M Theory Lesson 48
Urs Schreiber at The Cafe has developed a sudden interest in cubes, with an interesting post called The First Edge of the Cube. He says, "all that is required is playing Lego with these squares: two of them paste together to give a transition function. Six to give a triangle. Twelve a tetrahedron – these shapes are the differential cocycle representing the differential cohomology class of the n-transport....One aspect of the power of this formulation of differential cohomology is that I haven’t even had to mention the word 'differentiable' yet. Nor the word 'continuous'." Meanwhile Tao has a post on triple L series, which I would like to understand, but I'm dreaming.
In the realm of gauge theory this is all very complicated, which is why we like to do things with n-logoses instead. An n-logos should be defined long before one starts worrying about gauge symmetry, out of the simple combinatorics of cubes and Street's orientals and also n-operad polytopes. For example, we saw the 9 faced Stasheff polytope arise in $\mathbb{R}^3$ when tiling the (real) moduli space of the 6-punctured sphere. An example of this reduction of 6 dimensions to 3 dimensions appeared yesterday when Doug mentioned that typical Calabi-Yau spaces are fibred manifolds with 3-torus fibres, and he associated the 3-torus with the three Time dimensions of twistor triality. A 1-dimensional torus is just a loop, like the magnetic field loops around a current line, mused over long ago by Maxwell.
Mirror symmetry on such a Calabi-Yau space is a kind of T duality. So we see the interplay of T duality and S duality in the world of the prime 3. The Langlands picture is thereby related to the evenness issue (2 is for matrix), mentioned in Schreiber's post. The true primes are the odd primes, starting with three. An investigation of pure mass in this context leads one to the simple formula for Baryon fraction, as found by Louise Riofrio, who correctly predicted, amongst other things, the molten core of Mercury (a modern day perihelion).
Meanwhile, Carl's work on Schwarzschild orbits has progressed. Here is his Gravity applet. Carl says, "As an aside, I almost wasted my time programming up the usual geodesic equations. In fact, I actually sat down to do it. What stopped me was the realization of how much more difficult the usual geodesic equations are to use than the simple equations of motion I've found here".
In the realm of gauge theory this is all very complicated, which is why we like to do things with n-logoses instead. An n-logos should be defined long before one starts worrying about gauge symmetry, out of the simple combinatorics of cubes and Street's orientals and also n-operad polytopes. For example, we saw the 9 faced Stasheff polytope arise in $\mathbb{R}^3$ when tiling the (real) moduli space of the 6-punctured sphere. An example of this reduction of 6 dimensions to 3 dimensions appeared yesterday when Doug mentioned that typical Calabi-Yau spaces are fibred manifolds with 3-torus fibres, and he associated the 3-torus with the three Time dimensions of twistor triality. A 1-dimensional torus is just a loop, like the magnetic field loops around a current line, mused over long ago by Maxwell.
Mirror symmetry on such a Calabi-Yau space is a kind of T duality. So we see the interplay of T duality and S duality in the world of the prime 3. The Langlands picture is thereby related to the evenness issue (2 is for matrix), mentioned in Schreiber's post. The true primes are the odd primes, starting with three. An investigation of pure mass in this context leads one to the simple formula for Baryon fraction, as found by Louise Riofrio, who correctly predicted, amongst other things, the molten core of Mercury (a modern day perihelion).
Meanwhile, Carl's work on Schwarzschild orbits has progressed. Here is his Gravity applet. Carl says, "As an aside, I almost wasted my time programming up the usual geodesic equations. In fact, I actually sat down to do it. What stopped me was the realization of how much more difficult the usual geodesic equations are to use than the simple equations of motion I've found here".
posted by Kea | 9:39 AM
4 Comments:
Dear Kea,
I wish I could connect these n-transports to something having a concrete physical meaning! For years ago I tried to understand n-parallel transport (or perhaps it was something related;-)) in terms of simple geometric mental images. I try to formulate my mis-understandings using the ancient terminology still used by physicists like me and these mental images. No arrows nor commuting diagrams which make me mad!
n=1: One starts with a parallel transport from point a to b along curve C_1(a,b). 1-parallel transport defines a map between fibers.
n=2: 1-parallel transport along C_1(a,b) is parallelly transported to a 1-parallel transport along curve C_1(c,d). One can say that one parallelly transports curve instead of point. 2-connection would define this parallel transport of parallel transport. One obtains a kind of square like structure C_2(a,b|c,d).
n>2: One can continue this and obtains at n:th level parallel transport of parallel transport of.....
Some comments.
a) The ordered exponential representation for parallel transport suggests that n=1 parallel transport could define n-parallel transport. Probably something trivial and un-interesting.
b) If the n-connection is non-flat, the n-parallel transport depends on how the curve evolves from the initial state to the final state.
c) A physically highly attractive possibility is generalized general coordinate invariance stating that the parallel transport depends only on the n-surface spanned by the curve. Is n-parallel transport induced by 1-parallel transport the only solution to this requirement?
d) One can wonder about the counterparts of geodesic lines. 1-parallel transport leaves the tangent vector field of geodesic line invariant. n-parallel transport should leave invariant the n-form defining tangent spaces of a geodesic n-surface? For n-parallel transport induced by 1-parallel transport geodesic sub-manifolds would probably result. What is the n-counterpart for the equations of geodesic line? Could one model the behaviour of extended objects in gravitational fields using these kind of equations?
e) One could also generalize the notion of holonomy group. 2-holonomy group would be associated with cylinder-like surfaces C_2(a,b|a,b) with topology DxS^1. At higher levels you would have topology DxS^1xS^1 and so on. You could also consider closed curve at n=1 level and get hierarchy of n-holonomy groups associated with n-tori. Of course also other topologies can result if the parallel transport is such that the surface develops pinches. Could one generalize the notion so that one cold assign say 2-parallel transport to a 2-torus. What to do when the curve for 1-parallel transport decomposes into two separate pieces? Just hop? Why not?
For years ago I assigned this kind of hierarchical structure of parallel transports to a hierarchical structure defined by infinite primes. I believe that this kind of abstractions about abstractions about..., thoughts about thoughts about... , statements about statements about... and repeated second quantization, represent fundamental new physics especially relevant for quantum consciousness theories.
Cheers,
Matti
Amazing news! My calculating of orbits has been shut down at shut down by management at Physics Forums!
Well, Carl, I'm not surprised. As ZapperZ said, "based on all reports"... of course, that wasn't including my report, which was quite positive, I thought. Anyway, we have your website, and the thread at PF will no doubt be left up, so not to worry.
Thanks, Matti. The transport heirarchy idea is now quite popular amongst the trendy set, and fairly well developed in mathematical language. But as you say, the number theory is more illuminating.
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