photon~~<
electron / ............... \ positron
o ... o
.>|< where '|' represents a transition
o ... o
[Disregard the "." used to line up node ["o" or vertex] arc ["<" or "\" or edge]]
The transition is a source / sink in Petri nets.
The "~~" is the transition from annihilation transformation to creation transformation in Feynman diagrams.
Small glossary [from Max Plus Home Page, subsection]
“DES - discrete event system
“Dioid or idempotent semiring - algebraic structure endowed with an addition which is associative, commutative, with a "zero" element, with a multiplication which is associative, with a "one" element, the multiplication being distributive with respect to addition, zero being absorbing for multiplication (zero times a equals zero for all a); at last, addition is idempotent, namely a plus a equals a for all a.
“Event graph - Petri net in which each place has a single upstream (input) and a single downstream (output) transition, hence there is no competition in the supply of tokens to places nor in the consumption of tokens out of places; transitions may have several inputs and outputs, hence synchronization constraints can be represented in those graphs.”
http://www-rocq.inria.fr/MaxplusOrg/
WM McEneaney [Math & Mechanical and Aerospace Engineering, UCSD] has many PDF and a few LaTex Papers, on his publication list, including [4th from top]
‘Max-Plus Eigenvector Methods for Nonlinear H_infinity Problems: Error Analysis’ Siam J. Control and Opt. 43:379-412, 2004, which includes a discussion of truncation errors [only the 2003 submission on this site]
http://www.math.ucsd.edu/~wmcenean/pubs/
There is an arXiv paper on Max Plus 'How to find horizon-independent optimal strategies leading off to infinity: a max-plus approach' [math/0609243] by Marianne Akian, Stephane Gaubert, Cormac Walsh [13 pages, 5 figures, To appear in Proc. 45th IEEE Conference on Decision and Control] Abstract: A general problem in optimal control consists of finding a terminal reward that makes the value function independent of the horizon. Such a terminal reward can be interpreted as a max-plus eigenvector of the associated Lax-Oleinik semigroup. We give a representation formula for all these eigenvectors, which applies to optimal control problems in which the state space is non compact. This representation involves an abstract boundary of the state space, which extends the boundary of metric spaces defined in terms of Busemann functions (the horoboundary). Extremal generators of the eigenspace correspond to certain boundary points, which are the limit of almost-geodesics. We illustrate our results in the case of a linear quadratic problem.
http://arxiv.org/abs/math.OC/0609243
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