Thursday, September 29, 2011


The pole $\zeta (1)$ of the Riemann zeta function is associated to a Hagedorn temperature $T_{H}$. That is, when a particle's energy matches this special temperature, such as the CMB in the case of mirror neutrinos, we get the argument $s = E / kT_{H} = 1$, and the partition function diverges. For quarks, $T_{H}$ marks the transition to a quark gluon plasma.


  1. And for strings, even just in QCD, the Hagedorn temperature can mean the appearance of a tachyon. OK, I'm impressed. :-) I'm not sold, but I'm impressed.

  2. This paper describes BCFW recursion for open string tachyons (equation 6.24). That is, n-point tachyonic amplitudes are expressed in terms of simpler tachyonic amplitudes.

    This paper says that BCFW recursion for open string "gluons" (i.e. the spin-1 modes) should be the same thing but more complicated (page 9), and also mentions (page 38) that the vertex operators defining asymptotic string states obey a braiding relation.

    Apparently, stringy BCFW recursion is like field-theoretic BCFW recursion, except that certain infinite sums of poles appear in the formulae, corresponding to intermediate massive string states. It would be nice if someone had figured out field-theoretic tachyonic BCFW recursion already, but for now we only have the stringy version.

  3. Chapter 2 of this thesis constructs an action for a scalar tachyon field coupled to a gauge boson field, which might be useful in the development of BCFW recursion rules for tachyonic field theory.

    This talk and the accompanying paper, by Alexander Gorsky, open further connections to numerous juicy topics, like the BDS remainder function of N=4 Yang-Mills, which is where Goncharov's motivic formulae entered mainstream "amplitudeology". In the present context, I would point to section 3 of the paper, which talks about noncritical strings with central charge c=1. This is a (2+1)-dimensional space in which "The only physical modes are massless tachyons generically gravitationally dressed by the Liouville modes." This paper (and its reference 10) develop a very simple-looking diagrammatical calculus for the c=1 model - unfortunately the diagrams don't show up in the arxiv preprint, I had to look at the journal to see them... In Gorsky's words (page 9) "It turns out that tau-function of the Toda hierarchy serves as the generating
    function for the tachyonic amplitudes". You get multiple times from "noncompact FZZT branes"; the whole thing has a dual description in terms of fermions on a Riemann surface (possibly a moduli space for the c=1 model) which obey a "Baxter equation"... And then Gorsky wants to use all of this as a prototype for a deeper understanding of Yang-Mills theory (which of course includes QCD, bringing us back to my first comment).

  4. So in the end what do we have? Fermions that are dual to gravitationally dressed tachyons, with some twistorial (BCFW) and braid (Baxter) connections. (By the way, the tau function mentioned above is not the one from number theory; but it has something to do with Grassmannians. Also by the way, I read somewhere that Nima et al's twistor Grassmannian may have an interpretation as the moduli space of the twistor string.)

    In the present context, the message seems to be that it was wrong to just project the tachyons out of the bosonic string. They have a function, and somehow they're dual to fermions!!

  5. OK, Mitchell, and thanks for the good links. But the twistorial picture should be written in categorical motivic terms, and not in stringy terms. That is the key to why the stringy physics was wrong, as I have been trying to explain for 10 years.

  6. I suspect that you will never be sold by the theoretical stance. After all, I have been right, ahead of time, about quite a long list of things now, and you continue to ignore that fact. It is clearly more convenient to slot QG into the String Society, from which I am eternally excluded.


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