The First Edge of a Cube

I learned in Kea's blog about posting concerning something related to n-categories: the posting was The First Edge of the Cube. I did not understand much of it. I however tried to make something out of it and this boiled down to a posting to Kea's blog whose polished version with a couple of additional questions I attach below.

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 C₁(a,b). 1-parallel transport defines a map between fibers.

• n=2: 1-parallel transport along C₁(a,b) is parallelly transported to a 1-parallel transport along curve C₁(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 C₂(a,b|c,d).

• n>2: One can continue this and obtains at n:th level parallel transport of parallel transport of.....

Some comments.

• The ordered exponential representation for parallel transport suggests that n=1 parallel transport could define n-parallel transport. Probably something trivial and un-interesting.

• 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.

• 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?

• 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? Could one model the effect of non-gravitational forces on the motion by using n-connection not induced from 1-parallel transport?

• One could also generalize the notion of holonomy group. 2-holonomy group would be associated with cylinder-like surfaces C₂(a,b|a,b) with topology D×S¹. At higher levels you would have topology D×S¹×S¹ 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 could 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 (see this). 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