M Theory Lesson 22

The little disc diagrams of the last post are elements of the 2-discs 1-operad. To each $n \in \mathbb{N}$ we associate the diagram with $n$ distinct little discs inside the bigger disc. There is a $d$-discs 1-operad in any dimension $d$.

It is natural to wonder whether or not the boundary nature of operad mathematics can tell us anything about holography (this is Cambridge?). This has been discussed, such as in this article by Zois, which is motivated by Kontsevich’s original idea that the physical principle should be related to the higher dimensional Deligne conjecture. This conjecture now has many proofs (apparently), most notably via the operad methods of Batanin.

In other words, what we really care about are $n$-operad analogues of the 2-discs operad. Recall that 2-categorical TFTs were studied by Pfeiffer and Lauda. With objects, 1-arrows and 2-arrows, there is room for both open and closed strings. But rather than their stringy diagrams, we would like to invent higher operads of discs with extra data.

Moving up in dimension to 3-spheres (for building quaternions) we would naturally consider 3-sphere diagrams with embedded knots and links. The whole diagram is a 2-arrow, whereas the 3-sphere pieces are 1-arrows between the links as their local boundary. The nice thing about knots is that they have primes.

It sounds like we’re doing homotopy theory of spheres, doesn’t it? Well, that is in fact one way of looking at higher category theory, only we need to attach arrows to our diagrams.


3 Responses so far »

  1. 1

    Doug said,

    Hi Kea, I was reading the Runkel paper linking Frobenius algebras and TFTs.

    There is a Perron-Frobenius Theorem in Dynamic Noncooperative Game Theory [DNGT] that may even link to VOA.

    [In DNGT] Vertex is referred to as Node with the same properties.
    Edge is referred to as Arc which can be either a unit of time or a probability.
    Wasn’t Kneemo asking about what the edge represented in a previous thread?

    I would like to call your pants-faces pairing a STROOP diagram, but I think your diagram is historically like those in IEEE Phasor Theory developed by CP Proteus in the 1890s using Grassmann Algebra. This pairing reminds me of an architectural diagram with side and end-on views which is more powerful than a simple 2D view.

    With your diagrams, one might be able use a negative image.
    The apparent holes become particles or planets.
    The pants cylindrical surface becomes virtual with helical geodesic trajectories.
    In such a negative image format, one might view QM from a neutrino perspective and GR from a perspective in which space-time is the manifold with stars and planets tunneling through it rather than it being warped by massive objects.

    I am still looking for what I remember to be a dynamic rather than static representation of Phasors.

  2. 2

    Kea said,

    You mean Steinmetz? Hmm. That’s an interesting point. Takes me back to my youth, when I did quite a bit of electronics.

  3. 3

    Doug said,

    Yes Steinmetz, not his middle name Proteus’
    He was a master applied mathematician and electrical engineer. He is regarded as ‘the wizard who generated electricity from “i“’, from An Imaginary Tale: ‘The Story of “i” [the square root of minus one]’ (Paperback) by Paul J Nahin and page 7 this PDF.

    Background independent perspective in Architectural geometry:
    2D_pants_view + 2D_face_view = 3D_helis

    Perturbation perspective in Architectural geometry:
    2D_side_view + 2D_end_view + 2D_Top_view = 3D_helix
    virtual cylinder with apparent sine curve geodesic + loop + curved cylinder = helical geodesic on virtual curving cylinder

    Example Phasor Diagrams [some 2D. others as 3D in 2D]:

    1 – British Columbia Institute of Technology
    From Math examples: Trig & vectors
    Problem with solution by method of phasor addition.
    Phasor is a vector that represents a sinusoidal waveform.

    2- San Jose State U Physics 51
    Fig. 31.1 Phasors (Think of a phasor as a rotating vector.)

    3 – NI © 2007 National Instruments Corporation
    phase modulation Figure 19 A shows two sine waves that are identical in both frequency and amplitude. The only difference is that the blue wave would cross the center line before the red wave and is thus phase-shifted (is leading) a fixed amount from the red wave. B in turn, shows rotating vectors (phasors) representing these two sine waves. Since the vectors rotate in the conventional counterclockwise direction, the blue vector is also leading the red vector.

    Copyright 1998 All Rights Reserved C. Langton, mntcastle@earthlink.net
    Complex representation of Fourier series
    [Euler’s identity: -1 = e^{iPI}] = e^{jwt} = cos wt + i sin wt (1)
    Figure 3 – e^(jwt) plotted in three dimensions is a helix
    In Figure 3 cos wt is plotted on the Real axis and sin wt is plotted on the Imaginary axis. The function looks like a helix moving forward in time to the right. The X-Z and the Y-Z projections, if plotted, would be the sine and cosine functions

    5 – I was not able to find a phasor diagram is dynamically illustrated [synchronized side and end views] originally at a Kwamtlen U [BC, CA] website:
    Perhaps this can be obtained through Questions to mike.coombes@kwantlen.ca

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