M Theory Lesson 50
Fundamental to an n-logos is the concept of n-ality. Dualities belong to 2-logos structures, which are built upon the binary logic of the line element [0,1] and the parity square. In modern language, a duality expresses an equivalence of categories, which we take to be a bicategorical concept. In M Theory we see that triality is about the interplay of three parity cubes, based on ternary logic for the values 0,1 and 2.
Observe that connections between the logic values form a basic n-simplex, which may be labelled with directed faces as in Street's orientals, which describe strict n-categories. Unsurprisingly then, Kapranov's non-commutative Fourier transform is built upon simplices and cubes. This kind of Fourier transform should be a basic construction in M Theory, as it was for Heisenberg and Dirac.
Observe that connections between the logic values form a basic n-simplex, which may be labelled with directed faces as in Street's orientals, which describe strict n-categories. Unsurprisingly then, Kapranov's non-commutative Fourier transform is built upon simplices and cubes. This kind of Fourier transform should be a basic construction in M Theory, as it was for Heisenberg and Dirac.
posted by Kea | 3:27 PM
7 Comments:
Comment for kneemo: between Witten's new BH work, and your chat with Smolin ... now both camps will be 100% convinced that they're right, and ALL hell will break loose.
Witten’s ideas have a rather interesting relation to my own work, where the role of AdS3 is taken by a generic light-like 3-surface with boundary and itself defining an analog of blackhole horizon in 4-D case. I do not want however to write lengthy comments here so that I give a link to my blog.
Yes, it appears that quantum black holes provide the common ground for both camps. However, there still remain mysterious origins in quantum information theory. I suspect that the new "basic principle" behind quantum gravity is closely tied to the concept of information entropy.
Matti, thanks for the link.
closely tied to the concept of information entropy
Yes, kneemo, this is where the fancy new higher monads come in. Funny, I had a research proposal along these lines a couple of years ago, but nobody thought it was worth worrying about. Oh, well.
To me, this is all just a part of Mach's Principle for inertia, which is at the heart of causality. The right monads will form a 'causal diamond' for idempotent swapping structures. On one side of the diamond is Entropy and on the other is Information. The Entropy is about 'looking outward' (the arrow of time) whereas information gathering requires probing things more closely. But one must not mistake all this for the simple SpaceTime duality that crops up, because these monads are prior to everything else, including space etc.
It would be nice for Tony Smith to comment on the octonionic construction of the Leech lattice. From what I recall, the octonion integers play an integral role in the construction of the Gosset lattice (E8 root lattice), and Dixon has a similar construction for the Leech lattice using octonions with coordinates in Z/2.
To relate this to extremal black holes, we regard the exceptional Jordan algebra as a twenty-seven dimensional vector space over Z. This is the charge space for d=5 N=2 magical sugra, where the trace inner product takes values in Z (charge quantization). The off-diagonal octonion integers span twenty-four dimensions over Z, forming a charge lattice, from which the black hole can take its values. This twenty-four dimensional charge lattice is likely related to the Leech lattice via halving-sets.
Yes, I would like to hear from Tony Smith, especially with regard to the Surreal numbers and the Leech lattice. The tree picture of the surreals seems to crop up in 'complexifying' the 1-operad tilings. I mentioned this to Batanin, thinking he would just dismiss it, but he said it reminded him of the permutohedron work.
Oh, by the way, remember that we are trying to generalise the zeta function for integral values to other arguments, and what the surreal tree does is allow one to step in other directions, not just upwards towards omega.
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