String theory is made from Chew-ish strings, strings which follow Geoffrey Chew's S-matrix principle. These strings have the property that all their scattering is via string exchange, so that the amplitudes are entirely given by summing world sheets and by topology. This leads to a successful theory of gravity, we know, uniqueness, we know etc.

I would like to know exactly how much of the Chew-Mandelstam S-matrix philosophy, that proto-holographic principle which was so important for the development of string theory historically, is logically necessary for deriving the string interactions. This is obscured in more modern treatments, because these take for granted certain things that were historically difficult and came directly from the S-matrix.

One of those things which are taken for granted is that strings interact only by exchanging other strings, and that this interaction respects world-sheet duality. In a recent exchange (here: What are the details around the origin of the string theory?), Lubos Motl suggested that this may be derived only from the assumption that there is some sort of string excitation, and that this string interacts consistently. I would like to poke at this claim with examples, to determined exactly how many extra assumptions are hidden behind this claim.

The assumption that strings exchange strings with Doleh-Horn-Schmidt duality is equivalent to the assumption that the only interaction between strings is the topological exchange diagram from string theory. This assumption not only fixes the string interaction given the spectrum, it fixes the spectrum and even the dimension of space time, it fixes everything. So it is very weird physics. I believe that this assumption is Chew/Mandelstam physics, and that you can't derive strings without using it.

Stupid rubber band model

There are N scalar fields of mass m, and N U(1) gauge fields. Scalar k has a charge +1 under gauge field number k, and a charge -1 under gauge field k+1 (with periodic boundaries, so scalar N and 1 are interacting).

Each U(1)'s is broken from a separate SU(2) at some high energy by a technicolor mechanism which is irrelevant, but which embeds the U(1)'s in an asymptotically free theory at high energies, so that the theory is completely consistent.

These N scalars can form a matter-string, composed of a type 1 particle, bound to a type 2 particle, bound to a type 3 particle, etc around a loop closing back on particle 1. I will call this the rubber-band.

Further, the N different scalars are each separately in the fundamental representation of an SU(N) gauge theory, so that individual quanta of these scalars have infinite mass by confinement. The coupling of the SU(N) will be infinitesimal at the rubber-band scale, so that it will not affect the dynamics.

The rubber band is one of the baryons of the theory. If you just look at rubber-band dynamics, if you collide two rubber bands, you can exchange U(1) quanta between corresponding points on the rubber band, and get long range Van-der-Waals, splitting and rejoining interactions. But there is no duality.


Given a "baryon" in the model, it contains N scalars. If the scalars are not numbers 1...N, they don't bind together, and if the scalars are 1...N, then they bind together, releasing some U(1) photons and making a closed bound rubber-band at a lower energy by N times the binding energy of each pair (approximately). The isolated particles are infinitely massive, so the lightest baryonic state in the theory is the rubber band.

The rubber band has oscillation excitations. For large N, you get a spectrum which is by the frequency of a 1-dimensional fixed length string. There are modes which correspond to the N particles sitting on top of each other, rotating with 1 unit of angular momentum, moving apart and shrinking, etc.

The rubber band effective theory holds for energies above the bound-state baryon mass and below the mass of a cut rubber band, and it holds well above this too. But the interactions of the rubber band have nothing to do with string theory.


  1. Did I miss a failed conservation law which makes the rubber band decay?
  2. Would you consider the rubber band to be a relativistic string?
  3. How close to the real string spectrum can you get by mocking up rubber bands?
  4. Why do you say that string theory is only based on the assumption of "relativistic strings", when the relativistic rubber band is a "relativistic string" and does not obey duality?
  5. Is it obvious, given this model, that Chew bootstrap is necessary for deriving string theory, and that duality does not "obviously" follow from the existence of a string picture?
  • $\begingroup$ I am not sure where the question is here, instead it appears to be a statement about the authors idiosyncratic theory. Further, while the Smatrix program was indeed important in the history of string theory, there is a lot more mathematical structure than simply an SMatrix, as the AdS/CFT correspondance clearly shows. $\endgroup$ – Columbia Aug 22 '11 at 0:13
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    $\begingroup$ The question is to determine the exact assumptions which are implicitly used when making string theory. The "theory" above is just a stupid mathematical example, which has string-baryons with a fixed length and string harmonic excitations as the lowest energy baryon states, but which has absolutely none of the other properties of a real string theory. I will edit the question to make it clearer. AdS/CFT turns into S-matrix in the flat space limit, and I am only asking a mathematical question about relativistic theories with flat space stringlike spectrum. $\endgroup$ – Ron Maimon Aug 22 '11 at 0:40
  • $\begingroup$ What you describe sounds like a non-supersymmetric quiver gauge theory. So essentially you're asking whether this is in the "landscape" or in the "swampland", and why. $\endgroup$ – Mitchell Porter Aug 22 '11 at 3:41
  • $\begingroup$ To be more precise, your rubber band is a polygon made of 2-string junctions (as in arxiv.org/abs/hep-th/0403149 ), and may or may not have a realization in terms of a polygon made of open strings running between D-branes. Remember that string theory contains many other one-dimensional objects apart from fundamental strings, such as D1-branes and "little strings". $\endgroup$ – Mitchell Porter Aug 22 '11 at 4:02
  • $\begingroup$ I don't care if this is in the landscape (and its not a quiver theory because of the extra SU(N) on top, which is important). What I am asking is: here is a mathematical theory with a crappy stringlike spectrum. The rubber-bands in the theory are non-dual strings--- they don't obey the rule that their interactions are only through exchanging rubber-bands. So isn't this a counterexample to the statement that any string spectrum by itself requires duality? $\endgroup$ – Ron Maimon Aug 22 '11 at 6:28

In your construct, it's the segments of the rubber band which resemble the (open) strings of string theory - the flux lines connecting the scalars. The natural embedding of the rubber band into string theory would involve interpreting the scalars as zero-dimensional domain walls, with two wall-string junctions present at each such "0-brane". Since the walls would normally contain gauge theories, it seems it would have to be an N=2 gauge theory (in order to contain a scalar) broken to N=0 gauge theory. So the embedding of a rubber band into string theory, if it exists, might require a lot of structure invisible from your starting point; though perhaps that structure would show up in an AdS/CFT analysis of your technicolor theory?

edit: Yes, I think the "obvious" way to embed a field theory with "rubber bands" into string theory is to look for an appropriate N=2 superconformal quiver theory with an AdS dual, and then add masses / break susy with VEVs as in Seiberg-Witten 1994.

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    $\begingroup$ Yes, there are obvious brane deconstructions which get the rubber-band part of this as a low energy limit of some string theory (although remember that these things are also charged under an SU(N), and I don't know where you would put that brane stack). I am not asking about how to embed this in string theory. I am just pointing out that this theory has a string-looking spectrum and does not obey duality. So here is a rubber-band that does not scatter by dual string diagrams. Are you claiming that the embedding into string theory interprets the "rubber bands" as dual-strings? That's not right. $\endgroup$ – Ron Maimon Aug 22 '11 at 6:37
  • $\begingroup$ But the rubber band goes through a huge number of branes--- the "stringlike" part of it is just from the fact that it makes a polymer. The stringiness of each segment of the polymer is silly and uninteresting, that's just U(1) fluxes. $\endgroup$ – Ron Maimon Aug 22 '11 at 6:38
  • $\begingroup$ World-sheet duality is a property of string theory. If the segments of the rubber band do map to open strings, then world-sheet duality may be exhibited in their interactions with closed strings in the dual AdS space... Another possibility is that the scalars can be interpreted as world-sheet monopoles and that their presence "inside" the string interferes with the straightforward manifestation of world-sheet duality. $\endgroup$ – Mitchell Porter Aug 22 '11 at 6:38
  • $\begingroup$ Once a string hits a brane, then the interactions are by gauge fields, so all you are saying "this is a gauge theory", yeah ok. The point is that here is a non-fundamental silly rubber-band "string" sitting in this theory, and it doesn't work by string diagrams, whether the theory is supersymmetric or not. $\endgroup$ – Ron Maimon Aug 22 '11 at 6:44
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    $\begingroup$ But is it any different to a "QCD string"? The rubber band appears to be just a contrivance to link together a bunch of different QCD-like strings. I don't see anything in your argument - that this is a string that "doesn't work by string diagrams" - which doesn't also apply to QCD strings. And the story there is that we are still looking for the exact stringy dual to QCD; but most people think it does exist. $\endgroup$ – Mitchell Porter Aug 22 '11 at 7:07

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