M Theory Lesson 47

As Louise Riofrio likes to point out, Einstein was fond of discussing imaginary Time, so that the metric could be written with a bunch of plus signs. In octonionic M Theory, it can be useful to think about 3 Times and 3 dual Space directions. The reduction to complex numbers gives one piece of the triality, and so we would like to consider just one Space and one Time dimension as a complex plane, with a duality that interchanges them.

Recalling that the complex (unitary ensemble) ribbon matrix models are self-dual under T duality, it is natural to consider S duality, or rather electric-magnetic duality, as a candidate operation for the study of this Space Time interchange.

In the 19th century, J. C. Maxwell unified electricity and magnetism by observing the apparent duality in the known phenomena of (1) magnetic field loops about a current line and (2) electric fields emanating from a pointlike charge. He used hexagons in his description of the aether through which electromagnetic waves travel.

Today we are interested in problems such as the Riemann hypothesis for the zeta function, which has zeroes on both the $x$ axis of $\mathbb{C}$ and the line $x = \frac{1}{2}$, and most probably nowhere else. Recall that the (upper half of the) vertical line appears in the natural description of the moduli space for the 1-punctured torus (an elliptic curve), as the boundary of a fundamental domain for the modular group, which just happens to leave invariant some BPS black hole masses when one considers a transformation for the complex parameter $\tau = \frac{\theta}{2 \pi} + i \frac{4 \pi}{e.e}$, where $e$ is electric charge and $\theta$ is an instanton parameter. Chalmers has considered the connection between the four point amplitude in N=4 SUSY YM and the Riemann hypothesis.

5 Responses so far »

  1. 1

    Matti Pitkanen said,

    Dea Kea,

    a conservative view;-). Electric magnetic duality suggests that the 3+3-dimensional space could be assigned with electric and magnetic components of the antisymmetric field tensor of electrodynamics.

    This would not require giving up 4-D space-time and would allow to circumvent problems by SO(3,3) symmetry and 6-D momenta (tachyons, apparently continuous mass spectrum,…). Action density F^{munu}F_{munu} is proportional to E^2-B^2 and therefore one can think (E,B) as a point of 3+3-D space.

    This however raises the question, what are the additional 6-component fields assigned to em field tensor by the triality transformation. Is SU(2) gauge field in question?



    P.S. Cold fusion seems to be making the final breakthrough. For links see my blog.

  2. 2

    Kea said,

    Thanks, Matti. I have been enjoying your posts on your blog.

  3. 3

    Doug said,

    This may be only a figment of my concept of imaginary and complex numbers.

    If the visible planet Earth [or Mercury or other celestial object] is a volumetric curve in Real Space,
    if the invisible magnetosphere [relative to the corresponding planet or other celestial object]
    is a toric entity in Imaginary Space,
    is the composition of Earth and its magnetosphere [or appropriate celestial object and its magnetosphere] a curved entity in Complex Space?

    If so, then these compositions resemble what I conceive as a Calabi-Yau manifold, although much, much larger than those usually discussed?

  4. 4

    Kea said,

    You’re on to it, Doug. Cheers.

  5. 5

    Matti Pitkanen said,

    Doug’s visualization applies as a metaphor also to particle +”dark magnetic body” although mathematics is different.

    M-theorists would speak about copies of imbedding space H as analogs of 8-D branes with discrete bundle projection H/G_axG_b characterizing given copy. Elementary particles (in the sense of partonic 2-surface) would belong to their 4-D intersection and field bodies to the bulk of these branes like structures.


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