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Loop variables for string theory

  1. Aug 12, 2011 #1
    "Loop variables for string theory"

    That is the title of a http://arxiv.org/abs/hep-th/0207098" [Broken].

    I know quite a few people have wondered what string theory would look like in the framework of loop quantum gravity. Well, this is a glimpse of how it would be. Only a glimpse, because no-one else is building on Sathiapalan's work and so, like the early days of string theory, it's possible that he's overlooking the right constructions. But the spirit of it must be correct. I had vague thoughts along these lines myself, but this makes it all very clear and retrospectively obvious.

    String theory is equivalent to a field theory with an infinite number of fields of arbitrarily high spin, most of them massive. (These fields correspond to the different modes of the string.) So obviously, if you form a Wilson loop in such a theory, you'll be integrating an infinite tower of higher-spin fields around the loop. It's even more obvious that this is how it would be for Vasiliev higher-spin gauge theories, and something like this must be implicit in http://arxiv.org/abs/1102.3297" [Broken].

    At first glance, I'm finding Sathiapalan's algebraic constructions somewhat opaque and baroque. As I said, since he's doing this alone, he may be missing some details. (Perhaps it can be compared to http://arxiv.org/find/all/1/au:+kroyter/0/1/0/all/0/1" on lattice string field theory.) But I think that in essence this must be the right way to do "stringy LQG", and it offers yet another direction for comparison with "non-stringy LQG".
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Aug 12, 2011 #2
    Re: "Loop variables for string theory"

    No, string theory is "more" than an infinite collection of particle fields. If you look carefully, the "Feynman rules" are different from the Feynman rules of ordinary particle QFT. Hidden is the geometry of the moduli spaces of Rieman surfaces, which is intrinsinicly different to Schwinger parameter integration domains of QFT loop amplitudes. And this is precisely what makes string theory consistent (unitary scattering amplitudes) in contrast to particle theory. One may phrase this also in terms of the modular properties of string integrands which are at the heart of finiteness and absence of anomalies.

    The whole point is of course that one needs to go beyond particle QFT when one is dealing with quantum gravity.
  4. Aug 13, 2011 #3
    Re: "Loop variables for string theory"

    That was a very helpful remark... In that statement of mine, which you quoted, I took myself to be stating a fact, even though I couldn't quite see how it was true. I thought maybe there was a change of variables in string field theory that could somehow turn it into a QFT with an infinite number of fields. But the obvious problem is, how could that be, if there are no gauge-invariant observables except at infinity? So I did some reading and some thinking and I now have a much better sense of the various ways in which string theory resists being reduced to a local QFT in target space.

    As for Sathiapalan, I've now understood that his construction involves modifying the usual sum over Riemann surfaces, so that instead of asymptotic states being represented by operators at points, he excises a disk of finite area - so the Riemann surface has a boundary for each external state - and then he takes a limit in which the disk shrinks to a point. His Wilson lines or loops describe currents on that boundary. It seems like it ought to make sense in some form; as if you could describe string theory in flat space in terms of a "holographic LQG" based on Wilson loops at infinity.

    Here's something he says in his 1989 paper:
    Ref. 30 is just "D J Gross, Princeton University preprint (1988)" but it may refer to http://prl.aps.org/abstract/PRL/v60/i13/p1229_1" [Broken].
    Last edited by a moderator: May 5, 2017
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