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The wrong turn of string theory: our world is SUSY at low energies

  1. Jan 9, 2018 #261
    It's in this archive: SemisimpleLieAlgebras.zip -- I have Mathematica, Python, and C++ versions.

    Its user interface is not very fancy. One specifies algebras with it as {family, rank}, where family is 1, for A, 2 for B, up to 7 for G. One specifies irreps as highest-weight vectors, like {1,0,0,0}. Its outputs are lists of lists. For product reps, it outputs a list of {multiplicity, highest-weight vector}. It also does powers of reps (plethysms), breaking them down by symmetry. Each symmetry type has a list like for product reps.

    It does compound algebras, like SU(3)(color) * SU(3)(flavor) * SU(2)(spin), and in a file on notable physics results, I obtain the light-quark baryon spectra.

    For maximal subalgebras, it supports just about every one that I could work out projection matrices for. This includes root demotions (algebra root -> U(1) factor), extension splitting (add a root then remove another root), SO(even) -> SO(odd) + SO(odd) (the other parities are handled by the previous two types), SU(m*n) -> SU(m)*SU(n) and similar for SO and Sp, and Slansky's list of exceptional-algebra breakdowns, including my favorite, E8 -> G2*F4. However, I don't have ones like SU(6) -> SU(3), because I couldn't work out general formulas for them, though I've worked out (algebra) -> SU(2). One uses the heights of roots (sum of root components) in it.

    It specifies irreps as basis sets, something like magnetic-quantum-number values for angular momentum. It has a list of {multiplicity, root, weight}. It also breaks down into Weyl orbits, each one specified with its highest weight. For each orbit, one can find a list of {root, weight} in it.
  2. Jan 9, 2018 #262
    So for Sp(32), we have
    Extension splitting of Sp(32) at 1
    Sp(30) * SU(2)
    Root demotion of Sp(30) at 15
    SU(15) * SU(2) * U(1)
    Product splitting of SU(15) into 5*3
    SU(5) * SU(3) * SU(2) * U(1)
    Root demotion of SU(5) at 3
    SU(3)^2 * SU(2)^2 * U(1)^2
    more-or-less (Standard Model)^2

    One can get all the Standard Model's multiplets out of it, some with multiplicities that can be multiple generations.
  3. Jan 10, 2018 #263
    This also means that I have been wrong since #240, in my assertions that these latest excursions are meant to implement a chiral sbootstrap. I thought that the SU(2) was supposed to be SU(2)L, but in fact it's something like a (gauged?) flavor symmetry.

    OK. So what is actually going on? The original, "non-chiral" sbootstrap, looks at meson and diquark pairings of the five lighter-than-top flavors of quark in the standard model, and obtains electric charge values corresponding to all the elementary fermions of the standard model, so one asks if there is a implicit supersymmetry in the standard model, or if some supersymmetric QCD can unfold into the full standard model when super-composites are considered.

    Around #237, the 54 of SO(10) was introduced as a kind of unified heuristic model of these mesons and diquarks, with 15 and 15* standing for (anti)diquarks, and 24 standing for mesons. In the further branching under SU(3)xSU(2)xU(1) described in #239, the only part that is actually SM-relevant is the U(1), which will mix with other U(1)s to imitate SM electric charge.

    Then in #242 and #244, SU(3)c is introduced, as part of a scheme to obtain all these groups from certain large groups (SO(32), E8xE8, now USp(32)) appearing as the ten-dimensional gauge group in various string theories. Since SO(32) is the gauge group of the Type I open string, there may have been an intention to recover a stringy structure of the "mesons" and "diquarks", if these branchings could be implemented there.

    But SU(2)L has not been introduced, and the U(1)s are combining to imitate electric charge, not weak hypercharge. So even if the scheme can be implemented as intended, it won't give us the standard model, it will give us the non-chiral SU(3)xU(1)em effective theory that follows electroweak symmetry breaking.

    I am going to have to rethink where this has gotten us, but meanwhile I want to say something about how SM-like models are actually obtained in the string theories with a ten-dimensional SO(32) gauge group, heterotic SO(32) and the type I string. The key problem is how to obtain chiral fermions.

    This paper
    tells us that in the heterotic SO(32) theory, "orbifold and Calabi-Yau compactifications" and also "toroidal compactification with magnetic fluxes" can do this, and presents a model of the last type, in which SO(32) is broken to G_SM by the fluxes, and some but not all of the SM fermions are obtained from the 496 (see section 3.1). I believe that the 496 branching described by @arivero could very probably be obtained in one of these flux models, but as I have explained, it would at best give us that non-chiral effective theory and not the full standard model.

    As for the type I theory, the SO(32) open string actually derives from 32 space-filling D9-branes. There are type-I models where other D-branes are also introduced, but it seems like the models closest to the philosophy espoused in recent posts, would be those in which the D9-branes are the only ones. From the literature I have gleaned the following: these type I models are often dual to heterotic Z_n orbifolds. "D9-branes only" corresponds to n odd, while n even corresponds to D9s and D5s. D9+D5 is considered more promising phenomenologically, but type-I/heterotic duality can be easier to prove with D9s only, since a D5 maps to an NS5-brane in the heterotic theory, i.e. the M5-brane, whose worldvolume theory is not well understood.
  4. Jan 10, 2018 #264


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    More than wrong, lets say mainstream :-) But yep, our SU(5) here looks as a flavor symmetry, and so the same when decomposed in SU(3)xSU(2); in the diquark picture, SU(3) moves flavour between d,s,b "subquarks" and SU(2) moves between "u,c" subquarks. The only hint of chirality is the total content of the 15, were we have a pair number of objects of charge -1/3 and of charge +2/3, but only an odd number of objects of charge +4/3. This is a small hint, telling us that perhaps we need to reorganize objects to have _L and _R symmetries, and that during such reorganization the odd object, +4/3, should dissappear (of the low energy spectrum, at least).

    Your review is accurate; we see that breaking SO(32)/Usp(32) we get a "infrared standard model", this is, the limit where only colour and electromagnetism survive. And we need other way down where the "standard standard model" (uh, I need a better name) appears but also with three generations, or perhaps with mirror generations if we do not see how to produce different complex representations.

    My current speculation is that the heuristic of thinking in terms of the 54 of SO(10) could be complemented with a 27 + 27 from elsewhere (perhaps E6, perhaps some 26 plus a singlet, perhaps a 24+3) and that this view should be the one showing the L and R gauge symmetries. In the ten-dimensional heaven, it would correspond to the connection between SO(32) and E8xE8, or to a connection between USp(32) and some other type 0 theory.

    About getting "only" SU(3)xU(1), -by now-, I would not be very disappointed. At least it means that we are not claiming -yet- forbidden miracles such as to get chiral theories from real representations. And a lot of the game in string theories is about "effective theories", i.e, about the content of a theory in an extreme limit. Three generations of colored electromagnetism is the limit of the SM where the yukawas of the fermions are cero but the electroweak vacuum (or at least the mass of W and Z) is infinite.
    Last edited: Jan 10, 2018
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