As Kitaev will tell you, quantum codes are about things like ribbon categories. Ribbons and knots, knots and ribbons.
Back in the mid '90s, everyone was into quantum field theories with knots. I was supposed to be working in standard lattice QCD, in a Physics department, but being enamoured of knots I instead found myself lost amongst the deserted stacks of the mathematics library, and chatting to mathematicians like Bai-Ling Wang and Alan Carey. At that time, classical geometry was still boss, and everywhere I looked there were manifolds, manifolds, manifolds. The mathematicians weren't at all bothered by all the manifolds, but as a physicist who haunted the library stacks, I was looking for something more, let's say, background independent. Alan gave me some excellent advice, and told me to work on my thesis and forget about the knots (but naturally, the advice was bound to be ignored).
Anyway, a standard gadget in the knotty manifold world was a framed knot, where one thickened a knot into a kind of twisted torus so that it sat in a nice way in a larger space. By drawing lines along the torus, one could get ribbon diagrams. For example, quantum mathematicians liked symplectic spaces, such as $T^{*}S^3$, the cotangent bundle of the three dimensional sphere $S^3$. Such a six dimensional space has three dimensional Lagrangian submanifolds, good for putting knots in, and with which mathematicians can do some knot wizardry.
In time, the trend moved towards combinatorial alternatives. The only constant was the universal obsession with knots and ribbons. Six strands for three ribbons. I could no longer afford to listen to mathematics lectures in Princeton or Canberra, and nobody was willing to supervise my thesis proposal on a lovely $7$ dimensional Chern-Simons theory, so I went to Wanaka and bumped chairs and served chips to hungry skiers, and worried instead about the bottletops that my flatmate insisted on leaving all over the floor.
14 years ago
Is your "7 dimensional Chern-Simons theory"
ReplyDeletesomething that would fit well with
a (3+1) + 4 = 8-dim Kaluza-Klein model
in which
the 3 of the 3+1 corresponds to an associative calibration 3-form
and
the 4 corresponds to a coassociative calibration 4-form ?
Do you have a web reference for your 7-dim Chern-Simons ?
Tony
Hey Kea,
ReplyDeletehttp://snarxivblog.blogspot.com/2012/01/dharwadker-and-khachatryans-prediction.html
You are mentioned on this blog I follow. I, of course take this four color idea seriously too, perhaps for different reasons, one that ties together all the hierarchy of planes including the good Old Lagrangians Sines.
Anyway, in my childhood experience and mythology the Bottle cap is a powerful symbol that changes our paths in time. I think when we reach some mountain top we no longer care what the world thinks or what may have been or that detractors are stuck in their world of the receding past.
The PeSla whose best evidence of any abilities at all formally while the world excludes one in the background- Is that I see the significance of your work... Happy New Year.
Happy 2012, The Pe Sla. Tony, I don't think I ever wrote anything about the $7$ dimensional theory. Certainly nothing from before 2002 would be online, anyway. No, I was not thinking of a Kaluza-Klein idea. I was obsessed at the time with the homotopy groups of spheres, and $S^7$ seemed particularly pretty, so why not add knotty ($5$d) submanifolds. I don't even remember exactly what action I was considering, but there was some intuition from the octonions.
ReplyDelete