Level:Speculative physics, doctorate (or advanced graduate) level
I have been keeping studiying the Clifford V. Jonhson book (D-Branes) and I gues I hvae a good understanding of the first six chapters. Specially I think I, at last, begin to understand the subleties of D-brane theory.
I have made a bit of crosscheckingof my understanding with the first chapters on string theory of the book of Tomás Ortin "gravity and strings" of which I also read the posterior chapters "extended objects". To complete my cross reading (and geting a good idea of what I´ll read next) I have re-readed in good detail the (year 2000) review article of B.E Baaquie and L.C Kwek "superstrings, Gauge fields and Black holes" (arxiv:hep-th/0002165).
As it is well known the Jonshon books is intended as a book in d-branes and not in string theory, and assumes some previous knowledge of them. I had it (that is a good thing because i´ll go fast in the next few chapters about superstrings). Even so I have reestudied some aspects of "basic" string theory. Mianly the Polyakov integral, conformald fields, and the theory of Rieman surfaces. I had studied it mainly in the Lüst-Theisen book (and partially in the Kaku book).
Now I have used the chapters of theBrian Hatfield Book (quantum field thery of point particle and strings) wich has a whole chpater about Rieman surfaces in wchich he dedicates a half part oof it to introduce in a mathematically correct formalism the concepts neccesary for a good knowledge of Rieman surfaces (begining from basic things as what is a topoly , a minifold, a complex structere and so that). Later it has a chapter in the pPOlyakov integral and a third one in vertex operators.
Well, apaart of the always goo habbit of recording things, why ree-study such basic perturbative strings? Afther all d-branes,T-duality and S-duality are, to a certain stent, a mean to study non-perturbative string.
Well, you can guess the answer from the previous post. I have been triying to formalice a bit that idea of knotted strings. Beeing a concept tied to "elementary" string theory my lack of a perfect knowledge of advance topics on branes seems, at least for the basic consideratios, irrelevant. Whay did I get?
As far as I see (I´ll still think it more in deepd and sutudy better the relevant questions) there is no inconsistency in thinking that strings could knot betwen themselves. I still am clarifiying which diagramtic in perturbation theory could yield such knotted state but still without that I can avance a few.
The great part is to refelxionate about what such a knotted state could mean. Closed (bosonic and "fermionic" i.e. supersimmetric) strings, the ones wchich can knott, at least if we are not thinking about how open strings could knot when they are tied to a d-brane, have in their spectrum the well known graviton, antisymmetric tensor and graviton (a supergravity multiplete for the superstring). Well, a graviton knoting around other is curious idea but it is not too promising to be an observable quantity in a near future, do you agree? ;-).
The fun begins when we compactify. For the begining we have the basic compactification of a single dimension which leas as to the usual history of T-duality, and the branes. But appart of it we have the "old" kaluza Klein mechanism. From it we hve that in the spectrum of the compactified theory there is a graviton in the reduced dimesnsions and, in adition, gauge vectors. What are that vectors?
We begin withf left moving 26 dimensional, closed, bosonic string and right moving, closed, superstring, we compactify the 16 dimensions of the bosonic string that we need to reduce to 10 dimensions in a self-dual lattices you have, yeah, you know it, the heterotic strings ;-). The allowed groups asociated to that lattices are SO(32) and E8xE8. Both of that groups contain the well known SU(3)xSU82)xU(1) of the standard model. That is why people had so good hopes for the heterotic string. What is the exact spectrum of the heterotic string?
(i) The ten dimensional graviton, antisymmetric tensor and dilaton
(ii) Their supersimmetric partners (gravitino and dilatino)
(3) These are the interesting ones for the current ppurpose, the gauge bosons of E8xE8 (or SO(32)
(iv) Supsersymmetric partners of the guage bossons, i.e. gauginos.
So, and these is the fun we have closed stirngs, which so allow knotting, who have in their spectra gauge bosons. Of course that gauge bosons are not the ones on the standard model. It is neccesary an aditional compactification to 4 dimensions in something like a Calabi-yau or an orbifold.
Or maybe other mechanisms as the warped scenaries of Randal-Sundrum in whhich you have extended, but only ascesible to gravity, dimensions. The 4-d world would be a d-brane to which open strings would be tied and the extra bulk would be only transitable to open strings, i.e. gravitons. If you have readed carefuly from these it seems that the gauge fields would also be open strings. I don´t know why these is so, because it contradicts my previous assets about the heterotic string. Obviously I need to study that things carefully.
But forgeting by now that remianing questions let´s state, at last, the gfreat rsult. If you allow knoted strings these represent particles which must propagated "together". FOr all practical purposes you couldn´t distinguish from scatering experiments (suposedlly the only ones allowed) unknoteed and knoted states. But if you have a knoted state of, lets say as an example, two gluons the "resulting" state would be diferent in ther properties of any possible single gluon. The same would work (with care to the higgs mechanism) for W bosons, the responsible for the elektroweak force, you know.
So you could have some composed state of W bossons wich would be diferent of a single W boson. And as far as W bosons are observbles that would mean that if knoted states exist they predict what, for any practical purpose, would llok as a new kind of particle, which, from an stringy perspective would be a composed state.
Why am i publishing these in these, semengly totally unknown, blog and not in a good review (such as nuclear physics B) or at least in Arxiv. It is obvious, there are a lot of technical aspects to be checked. But in prevention of someone else having the same idea and pubising it in a more precise way i left here these preliminar version as a proof that at least partially I was the first one (asfar as I know) to have the idea ;-).
Of course I will have made some stupid mistake and it will be mostlly a proof of my missunderstanding of the theory, but anyway here it is :-).
P.S. The idea of knotting strings has a few subleties. For example, it is not totally stated in the intuitive picture of the theory what happens when two strings are bringed together. The ony picture is the well known "pant diagram". The existence of knotted states would require that, at least to some stage, two strings couldn´t cross each other (something coherent with the pant diagram). These would mena that once formed the knoted state couldn´t break when an string simply cross each other. theformation of the knoted stated would require some mechanism, maybe implying itemediary open string states.
P.S. 2. In the observational side there is an even more interesting posibitie. If for some reason a closed string couldn´t easilly go around the d-brane where an open string is tied that would mean that open strings could somewhat tied up gravitons. That could result in some kind of gravity shielding. Also it could mean that ther could be unexpecteds relationships betwen gravity and elecromagnetism. It would be interesting to examinate from these viewpoint the controversal experiments of podkeltnov and a few other relted to gravitomagnetism.
P.S. 3 Too much speculation going here. Yeah, sure. I´ll try to go into more precise statements, but it I cant´go too further in that purpose at least I have a very good speculative ideas to include in science fiction writngs, which is another thing I like to do. In fact string theory is a lot better that LQG for writing science fiction. It is an open question (at least for me, even beeing aware of the interestings drawbacks Lubos Motl usually post about LQG) that it is so much better than LQG to do real physics, that´s why I try to learn both of them.