Loop variables for string theory

[mitchell porter]

"Loop variables for string theory"

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

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" .

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".

 

[suprised]

String theory is equivalent to a field theory with an infinite number of fields of arbitrarily high spin, most of them massive.

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.

 

[mitchell porter]

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:

Since the discovery of renormalizability of gauge theories we have come to treat the infinities in quantum field theory as a problem with the existing mathematical formalism rather than as a serious physical or phenomenological problem. However, the discovery of asymptotic freedom and the resulting dimensional transmutation in QCD, both facts of nature, seems to indicate that these infinities have a physical cause. They seem to suggest that the idea of a space-time continuum must be discarded and be replaced by some discrete lattice-like entity. This is of course what is done in lattice gauge theory but there the lattice is treated as a temporary scaffolding to be discarded at the end. It is possible to do this because the theories have a symmetry, namely, renormalizability...

We would like to suggest here, as a first step, that one should take the lattice seriously and not as a formal device This is not a new idea, particularly in the context of quantum gravity. Another way to regularize a field theory is to embed it in a string theory. We make the speculative conjecture that these two methods are in fact the same. We will assume that the links of the lattice are dynamical and identify them with the string. After all, in lattice gauge theory the gauge fields are placed on the links and if we place an infinite tower of higher-spin fields, the links would, so to speak, come to life. In fact, if we are to keep a finite lattice size it seems hard to maintain Lorentz invariance without making the whole lattice dynamical... The idea of a dynamical lattice is not new, only the connection with string theory is new.

It would be contrary to the spirit of things if we had to specify what kind of lattice one must have. In fact it is natural to require that physics should be essentially independent of the nature of the lattice ("universality"). We should perhaps elevate this to a gauge principle analogous to general coordinate invariance. The gauge invariant elements are of course the S-matrix elements of string theory, although their geometrical significance in the context of a lattice is not clear. Presumably we live in a phase where this symmetry is broken down to the usual gauge symmetries. Perhaps the symmetries found in ref [30] are remnants of this generalized "renormalization group" symmetry. Symmetry under global renormalization group transformations in the limit of infinite cutoff (renormalizability) is a requirement that we usually make in field theories. We are suggesting that if we strengthen this requirement in the manner outlined above we could end up with string theory.

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" .